The rational homotopy type of a blow-up in the stable case

The rational homotop y t yp e of a blo w-up in the stable case. (May 18, 20 07) P ascal Lam brec h ts 1 and Don Stanley 2 Univ ersit´ e de Louv ain Univ ersity of Regina Abstract Suppo se f : V → W is an embedding o f closed oriented manifolds whose normal bundle has the s tructure o f a co mplex vector bundle. It is well known in b oth complex and symplectic g eometry that one can then construct a manifold f W which is the blow-up of W along V . Assume that dim W ≥ 2 dim V + 3 and that H 1 ( f ) is injective. W e cons tr uct a n algebra ic mo del of the rational ho motopy t yp e of the blow-up f W from an algebr aic mo del of the em bedding and the Chern classes o f the nor mal bundle. This implies that if the spa ce W is simply connected then the r ational homotopy type o f f W de p ends only on the rationa l homotopy c lass of f and on the Che r n cla sses of the no rmal bundle. Key words : blo w-up - shriek map - rationa l homotopy - symplectic manifold AMS-classiﬁcation (2000) : 55P6 2 -14F35 - 53C15 - 53D05 1 P . L. is a Chercheur Qualiﬁ´ e at FNRS. 2 Support of the Institut M ittag-Leﬄer is gratefully ackn owledge d 1 1 In tro duction The blow-up co nstruction comes from complex algebraic ge ometry; Gromov [12] and McDuﬀ [20] constructed the blow-up for symplectic ma nifolds. McDuﬀ used it to construct the ﬁrst examples o f simply co nnected non- K ¨ ahler sy mplectic manifolds. In this pap er w e study the ra tional homotopy t yp e of the blow-up construction. It is well k nown that all K¨ ahler manifo lds are symplectic a nd fundamental problem is to ﬁnd closed symplectic manifolds which ca nnot be given a K¨ ahler structure. A no n-simply connec ted ex a mple was ﬁrst found by Thurston [28], but to ﬁnd simply c o nnected exa mples proved more diﬃcult. In fact McDuﬀ int ro duced the blow-up cons truction to r esolve this pr oblem. She show ed that the blow-up of Thurston’s examaple in a c o mplex pro jective spa ce has an o dd third Betti num ber and is thus not K¨ ahler. This leads to more prec is e structura l questions. A space is formal if its rationa l coho mology algebr a serves as a ra tional mo del of it. In par ticular this implies its cohomology algebr a determines its rational homoto py type and that it has no nont rivial Mas s ey pr o ducts in its cohomolog y . It has b een shown [5] that all closed K¨ ahler ma nfolds ar e formal. Thu rston’s example was known to b e non forma l. By constructing non-trivia l Massey pr o ducts Bab enko and T a ima nov [1] show ed that the McDuﬀ exa mple is not forma l. Rudy ak a nd T r alle [2 5] extended these results for the blow-ups along many other manifolds. One of the applications of o ur mo del is to prove [19] that the blow-up alo ng a manifo ld M symplectically e mbedded in a la r ge enough C P ( n ) is for mal if a nd only if M is formal. Although McDuﬀ w as interested in the blow-up for symplectic manifo lds , the construction is mo re general. Suppo se f : V → W is an embedding o f smo oth clos e d oriented manifolds and that the nor mal bundle o f the embedding has b een given the str ucture of a complex vector bundle. This is eno ugh data to construct the blow-up f W o f W along V as des c r ibe d b y McDuﬀ [20] a nd outlined in Section 2 below. If f is an em b edding of complex manifolds with the cano nical complex structur e on the normal bundle then this blow-up is ho meomorphic to the classical blow-up of W a long V . Under some r estrictions w e give a complete description of the rationa l ho- motopy type of the blow-up f W using only the ra tio nal homo to p y class of the map f : V → W and the Cher n classe s of the nor mal bundle of V . The r ational homotopy t yp e of a space is deﬁned as long as all the spaces in volv ed a r e nilp o- ten t (see Section 3.3). Note that s imply connected closed manifolds are always nilpo ten t. Theorem 1. 1 (The or em 7.8) L et f : V → W b e an emb e dding of close d ori- entable smo oth manifolds and supp ose that the normal bund le ν is e quipp e d with the stru ctur e of a c omplex ve ctor bund le. Assume that dim W ≥ 2 dim V + 3 , that W is simply c onne cte d and that V is nilp otent. Then the r ational homo- 2 topy typ e of the blow-up of W along V , f W c an b e explicitly determine d fr om the r ational homotopy typ e of f and fr om t he Chern classes c i ( ν ) ∈ H 2 i ( V ; Q ) . More generally o ur des c ription ho lds as long as V , W and f W are nilp otent, H 1 ( f ; Q ) is injective and dim( W ) ≥ 2 dim( V ) + 3 (Corollar y 7.7). It is not so easy to determine if f W is nilp otent, so the usual case is W simply connected. W e also note that witho ut the dimens io n restr ic tion, dim( W ) ≥ 2 dim( V ) + 3 , there exists manifolds V and W , and homotopic em be ddings V → W with isomorphic complex normal bundles whose blow-ups are not rationally equiv ale nt [17]. Thus our dimension restriction, dim( W ) ≥ 2 dim( V ) + 3, ca nnot b e discarded. Sulliv an [26] studies ra tional homotopy using his piece wise linear forms A P L ( ) functor, which is analag o us to the de Rham diﬀer e n tial forms functor , Ω ∗ ( ). In par ticular it is co n trav ariant a nd for a top olo gical space X , A P L ( X ) is a commutativ e diﬀerential graded a lgebra or CDGA (deﬁned in Section 3.2 ). Another wa y to think of A P L ( X ) is as a comm utative version of the c o c hains on X with co eﬃcients in Q , C ∗ ( X ; Q ). A mo del of X is any CDGA weakly equiv- alent to A P L ( X ) (see Section 3.1). Similarly a mo del o f a map f : X → Y is any map of CDGAs weakly equiv alent to A P L ( f ) : A P L ( Y ) → A P L ( X ). Under some ﬁniteness conditions a ny mo del of X completely determines the r ational homotopy type of X (see Section 3.3 ). It is in this sense that we deter mine the rational ho motopy t yp e of f W . In fact we determine the homotopy type of the CDGA A P L ( f W ) witho ut a ny ﬁnitenes s re strictions. In Section 7.2 and Theorem 7.6 using only a mo del for f a nd the Chern classes of the no r mal bundle we construct an explicit mo del for f W . This is our main result and T he o rem 1 .1 follows directly from it. There are a num b er of int eresting bypro ducts pro duced along the wa y to the pro of. F o r example we derive a lgebraic mo dels for the co mplimen t of the embedding (see also [18]) and for the pro jectivizatio n of a complex bundle. In the las t section of the pap er w e g ive a few applicatio ns of the mo del of the blow-up. First we study the s pecia l case o f a blow-up o f C P ( n ), a nd lo ok- ing a t McDuﬀ ’s example of the blow-up of C P ( n ) a long the Ko daira -Thurston manifold we prov e the existence of non-tr iv ial Massey pro ducts by direct cal- culation. Our next application is to calculate the cohomolog y algebr a of the blow-up along f : V → W under our dimensio n res trictions (Section 8.4). This is co mplemen tary to work o f Gitler [10] who gav e a diﬀerent description o f this algebra when H ∗ ( f ) is surjective. In Section 8.5 w e use this calculation to show that there are inﬁnitely many distinct rationa l homotopy types of s y mplectic manifolds that can be constructed as the blow-up of C P (5) alo ng C P (1). 1.1 Con t en ts 1) Introduction 2) Mo delling the blow-up 3) Background and notation 3 4) Thom class and the shriek map 5) Mo del of the complement of a submanifold 6) Mo del of the pro jectiviza tion of a complex bundle 7) The mo del of the blow-up 8) Applications A more detailed list of conten ts app ears at the b eginning of each section. 1.2 Ac kno wledgemen ts The author s would like to thank Professor Hans Ba ues for help with homolog ical algebra and for p ointing o ut that there is a subtle diﬃculty in ﬁnding the mo del of a pushout from the pullbac k of the mo dels. W e would also like to thank Yves F´ elix, F ranco is Le s cure and Y uli Rudy ak for enlightening conv er sations regar ding this work. W e ar e very grateful to Bar r y Jessup for rea ding a dra ft of the pap er and suggesting many improv ement s. The seco nd author would lik e to thank T err y Gannon for his encourag ement and supp ort. A larg e p ortion of this work was done while the authors were at the Max Plank Institute, Universit ´ e de Lille-1, Universit ´ e de L o uv ain and the Mitt ag-L e ﬄer Institut. W e would like to thank them for providing gr eat environmen ts for resear c h. Both authors also b eneﬁted fro m the funding o f FNRS, CNRS and NSERC without whose generosity this work could never hav e b een completed. 2 Mo delling the blo w-up In this section we ﬁr st describ e the top ology of the blow-up construction and then des crib e the model o f the blow-up. The precise s tatement may b e found in Theorem 7.6. Again s uppo s e f : V → W is an embedding of connected closed or ien ted manifolds and suppose the normal bundle ν of f ha s b een giv en a co mplex structure. Let T b e a tubular neig hborho o d o f V in W . Let ∂ T b e the boundar y of T and B = W \ T . Then T ∪ B = W and T ∩ B = ∂ T . Hence we have a pushout ∂ T k / /   B   T / / W . (1) By the T ubular Neighborho o d Theorem [22, Theo r em 1 1.1] there is a diﬀeomo r- phism betw een T and the disc bundle Dν that s ends V to the zero section of D ν and s ends ∂ T to the s phere bundle S ν . Since ν is a complex bundle we can quotient by the S 1 ⊂ C ∗ -action on S ν ∼ = ∂ T . W e obtain a complex pro jectiv e 4 bundle P ν ov er V and a commutativ e dia gram ∂ T q / /   P ν } } z z z z z z z z V Next we can r emov e T fr o m W and instead o f putting it back as in (1) we ca n replace it by P ν . This gives us a pushout ∂ T k / / q   B   P ν / / f W . (2) The spac e f W is called the blow-up of W along V . This actually o nly gives us the homeo mo rphism type of the blo w-up but with slightly mo re care w e can get the diﬀeomorphism type. Since we are only studying the rational homotopy t yp e, homeomorphism type is more than enough. An imp or tant po in t is that (2) is also a homotopy pushout. In the next few parag raphs w e analy ze Diagra m (2 ) and g ive an idea of how we co nstruct the mo del B ( R, Q ) of A P L ( f W ). W e b egin with the fact that A P L ( ) takes homotopy pushouts to homotopy pullbacks. So we are left with analyzing the homotopy pullback of the diagra m A P L ( ∂ T ) A P L ( B ) A P L ( k ) o o A P L ( P ν ) A P L ( q ) O O which we ge t from applying A P L ( ) to Diagr am (2). T o do this we c onstruct mo dels of the tw o “ legs” of the dia gram, the maps A P L ( B ) → A P L ( ∂ T ) and A P L ( P ν ) → A P L ( ∂ T ). Th e pullback of thes e models should b e a mo del of the pullback o f the diagra m. Ho wever we need to b e careful how the mo dels are “glued” together b ecause diﬀerent gluing s corres po nd to homotopy automor - phisms of A P L ( ∂ T ) and can lead to pullbacks tha t are no t homo top y e q uiv alent. T o construct a mo del o f k we ma ke use of our co chain level version of the classical shriek map f ! : H ∗− r ( V ) → H ∗ ( W ), where r is the co dimension of V in W . Under our hypothesis that dim( W ) ≥ 2 dim( V ) + 3 , a mo del of k can b e constructed from any mo del of f and so the rational homotopy type of ∂ T and B and the ra tio nal homo topy class of k dep end only on the r a tional ho motopy class of f . This is to b e exp ected since we a re in the s table ra nge wher e homoto pic maps are iso topic. It is shown in [18] that without our dimension restriction 5 the rational homo to p y clas s of k can dep end on more than just the ra tional homotopy t yp e of f , and so it is at this p oint that the dimensio n res triction for the mo del of the blow-up a rises. In [17] it is shown that the blow-up along homotopic embeddings with the s ame Chern classes on their normal bundles can have diﬀerent homotopy types. Next we more fully describ e the mo del of B . Supp ose φ : R → Q is a mo del o f A P L ( f ): A P L ( W ) → A P L ( T ). W e can c onsider Q as a diﬀerential graded R -mo dule (or R -dgmo dule, see Section 3.2). Suppos e u V ∈ H ∗ ( Q ) and u W ∈ H ∗ ( R ) are or ien tation cla sses, where m = dim( V ) and n = dim( W ). Let φ ! : s − r Q → R be a map of R -dgmo dules such that H ( φ ! ∗ )( u V ) = u W , wher e r = n − m . (Recall that the sus p ensio n op era tor gives ( s − r Q ) j + r = Q j , so u V ∈ H n ( s − r Q ).) Suc h a map will be ca lled a shriek map . A shriek ma p alwa ys exists (Prop os ition 4.5) and is unique up to homotopy (Pro po sition 4 .4 ). Let m = dim V , n = dim W and r = n − m . W e describ e a mo del of k : ∂ T → B , and r efer the rea der to Lemma 5 .8 fo r the deta ils. Lemma 2.1 Assume n ≥ 2 m + 3 . If R ≥ n +1 = 0 and Q ≥ m +2 = 0 t hen ther e exists explicit CDGA structu r es on R ⊕ ss − r Q and Q ⊕ ss − r Q determine d by the CDGA structu re s on R and Q and by the shriek map φ ! such that the map φ ⊕ id : R ⊕ ss − r Q → Q ⊕ ss − r Q (3) is a CDGA m o del of A P L ( k ): A P L ( B ) → A P L ( ∂ T ) . F rom any mo del φ : R → Q , one satisfying the deg ree res trictions can always b e constructed (Prop os ition 4.5). The diﬀerential o n R ⊕ ss − r Q comes fr om the fact that it is actually the mapping cone on φ ! (see Section 3.8) and the CDGA structure is wha t we ca ll the semi-trivial CDGA structure (see Deﬁnition 3.19). Constructing a mo del of q : ∂ T → P ν is mor e stra ig ht forward. The co ho- mology algebra o f the pro jectiv e bundle can b e used to deﬁne the Chern class e s c i ( ν ) of the norma l bundle ν [2, IV.20]. T his descr iption allows us to co nstruct a mo del of P ν . The free gra ded commutativ e a lg ebra on the gra ded generato rs a i is denoted by Λ( a 1 , . . . , a n ). Next we describ e a mo del o f q : ∂ T → P ν , and refer to Theorem 6.9 for the details. Theorem 2. 2 Assume n ≥ 2 m + 3 .L et 2 k = n − m and let γ 0 = 1 and γ i ∈ Q b e r epr esentatives of t he Chern classes c i ( ν ) . S upp ose | x | = 2 and | z | = 2 k − 1 . Deﬁne CDGA s ( Q ⊗ Λ ( x, z ); D ) by D x = 0 and D z = Σ k − 1 i =0 γ i x k − i with and ( Q ⊗ Λ z ; D ) by D ( z ) = 0 . Then t he pr oje ction map pro j: Q ⊗ Λ( x, z ) → Q ⊗ Λ z (4) sending x to 0 is a CDGA mo del of A P L ( q ) : A P L ( P ν ) → A P L ( ∂ T ) . 6 In the last theor em the dimension restr ic tio n n ≥ 2 m + 3 arise s sinc e w e require that D z = 0, which represents the Euler cla ss of the bundle, but o ther than that are not really nee ded and probably a similar theor em without the dimension r estrictions is tr ue. Having co nstructed mo dels for b oth k and q of Diagram (2) we ca n use them to construct a mo del B ( R, Q ) of the pushout f W by ta king the pullback of these tw o mo dels. It is des crib ed precisely in Section 7.2. The fo rm o f B ( R, Q ) is: B ( R, Q ) =  R ⊕ Q ⊗ Λ + ( x, z ) , D  . As mentioned b efore, to get B ( R, Q ) we hav e to b e care ful how the mo dels of k and q ﬁt together . In the mo del (3 ) o f k the mode l of A P L ( ∂ T ) is Q ⊕ ss − r Q whereas in the mo del (4) of q it is Q ⊗ Λ( z ). These are alwa ys isomo r phic as Q -mo dules and o ur dimension restrictions imply that they are isomo rphic a s CDGAs. The pr oblem is to ma ke s ure we pick the co rrect iso morphism. While we are co nstructing o ur mo dels of k and q , we keep track of the isomo rphism with the help of orientation classes on b oth of our manifolds and on the normal bundle. In our sp ecial situation, this o rientation information together with the Q -dgmo dule structures is enough to determine the isomorphism. Once we have this nailed down it is stra ightf orward to co ns truct a mo del o f the blow-up from our mo dels o f k a nd q using a pullback. 3 Bac kground and nota tion F or this pa per the ground ﬁeld and the co eﬃcient ring will b e the rational nu mbers, unless otherwise stated. F or a top olo gical space X , H ∗ ( X ) refers to its singular coho mo logy . W e deno te a lso by H ( ) the functor taking homology of a diﬀerential complex. W e w ill use H n ( ) to refer bo th to the n -th sing ular cohomolog y gr oup and to the homology in degr ee n of any diﬀerential g raded ob ject. 3.1 Categorical preliminaries F or a small category I and any catego ry D , let D I be the diagr am c ate gory deﬁned as follows: the ob jects of D I are the functor s I → D and the mo r phisms are na tural tra nsformations. W e often r efer to the o b jects in such a categor y as diagrams. F or example o ur categor y I can consist o f tw o ob jects with exactly one non-identit y map joining them (this category ca n b e depicted as • → • ) and each diagra m in D I corres p onds to a single map in D . Simila rly I could b e the categ ory • • o o / / • which corr esp o nds to the data of a pushout in D or its dua l • / / • • o o (data for a pullback). W e a lso use squa r es of 7 ob jects which corr esp ond to the categor y • / /   •   • / / • If A = A 1 / /   A 2   A 3 / / A 4 and B = B 1 / /   B 2   B 3 / / B 4 are squares of ob jects and α i : A i → B i are maps, then  α 1 α 2 α 3 α 4  : A → B denotes the map b et ween the square s that would tra ditionally be denoted a s a commutativ e cub e: A 1 α 1   / / B B B B B B B B A 2 α 2   ! ! C C C C C C C C A 3 α 3   / / A 4 α 4   B 1 / / B B B B B B B B B 2 ! ! C C C C C C C C B 3 / / B 4 . If D is the categ ory of top o logical s pa ces a nd F , G ∈ D I then a morphism η : F → G is a home omorphism if for each i ∈ I , η ( i ): F ( i ) → G ( i ) is a homeo - morphism for each element. If D ha s weak equiv a le nces then η is called a we ak e quivalenc e if η ( i ) is a weak equiv a lence for ea ch i ∈ I . Two diag rams are w eakly equiv alent if they are connected by a chain of weak equiv alences. So F , G ∈ D I 8 would b e weakly equiv alent if there exists a diagra m in D I as b elow with all the maps weak equiv alences. F F 1 ≃ / / ≃ o o F 2 · · · ≃ o o F n ≃ / / ≃ o o G Notice that there may not b e a direct map b etw ee n F and G . 3.2 Homotop y theory of CDGA and R -DGMo d A go o d refere nc e for the categor ies CDGA a nd R -DGmo d is the b o ok o f F elix- Halp e rin-Thomas [9]. Next we review the notion of CDGA. Let A = ⊕ ∞ i =0 A i be a graded vector s pace together with an asso cia tive m ultiplication µ : A ⊗ A → A , a unit 1 ∈ A 0 and a linear map of degree +1, d : A → A , called the diﬀerential. W e denote µ ( a, b ) by ab or sometimes a · b . If a ∈ A n we write | a | = n and say that a has deg r ee n . W e r equire that the multiplication µ is g raded co mm utative, that d 2 = 0 and tha t d is a der iv ation . In other words we requir e tha t | ab | = | a | + | b | , that ab = ( − 1) | a || b | ba and tha t the Leibnitz law d ( ab ) = ( da ) b + ( − 1) | a | a ( db ) holds. Suc h an ( A, d ) is called a commutativ e diﬀer e n tial graded alg e bra or CDGA. Notice that the multiplication is suppressed. Sometimes we just write A also suppressing the diﬀer e n tial. W e ca ll A c onne cte d if A 0 = Q · 1 . Maps betw een CDGAs ar e graded vector spa c e homomo r phisms that commute with the mu ltiplication and the diﬀerential. Clearly we get a c a tegory which we will also denote CDGA. A sp ecial case of CDGA a re Sulliv an algebras (Λ V , d ) where Λ V is the free g raded c o mm utative algebra o n a g raded vector space V and d is a diﬀerential on Λ V satisfying a cer tain nilp otence condition as descr ibe d in [9, Part I I]. How ever we will relax their condition that V should be co ncent rated in p ositive degr ees and just demand that V b e concentrated in non- neg ative degrees. If V = Q h a i , b j i is a gr aded vector space with ba sis { a i } ∪ { b j } where the a i are homogeneous in even degrees and the b j are homogeneous in o dd degree then as a gr aded alge br a Λ V ∼ = P ( a i ) ⊗ E ( b i ) where P ( a i ) is the free graded p olynomial algebra on the a i and E ( b i ) is the free graded exterior algebr a on the b i . W e can a lso deﬁne a catego ry of diﬀerential graded mo dules over some ﬁxed CDGA. Le t ( R, d ) be a CDGA. Let M = ⊕ ∞ i = −∞ M i be a diﬀerential g raded R -mo dule with structure map µ : R ⊗ M → M and diﬀerential of degr e e +1, d ′ : M → M . W e require that d ′ ( rm ) = ( dr ) m + ( − 1 ) | r | d ′ m . Of course M is a n R -mo dule in the ungraded sense as well and | r m | = | r | + | m | . W e call such a n ob ject a diﬀer en tial graded R -mo dule or an R -dgmo dule. Maps ar e R -mo dule ma ps that pres e rve the grading a nd commute with the diﬀerential. The ca tegory of R -dg modules and R -dgmo dule maps is deno ted by R - DGmo d. W eakly equiv alent diag rams of CDGA or R -dg mo dules are referred to a s mo dels of eac h other. F or e xample let f : X → Y be a ma p of spaces. If a diagram φ : A → B of CDGA is weakly equiv alent to the dia gram A P L ( f ) : A P L ( Y ) → A P L ( X ), then we will say that φ : A → B is a mo del of A P L ( f ). 9 In this case, as is common, we also call φ a mo del of f . Similarly if A is a mo del of A P L ( X ) we call A a mo del of X . T he concept of a CDGA mo del of a spa ce or a map is well established. Note howev er that ma n y author s reserve the term mo del to refer to free CGDA wherea s we use it mor e gener ally . F or our purpo ses we a ls o need mo dels of more g eneral diagrams. In fact since we will b e gluing dia grams together we will sometimes need that the equiv alences betw een our diagra ms preser ve certain extra str ucture. Both CDGA and R -DGMo d are closed mo del categ ories (see [6] for a re v iew of mo del catego ries and [3] fo r a pr o of of the fact that CDGA satis ﬁe s the ax io ms of a closed mo del categ o ry). It is not necessary for the r eader to b e familiar with closed model catego r ies sinc e a ll of the r elev ant results can be proved directly in these categor ies. In b oth ca teg ories the clo sed mo del structure is determined by the following families of maps: • ﬁbrations a re surjections, • weak eq uiv alences are quasi-is omorphisms. W e use the terms ﬁbration and surjection in terchangeably w he n working in CDGA or R -DGMo d. By a c el lular c oﬁbr ation we mean: • in CDGA, a relative Sulliv an alg ebra B → B ⊗ Λ V as deﬁned in [9, Chapter 14] except that we allow non- negatively gr aded V instead of just p ositively graded V . • in R -DGMo d, a se mi-free extension M → M ⊕ R ⊗ V a s deﬁned in [8, § 2 ] W e will gener ally deal with coﬁbratio ns that are cellula r ones. Note that no t all coﬁbrations ar e of this fo r m (see [3, § 4.4]) but at le ast all coﬁbr ations be tw een connected CDGA ar e. A map tha t is b o th a coﬁbra tio n and a weak equiv alence is called an acyclic coﬁbr ation. Simila rly a map that is b oth a ﬁbra tion and a weak eq uiv alence is called an acyclic ﬁbration. If ∅ denotes the initial ob ject of the ca tegory (it is Q concentrated in dimension 0 in CDGA and 0 in R -DGMo d) then a c el lular c oﬁbr ant obje ct is an ob ject X suc h that the map ∅ → X is a cellular coﬁbration. In CDGA cellular coﬁbra n t ob jects are the Sulliv an algebr as [9, § 12] and in R -DGMo d they are the semi-fr ee R -dg mo dules [8, § 2]. Dually if ∗ denotes the terminal o b ject then a n ob ject X is ca lled ﬁbr ant if X → ∗ is a ﬁbration. Note that all ob jects in CDGA and R -DGMo d ar e ﬁbrant. The following lemma is a slig ht mo diﬁcation o f one of the ax ioms of a closed mo del categor y . Lemma 3.1 Supp ose A and B ar e CDGA s such t hat H 0 ( A ) = H 0 ( B ) = 0 . Any CDGA map f : A → B c an b e factor e d into a c el lular c oﬁbr ation fol low by an acyclic ﬁbr ation. Pro of. This is prov e d as part of [13, Theo rem 6.1] with the added condition that A is augmented. How ever the augmentation is no t used to get the res ult of the lemma. 10 The analog ous r esult in R -DGmo d is a lso true and follows b y standa r d ar - guments. Lemma 3.2 A ny map in R -DGmo d c an b e factor e d into an acyclic c el lular c oﬁbr ation fol lowe d by a ﬁbr ation and also into a c el lular c oﬁbr ation fol lowe d by an acyclic ﬁbr ation. Homotopies in a closed mo del ca tegory c a n b e deﬁned with the help of a cylinder ob ject which we descr ibe no w. Let ` denote the copro duct, and ∇ : A ` A → A the fold map. F a ctor ∇ into a c e llula r coﬁbra tion i 0 + i 1 : A ` A → C y l A followed by a n acyclic ﬁbratio n. This implies in pa rticular that the maps i 0 , i 1 : A → C y l A are w eak equiv alences. The ob ject C y l A is called the cylinder obje ct of A , it is unique up to homotopy . Two morphisms f 0 , f 1 : A → X are homotopic if there ex ists a ma p H : C y l A → X such that H i 0 = f 0 and H i 1 = f 1 . W e wr ite f 0 ≃ f 1 , or f 0 ≃ R f 1 if we wish to emphasize the fact tha t the homo topy is in the ca tegory R -DGMo d. Then it is ea sy to chec k from the deﬁnition of a homo top y in R -DGMo d: Lemma 3.3 Two m orphisms f 0 , f 1 : A → X in R - DGMo d with A c oﬁbr ant ar e homotopic if ther e exists an R -mo dule de gr e e − 1 morphism h : A → X such that d X h + hd A = f 0 − f 1 . Note t hat such a homotopy c an also b e se en as an R -mo dule de gr e e 0 morphism h : sA → X wher e s is the susp ension (se e Deﬁnition 3.10). In CDGA our notion of homo to p y is also equiv alent to the mor e traditio na l one ([9, Chapter 12(b)]). W e reca ll the notion of sets o f homotopy clas ses of maps in CDGA and R - DGmod. Let X , Y ∈ CDGA or R − DGMo d. F actor ∅ → X as a coﬁbratio n follow ed b y a n acy clic ﬁbration ∅ → ˆ X → X . W e deﬁne the set of homo- topy cla sses of maps from X to Y , [ X , Y ] as the set of equiv alence classes of Hom( ˆ X , Y) under the ho motopy relation in either CDGA or R - DGMod. [ X , Y ] = Hom( ˆ X , Y) / ≃ . W e will write [ X, Y ] R if w e wis h to emphasize we are looking at homo to p y classes of R - dgmo dule maps. The Lifting Le mma (Lemma 3.4 b elow) implies that [ X , Y ] is indep endent of the choice of ˆ X (any tw o choices g ive naturally isomorphic sets) and there is an obvious map Hom(X , Y) → [X , Y]. W eak equiv alences in the range and domain of [ , ] induce bijectio ns o f sets. T o those already familiar with closed mo del categor ies we p oint out that we do n’t hav e to r eplace the ra nge by a ﬁbrant ob ject since all ob jects in o ur catego ry ar e already ﬁbrant. 11 3.3 The functors A P L ( ) and | | A connected spac e X is nilp otent if π 1 ( X ) is a nilp otent g roup and π n ( X ) is a nilp otent π 1 ( X )-mo dule for ea c h n ≥ 2 (See [14, I I.2]). Note that all simply connected spaces ar e nilp otent. A nilpo ten t space X is r ational if π n ( X ) is a rational vector space for each n ≥ 1. A nilp otent space X has ﬁ n ite Q -typ e if H n ( X ; Q ) is ﬁnite dimensio nal for ea ch n ≥ 1. Let T op b e the catego ry of top ologica l spaces. There are adjoint functors A P L : T op → C D GA | | : C D GA → T op int ro duced by Sulliv an [26]. W e re fer the reader to [9, I I 10 and I I 17] for their deﬁnitions (se e als o [3, Section 8 ] and [26, Section 7 and 8]). These functors induce a n equiv a lence b etw een the homotopy types of ﬁnite type CDGA and of nilp otent ra tional top ologica l spa ces of ﬁnite Q -type (see [3, 9.4] for a mor e precise statement). It is in this sense that the homotopy type of A P L ( X ) as a CDGA determines the r ational homotopy type o f X . A key pro per t y of A P L that we will use rep eatedly is the Q - version of the na tural a lgebra de Rham isomorphism H A P L ( ) ∼ = H ∗ ( ; Q ) (5) This isomor phism holds for all spaces and will b e ﬁxed througho ut the pap er. In order to reduce clutter in our equa tions, for a map f o f spaces we will often write f ∗ for A P L ( f ). 3.4 Standard abuses of notation Because of the isomo rphism (5), to any cohomolo gy class µ ∈ H n ( X ; Z ), using the change of co eﬃcients morphism H ∗ ( , Z ) → H ∗ ( , Q ), we can asso ciate a cohomolog y class in H n ( A P L ( X )). Abusing notation we also deno te this cla ss by µ ∈ H n ( A P L ( X )). Also if R is a CDGA mo del of another CDGA R ′ through a ﬁxe d chain of weak equiv alences R ≃ / / . . . R ′ ≃ o o and if µ ∈ H n ( R ) is a cohomolog y class we will denote the cor r esp onding clas s µ ∈ H n ( R ′ ) by the sa me name. Note that which classes corre s po nd o nly dep ends on which element of [ R, R ′ ] is represented b y the chain of weak equiv a lences. W e will often abuse “inc” to deno te any inclusion map and “pr o j” to deno te any pro jection ma p. The abuse is t wofold since not only will we often not explicitly deﬁne o ur inclusions but als o we will have many diﬀerent maps referred to b y the s a me nota tio n. W e will a lso sometimes use “   / / ” to denote coﬁbrations or inclusions. 12 3.5 Closed mo del category facts The following result is standard. Let C b e a closed mo del catego ry . Lemma 3.4 (Lifting Lemm a) Supp ose given the fol lowing solid arr ow dia- gr am in C A f / / g   B v   C u / / l > > } } } } D . (i) If t he diagr am is c ommutative, g is a c oﬁbr ation, v is a ﬁ br ation and g or v is a we ak e quivalenc e t hen ther e ex ists a lift l making e ach triangle c ommutative. Mor e over su ch a lift l is unique up to homotopy. (ii) If the diagr am is c ommutative (r esp e ctively, c ommutative up t o homotopy), g is a c oﬁbr ation and v is a we ak e qu ivalenc e then t her e exists a lift l s uch that l ◦ g = f (r esp e ctively, l ◦ g ≃ f ) and v ◦ l ≃ u ; that is, the upp er triangle is c ommutative and the lower t riangle is c ommut ative up t o homotopy (r esp e ctively, b oth triangles ar e c ommu tative up to homotopy.) Mor e over such a lift l is unique up to homotopy. Pro of. See [9 , Chapter 1 4] for the CDGA case, and [7, Lemma A.3] for the sp e- cial cas e of A = 0 in R -DGmo d. The ca se for g eneral A follows by standard techn iques. The following next tw o lemmas allow us to co n vert ce r tain homo topy com- m utative diagr a ms into strictly commutative dia grams. Lemma 3.5 L et C b e a close d m o del c ate gory and supp ose ˆ A f           f   f ′   ? ? ? ? ? ? ? ? B ˆ B β o o β ′ / / B ′ is a homotop y c ommutative diagr am in C . If ˆ A is a c oﬁbr ant obje ct and if ( β , β ′ ): ˆ B → B ⊕ B ′ is a ﬁbr ation (in other wor ds a surje ction in CDGA or R -DGMo d), then ther e exists a morphism ˆ f : ˆ A → ˆ B such that f ≃ ˆ f and the diagr am ˆ A f           ˆ f   f ′   ? ? ? ? ? ? ? ? B ˆ B β o o β ′ / / B ′ is s t rictly c ommutative. 13 Pro of. Consider the fo llowing solid arrow diagram ˆ A i 0   f / / ˆ B ( β ,β ′ )   C y l ˆ A H / / G : : v v v v v B ⊕ B ′ where H is a homotopy between ( β , β ′ ) f a nd ( f , f ′ ). The map i 0 is alwa ys a weak equiv alence and here is a coﬁbratio n since ˆ A is coﬁbrant. Also ( β , β ′ ) is a ﬁbration by hypothesis and hence there is a lift G by L e mma 3 .4(i). Let ˆ f = Gi 1 . This makes the diagr a m of the conclusion commute since H i 1 = ( f , f ′ ) and Gi 1 ≃ Gi 0 = f . Lemma 3.6 Supp ose in CDGA or R -DGMo d that f : A → B is a m o del of f ′ : A ′ → B ′ . If we ar e in CDGA then also assume that H 0 ( A ) = H 0 ( B ) = H 0 ( A ′ ) = H 0 ( B ′ ) = Q . Then ther e exists a c el lu larly c oﬁbr ant ˆ A , a c el lular c oﬁbr ation ˆ f : ˆ A / / ˆ B , and we ak e qu ivalenc es α : ˆ A ≃ → A , α ′ : ˆ A ≃ → A ′ β : ˆ B ≃ → B and β ′ : ˆ B ≃ → B ′ such that ( β , β ′ ): ˆ B → B ⊕ B ′ and ( α, α ′ ): ˆ A → A ⊕ A ′ ar e surje ctive and the fol lowing diagr am is strictly c ommu tative A f   ˆ A ˆ f   ≃ α o o ≃ α ′ / / A ′ f ′   B ˆ B ≃ β o o ≃ β ′ / / B ′ . In addition the isomorphisms H ∗ ( α ′ ) H ∗ ( α ) − 1 ∈ Hom(H ∗ (A) , H ∗ (A ′ )) and H ∗ ( β ′ ) H ∗ ( β ) − 1 ∈ Hom(H ∗ (B) , H ∗ (B ′ )) ar e the same as the one determine d by t he original string of we ak e quivalenc es making f a mo del of f ′ . Pro of. If we let I b e the c a tegory ( • → • ) with t wo ob jects a nd o ne non-identit y map, then C I is the catego ry of maps in C . According to [16, Section 5 .1] for any mo del c a tegory C , C I can b e given a mo del structure such that if A 1 → B 1 and A 2 → B 2 are in C I then a ma p be t ween them A 1   α / / A 2   B 1 β / / B 2 14 is a weak equiv a lence if α and β are b oth weak equiv alences and a ﬁbr ation if bo th α and β are ﬁbrations. Since f is a mo del of f ′ they ar e connected by a s equence of weak equiv a- lences, f = f 0 g 0 ≃ / / ≃ o o f 1 · · · ≃ o o g n − 1 ≃ / / ≃ o o f n = f ′ with each g i , f i ∈ C . W e can factor g i → f i ⊕ f i +1 as an acyclic coﬁbration follow ed by a ﬁbra tion. and therefor e we can as sume that all of the maps in the s equence a r e ﬁbra tions (since they are surjections when ev aluated at ea ch ob ject o f I ). Also if g i was coﬁbrant we replace d it with something coﬁbr ant. Now let ˆ f ≃ → f 0 be a weak equiv a lence such that ˆ f is coﬁbrant. Since they are ﬁbrations, we can lift along the maps of the sequence to get a dia gram f ˆ f ≃ / / ≃ o o f ′ As b e fore we can assume tha t the map ˆ f → f ⊕ f ′ is a ﬁbratio n. So we get a diagram A f   ˆ A ˆ f   ≃ α o o ≃ α ′ / / A ′ f ′   B ˆ B ≃ β o o ≃ β ′ / / B ′ . Since ˆ f → f ⊕ f ′ is a ﬁbra tion, ( α, α ′ ): ˆ A → A ⊕ A ′ and ( β , β ′ ): ˆ B → B ⊕ B ′ are surjections. Finally it follo ws by a diagram chase that the isomor phisms H ∗ ( α ′ ) H ∗ ( α ) − 1 ∈ Hom(H ∗ (A) , H ∗ (A ′ )) and H ∗ ( β ′ ) H ∗ ( β ) − 1 ∈ Hom(H ∗ (B) , H ∗ (B ′ )) are the same a s the o ne determined by the or iginal string o f weak equiv alences making f a mo del of f ′ . 3.6 Homotop y pullbac ks Homotopy pullbacks ar e homotopy inv a riant versions of pullbacks. They exis t in any closed mo del ca tegory ([6, Section 10]). W e now give a de s cription that is adapted to our catego ries CDGA and R -DGMo d. Let f : D → C b e a map in CDGA or R -DGMo d a nd D i / / D ′ f ′ / / C be a factorization into an acyclic coﬁbr ation follow ed by a ﬁbration. Then the homotopy pul lb ack o f the diagram B g / / C D f o o is the pullback of B g / / C D ′ . f ′ o o 15 There is a map induced by id B , id C and i from the pullback to the homotopy pullback. Both the ho motopy pullback and this induced map a re unique up to homotopy . A homotopy pullback in CGDA o r RDG-Mo d g ives r is e to a Mayer- Vietoris sequence , cor r esp onding to the fac t that pushouts of spaces hav e such sequences and A P L ( ) takes homo topy pushouts to homotopy pullbacks. The following lemma states some standard facts ab out homotopy pullbacks. Lemma 3.7 L et B 1 / /   C 1   D 1 o o   B 2 / / C 2 D 2 o o (6) b e a c ommu tative diagr am in CDGA or R -D GMo d. L et E i b e the pul lb ack of B i / / C i D i o o and E ′ i its homotopy pul lb ack. (i) Ther e is an induc e d map E ′ 1 → E ′ 2 such that the fol lowing diagr am o f induc e d maps c ommu tes E 1   / / E ′ 1   E 2 / / E ′ 2 . (ii) If the vertic al maps in (6) ar e we ak e quivalenc es then the induc e d map E ′ 1 → E ′ 2 is a we ak e quivalenc e. Pro of. Part ( i ) follows from the Lifting Lemma and Part ( ii ) follows fro m May er- Vietoris. The following lemma follows directly from Lemma 3.7. Lemma 3.8 Supp ose that in CDGA or R -D GMo d, the diagr am A 1 → B 1 → C 1 ← D 1 is a mo del of the diagr am A 2 → B 2 → C 2 ← D 2 . L et E i , F i b e t he homotopy pul lb acks of A i → C i ← D i and B i → C i ← D i r esp e ctively. T hen the induc e d map E 1 → F 1 is a mo del of the induc e d map E 2 → F 2 . 16 3.7 Making a homotopy comm utativ e diagram strictly com- m utative In general if we have a homoto p y c o mm utative diagra m we cannot a lw ays r e- place it with a strictly co mmuting o ne. Ho wev e r in the following particularly simple case we can. Recall that weakly e quiv alent dia grams ar e connec ted by a sequence of weak equiv alences and weak equiv alences b etw een diag r ams a re strictly commuting diagr ams (see Section 3.1 ). Lemma 3.9 Assume that we have a homotopy c ommu tative diagr am in CDGA or R -DGMo d A ′ ≃   f ′ / / B ′ ≃   C ′ ≃   g ′ o o D ′ h ′ o o ≃   A f / / B C g o o D h o o (7) such that al l vertic al arr ows ar e we ak e quivalenc es. Then the dia gr ams that make up the top and b ottom r ow of (7) ar e we akly e quivalent. Pro of. Replacing the top row by something weakly equiv alent we can assume that the ob jects in the top row a re coﬁbr ant a nd the middle tw o vertical ar r ows are ﬁbrations. Consider ﬁrst the left square of Diagra m (7): A ′ α   f ′ / / B ′ β   A f / / B . Since β is a sur jection a nd A ′ is co ﬁbrant the map f α can be lifted thro ugh β to ge t a new map f : A ′ → B ′ . Also by Lemma 3 .4 there is a homoto p y H : C y l ( A ′ ) → B ′ betw een f ′ and f . So the following diagr am is strictly com- m utative with all hor izontal a r rows weak equiv alence s: A ′   f ′ / / B ′ =   C y l ( A ′ ) H / / B ′ A ′ O O f / / α   B ′ = O O β   A f / / B . (8) 17 Similarly , we can replace the c e nter a nd right squares by a strictly commuting diagram B ′ =   C ′ g ′ o o   D ′ h ′ o o   B ′ C y l ( C ′ ) H ′ o o C y l D ′ C y lh ′ o o B ′ = O O =   C ′ O O =   g o o D ′ O O   h ′ o o B ′ C ′ g o o C y l D ′ H ′′ o o B ′ = O O β   C ′ = O O   g o o D ′ h o o O O   B C g o o D h o o (9) where C y l h ′ is any lift in the following solid ar row diagr am D ′ ` D ′   / / C y l ( C ′ )   C y l D ′ C y lh ′ 9 9 s s s s s H ′′ / / C ′ Notice that a lift exists s inc e in this diagram the le ft map is a coﬁbra tion and the right an a cyclic ﬁbra tion. Now glue the b ottom of Diagr am (8) to the top of Diagram (9) to g et a sequence of weak equiv alences connecting the left a nd right columns of the new diagram. This completes the pro of of the lemma. 3.8 Mapping cones Deﬁnition 3.10 L et k ∈ Z . Th e k -th susp ension of an R -dgmo dule M is the R -dgmo dule s k M deﬁne d by • ( s k M ) j ∼ = M k + j as ve ct or sp ac es for j ∈ Z and this isomorphism is denote d by s k , • r · ( s k x ) = ( − 1) | r || k | s k ( r · x ) for x ∈ M and r ∈ R , • d ( s k x ) = ( − 1) k s k ( dx ) for x ∈ M 18 Deﬁnition 3.11 L et R b e a CDGA. The ma pping c o ne of an R -dgmo dule m or- phism f : A → B is the R -dgmo dule ( B ⊕ f sA, d ) deﬁne d as fol lows: • B ⊕ f sA = B ⊕ sA as gr ade d R -mo dules, • d ( b, s a ) = ( d B ( b ) + f ( a ) , − sd A ( a )) for a ∈ A and b ∈ B . Recall that a mapping cone gives rise to a long exa ct cohomology sequence, therefore the following tw o lemmas follow eas ily fro m the ﬁve lemma. Lemma 3.12 L et 0 / / A i / / B p / / C / / 0 b e a short exact se quenc e of R -dgmo dules. Then ther e is a quasi-isomorphi sm of R -dgmo dules p ⊕ 0: B ⊕ i sA ≃ → C. Lemma 3.13 L et A f / / α   B β   A ′ f ′ / / B ′ (10) b e a homotopy c ommu tative diagr am of R - dgmo dules and let h : sA → B ′ b e an R - dgmo dule homotopy b etwe en β f and f ′ α as in L emma 3.3. Then ther e exists a c ommutative diagr am 0 / / B   / / β   B ⊕ f sA / / β ⊕ h sα   sA / / sα   0 0 / / B ′   / / B ′ ⊕ f ′ sA ′ / / sA ′ / / 0 of R -dgmo dules in which e ach line is a short exact se quenc e and wher e β ⊕ h sα is t he R -dgmo dule morphism deﬁne d by ( β ⊕ h sα )( b, sa ) = ( β ( b ) + h ( sa ) , sα ( a )) . Mor e over if α and β ar e quasi-isomorphisms t hen so is β ⊕ h sα . If the diagr am (10) is st rictly c ommutative then we c an take h = 0 and β ⊕ h sα = β ⊕ sα . 19 3.9 Sets of homotop y classes in R -DGMo d Let R b e a CDGA a nd M and N be R -dg mo dules with diﬀer en tials d M and d N resp ectively . F or getting ab out the diﬀerentials we denote by Hom i R (M , N) the v ecto r space of R - mo dule maps from M to N r aising degr ee by i . W e can put a diﬀerential D on Hom R (M , N) = ⊕ i ∈ Z Hom i R (M , N) as follows. F o r f ∈ Hom i R (M , N) and m ∈ M deﬁne ( D f )( m ) = d N ( f ( m )) + ( − 1) i f ( d M ( m )) . D is of degree +1 and turns Hom R (M , N) into a ch ain complex. It is easy to chec k that the cycles ar e exactly the chain maps. Let ρ : H 0 (Hom R (M , N)) → [M , N] R (11) be any map which when restricted to cycles in Hom 0 (M , N) gives their equiv a- lence cla s s in [ M , N ] R . Since a map is a cy c le if a nd only if it is a chain map and a bounda ry if and only if it is ho motopic to 0, if M is coﬁbrant then ρ is an is omorphism. W e consider R and H n ( R ) as chain co mplexes with 0 diﬀerential, with R concentrated in degr ee 0 a nd H n ( R ) concentrated in degree n . Le t ǫ : R → s − n H n ( R ) b e any chain map that induces the identit y isomor phism after taking H n . In o ther words such that ǫ maps dimension n co cycles to their equiv alence classes in cohomolo gy . The co nditio n that ǫ is a chain map of course implies that all other dimensions go to 0. An y such ǫ gives us a map ǫ : R → Hom Q (R , s − n H n (R)) , r 7→ ǫ (r · ) (12) It is straightforward to chec k that ǫ is a degree 0 map of R - dgmo dules. Note that in the gener al case where we don’t a ssume co mm utativity , for s ∈ R we would hav e ǫ ( r )( s ) = ( − 1) | r || s | ǫ ( sr ). Since we are working in the g raded commutativ e situation ( − 1) | r || s | sr = r s . Th us we g et the same formula in our situation but with fewer signs. Deﬁnition 3.14 A c onne cte d gr ade d algebr a R is a Poincar´ e dua lit y algebr a of formal dimension n if ǫ is a quasi-isomorphism. F or M a left R -dgmo dule, N a rig ht R -dg mo dule and L a chain co mplex, deﬁne φ : Hom R (M , Hom Q (N , L)) → Hom Q (N ⊗ R M , L ) (13) by φ ( f )( n ⊗ m ) = ( − 1) | n || m | f ( m )( n ). Clea rly φ is a degr ee 0 isomo rphism of chain complexes. Lemma 3.15 L et R b e a CDGA su ch that H ∗ ( R ) is a Poinc ar´ e duality algebr a of formal dimension n . L et P b e an R -dgmo dule. Th en the map H n : [ P, R ] R → Hom(H n (P) , H n (R)) , [f ] 7→ H n (f ) . is an isomorph ism of Q -mo dules. 20 Pro of. By the deﬁnition of [ P , R ] R we may assume that P is coﬁbra n t. Since R is a Poincar´ e dua lit y algebr a and P is c o ﬁbrant we have a qua si-isormo r phism ǫ ∗ : H 0 (Hom R (P , R)) → H 0 (Hom R (P , Hom Q (R , s − n H n (R)) induced by Equation (12). Also φ and ρ from equa tions (11) and (13) are iso morphisms. So we get a string of isomor phisms. [ P, R ] R ρ − 1 → H 0 (Hom R (P , R)) ǫ ∗ → H 0 (Hom R (P , Hom Q (R , s − n H n (R)) φ ∗ → H 0 (Hom Q (R ⊗ R P , s − n H n (R))) = H 0 (Hom Q (P , s − n H n (R))) = Hom Q (H n (P) , H n (R)) It is str a ightforw a rd to chec k that the comp osition of thes e maps is the same a s the map induced by taking H n . Lemma 3.16 L et P and X b e R -dgmo dules. If H n ( ˜ R ) = 0. Similarly J is an acyclic ideal in ˜ Q . Deﬁne R = ˜ R/I and Q = ˜ Q/J . Since n ≥ m + 2, ˜ φ induces a map φ : R → Q which is a model of f and since m + 1 + r ≥ n + 1, ˜ φ ! induces an R -dgmo dule morphism φ ! : s − r Q → R which is a s hriek map. Obviously R and Q are connected and R ≥ n +1 = Q ≥ m +2 = 0. 4.4 Preliminaries with A P L Recall the no tation of Diagram (14) a t the b e ginning of the s ection. Consider the ladder of maps b etw een short exact sequences of pa irs: A P L ( W , B ) ι / / ρ   A P L ( W ) l ∗ / / j ∗   A P L ( B ) k ∗   A P L ( T , ∂ T ) ι ′ / / A P L ( T ) i ∗ / / A P L ( ∂ T ) where ι is the kernel o f l ∗ and ι ′ is the kernel of i ∗ . Also ρ is the induced map which is well deﬁned since Diagram (14) co mmutes and A P L is a functor. As each is a k ernel of a CDGA map, each is a diﬀerential idea l - A P L ( W , B ) inherits an A P L ( W )-dg mo dule structure and A P L ( T , ∂ T ) inherits an A P L ( T )-dgmo dule structure. Since j ∗ is an algebra map, ρ is an A P L ( W )-dg mo dule map. Note that l : B → W and i : ∂ T → T are coﬁbrations a nd so l ∗ and i ∗ are s urjections [9, P rop osition 1 0.4 and Lemma 1 0.7]. This w ill b e used la ter. Lemma 4.6 The r estriction map ρ : A P L ( W , B ) → A P L ( T , ∂ T ) is a su rje ctive we ak e quivalenc e of A P L ( W ) -dgmo dules. 27 Pro of. As obser ved ab ov e ρ is an A P L ( W )-dg mo dule map. I t is a weak equiv alence by excision. T o see that it is surjective, let α ∈ A P L ( T , ∂ T ). Since j is a coﬁbra- tion we know b y [9, P rop osition 10 .4 a nd Lemma 10 .7] that j ∗ is a surjection. Let β ∈ A P L ( W ) b e such that j ∗ ( β ) = ι ′ ( α ). Since i ∗ ι ′ = 0, k ∗ l ∗ ( β ) = 0. So we can extend l ∗ ( β ) by 0 on simplices contained in T a nd a r bitrarily to the rest of the s ing ular simplices in W (the ones whose image is contained in neither B or T ) to ge t β ′ ∈ A P L ( W ). W e know that l ∗ ( β ′ ) = l ∗ ( β ) and j ∗ ( β ′ ) = 0. Th us l ∗ ( β − β ′ ) = 0 and j ∗ ( β − β ′ ) = j ∗ ( β ) = ι ′ ( α ). So β − β ′ lifts to ˜ β ∈ A P L ( W , B ) such that ρ ( ˜ β ) = α . Since α was arbitra ry ρ is surjective. See 3 .1 for no ta tion co ncerning squa res. Lemma 4.7 Consider the fol lowing squar es: F ′ = A P L ( W ) j ∗ / / inc   A P L ( T ) inc   A P L ( W ) ⊕ ι sA P L ( W , B ) j ∗ ⊕ sρ / / A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) and F = A P L ( W ) j ∗ / / l ∗   A P L ( T ) i ∗   A P L ( B ) k ∗ / / A P L ( ∂ T ) (i) With the semi-trivial CDGA structu r e (se e Deﬁnition 3.19) on the mapping c ones, F ′ is a CDGA s qu ar es. (ii) Θ 6 =  id id l ∗ + 0 i ∗ + 0  : F ′ → F is a we ak e quivalenc e of CDGA squar es. Pro of. (i) By P rop osition 3.18 (i) there a re semi-triv ial CDGA-str uctures on the mapping cones: A P L ( W ) ⊕ ι sA P L ( W , B ) a nd A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) . It is then e a sy to see that F ′ is a CDGA squar es. (ii) It is immediate that the following diagra m is o ne in CDGA and is co m- m utative: A P L ( W ) ⊕ ι sA P L ( W , B ) j ∗ ⊕ sρ / / l ∗ +0   A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) i ∗ +0   A P L ( B ) k ∗ / / A P L ( ∂ T ) . (19) 28 W e need only to pro ve that the v er tical arrows in Diagram (19) are quasi- isomorphisms. As no ted ab ov e Lemma 4.6, l ∗ is surjective. Thus we have a short exact sequence 0 / / A P L ( W , B ) ι / / A P L ( W ) l ∗ / / A P L ( B ) / / 0 . By Lemma 3.12 this implies tha t the vertical map l ∗ + 0 is a qua si-isomorphis m. The pro of that i ∗ + 0 is a quasi- isomorphism is similar . Lemma 4.8 L et θ ∈ A r P L ( T , ∂ T ) ∩ ker d b e a r epr esent ative of the Thom class ¯ θ chosen in Se ction 4.1. Then multiplic ation by θ , · θ : s − r A P L ( T ) ≃ → A P L ( T , ∂ T ) is a quasi-isomorphism of left A P L ( T ) -dgmo dules. Pro of. This follows from the Thom is o morphism of E quation (18). Lemma 4.9 The fol lowing se quenc e of arr ows s − r A P L ( V ) / / s − r A P L ( T ) · θ ≃ / / A P L ( T , ∂ T ) A P L ( W , B ) ρ ≃ o o ι / / A P L ( W ) induc es an isomorphism H n − r ( V ) ∼ = H n ( W ) . that takes s − r u V to u W . Pro of. Apply Lemma 4.1 and the natural equiv alence of Equation (5) b etw een H ∗ ( ) and H ( A P L ( )). 5 A mo del of the comple men t of a submanifol d As in Section 4 w e suppos e we are giv en an em b edding of closed manifolds f : V ֒ → W as w ell as orientation clas s es u V ∈ H m ( V ; Z ) and u W ∈ H n ( W ; Z ). Again W is of dimension n and V of dimension m a nd co dimension r = n − m . W e also use the notatio n from Dia g ram (14), the CDGA ma ps ι , ι ′ , and ρ from Section 4.4 and the repres en tative θ of the Tho m clas s fro m Lemma 4 .8. Assume that H 1 ( f ) is injective and dim W ≥ 2 dim V + 3. W e ﬁx a CDGA- mo del φ : R → Q 29 of j ∗ : A P L ( W ) → A P L ( T ) such that R and Q are connected, R ≥ n +1 = 0 and Q ≥ m +2 = 0. Notice that φ is als o a model of A P L ( f ). By our standar d a buse of notation (s e e Sec tion 3.4) this determines orientation classes u W ∈ H ( R ) a nd u V ∈ H ( Q ). W e suppo se also that we hav e b een given an asso cia ted shr iek map φ ! : s − r Q → R. Notice that by Pr op o sition 4.5 we ca n alwa ys build such a CDGA mo del φ and shriek map φ ! . Our a im in this section is to describ e a CDGA-model of the ma p k : ∂ T → B using a CDGA mo del of the embedding f under the hypo theses that n ≥ 2 m + 3. In fact we will give a CDGA mo del of the Diagram (14) of the last section (Lemma 5.8 ) using only the mo del φ and the shriek ma p φ ! . This extr a precision is necessar y to get the mo del of the blow-up. The section is o rganized a s follows. In Section 5.1 we ﬁx a common mo del ˆ φ : ˆ R → ˆ Q of φ a nd j ∗ . In Section 5.2 we c o nstruct a common model ˆ φ ! of φ ! and ι . The co mmo n mo del ˆ φ ˆ φ ! of φφ ! and j ∗ ι c o mes with a ˆ R -dgmo dule structure. In Section 5 .3 we show that it is homoto pic to a ˆ Q -dgmo dule map χ . This extension of structure would not b e necessa ry if we just w anted a mo del of A P L ( B ), but it is necessa ry to o btain a mo del of A P L ( k ). In Section 5.4 (Lemma 5.4) we show that the co ne o n φ ! is a ˆ R -dgmo dule mo del of the cone on ι and that the cone on φφ ! is a ˆ Q -dgmo dule mo del of the cone o n j ∗ ι . W e alr e a dy know fr om the las t section (Lemma 4 .7) that the cones on ι and j ∗ ι are CDGA mo dels of A P L ( B ) and A P L ( ∂ T ) r esp ectively . W e also construct the maps betw een our mo dels that we will ne e d. In Section 5.5 (Lemma 5.6 ) we g ive conditions under which a map betw een diagra ms with certain dg mo dule str ucture c an b e extended to a CDGA map. Next we put everything to g ether and construct a CDGA mo del of Diagram (14 ), (Lemma 5.8). Since we are constructing the mo del of the blow-up we a lso keep track of the weak equiv alences connecting the mo dels. 5.1 A common mo del of φ and j ∗ Solely for purp oses o f the pro o f we use Lemma 3 .6 to ﬁx a c omm uting diag ram of CDGAs R φ   ˆ R α ≃ o o α ′ ≃ / / ˆ φ   A P L ( W ) j ∗   Q ˆ Q β ≃ o o β ′ ≃ / / A P L ( T ) (20) such that α, α ′ , β and β ′ are quasi-is omorphisms, ˆ R is cellular, ˆ φ is a cellular coﬁbration and the maps ( α, α ′ ) : ˆ R → R ⊕ A P L ( W ) ( β , β ′ ) : ˆ Q → Q ⊕ A P L ( T ) 30 are surjective. As a consequence R and A P L ( W ) are ˆ R -dgmo dules and Q and A P L ( T ) a r e ˆ Q -dgmo dules. Notice also that since ˆ φ is a cellular coﬁbration and ˆ R and ˆ Q a re connected, ˆ Q is a semifr e e ˆ R -dgmo dule. By the second part o f Lemma 3.6 the ho mology class e s u W ∈ H ( R ) a nd u V ∈ H ( Q ) co r resp onding to the o r ientations u W ∈ H ∗ ( W ) = H ( A P L ( W )) and u V ∈ H ∗ ( V ) π ∗ ∼ = H ∗ ( T ) = H ( A P L ( T )) using the quasi-isomor phis ms of Diagr am (20) are the same as those given by the original string of weak equiv alences that ma de φ a mo del of j ∗ . 5.2 A common mo del of φ ! and ι Next we cons truct a common mo del ˆ φ ! of φ ! and of ι : A P L ( W , B ) → A P L ( W ) deﬁned in Section 4.4. Recall also fro m Section 4.4 the deﬁnition ρ and the co cycle θ fro m Lemma 4.8. W e consider the following commut ative solid arrow diagram of ˆ R -dgmo dules s − r A P L ( T ) ≃ · θ ' ' O O O O O O O O O O O s − r Q φ !   s − r ˆ Q s − r β ≃ o o γ ′ ≃ / / _ _ _ ˆ φ !      s − r β ′ ≃ 9 9 s s s s s s s s s s A P L ( W , B ) ι   ρ ≃ / / A P L ( T , ∂ T ) R ˆ R α ≃ o o α ′ ≃ / / A P L ( W ) . (21) In the ne x t lemma we construct ˆ R -dgmo dule maps γ ′ and ˆ φ ′ making the dia gram commute. Notice that all horiz o nt al and diago nal maps a re weak equiv alences. Some of the maps ab ove are more than just ˆ R -dgmo dule ma ps and this ex tra structure will be used when we enhance our ˆ R -dgmo dule s tructure to get a CDGA structure in Section 5.5. Lemma 5.1 Ther e exists an ˆ R -dgmo dule we ak e quivalenc e γ ′ : s − r ˆ Q → A P L ( W , B ) and a ˆ R -dgmo dule map ˆ φ ′ : s − r ˆ Q → ˆ R making Diagr am (21) ab ove c ommute. Pro of. The quasi- isomorphism γ ′ is just any lift in the following solid ar row diagra m A P L ( W , B ) ρ ≃   s − r ˆ Q γ ′ 6 6 l l l l l l l ( · θ ) s − r β ′ ≃ / / A P L ( T , ∂ T ) . The Lifting Lemma (Lemma 3.4) implies the lift γ ′ exists s ince by Lemma 4.6 ρ is an a cyclic ﬁbr ation and s − r ˆ Q is a coﬁbrant ˆ R -dgmo dule. 31 By the same argument ther e exists a lift ˜ φ ! in the solid arrow diagram ˆ R α ′   s − r ˆ Q ιγ ′ / / ˜ φ ! : : t t t t t A P L ( W ) It follows fr om Lemma 4.9 that H ( α ) H ( ˜ φ ! ) H ( s − r β ) − 1 ( s − r u V ) = u W . So since by Prop os ition 4.4 φ ! represents the unique homoto p y class of maps with that prop erty , we hav e that φ ! s − r β ≃ α ˜ φ ! . Since ( α, α ′ ) : ˆ R → R ⊕ A P L ( W ) is a surjection we ca n use L e mma 3 .5 to replace ˜ φ ′ by a map ˆ φ ! : s − r ˆ Q → ˆ R ma king Diagram (2 1) commute on the nose. 5.3 Replacing ˆ φ ˆ φ ! b y a ˆ Q -dgmodule morphis m In this subs e ction w e s uppos e ﬁxed Diag rams (20 ) a nd (21 ) with the maps γ ′ and ˆ φ ! constructed in Lemma 5.1. Here we show that the ˆ R -dgmo dule ma p ˆ φ ˆ φ ! can b e repla ced by a ˆ Q -dgmo dule map χ which is homotopic to ˆ φ ˆ φ ! in a controlled wa y . Lemma 5.2 (i) ργ ′ : s − r ˆ Q → A P L ( T , ∂ T ) is a morphism of ˆ Q -dgmo dules. (ii) φφ ! = 0 and is t hu s a morphism of ˆ Q -dgmo dules. Pro of. (i) Lemma 5 .1 s ays that γ ′ makes Diag ram (21) commute s o we have that ργ ′ = ( · θ ) s − r ( β ′ ) Of co urse ( · θ ) is an A P L ( T )-dgmo dule map and hence a ˆ Q -dgmo dule map. Also β ′ and thus s − r β ′ are ˆ Q -dgmo dule maps. Hence the c o mpo sition ( · θ ) s − r β ′ = ργ ′ is a ˆ Q dg mo dule map. (ii) W e hav e assumed that n ≥ 2 m + 3, hence r = n − m ≥ m + 3. Since Q ≥ m +2 = 0 the statement follows. In g eneral ˆ φ ˆ φ ! : s − r ˆ Q → ˆ Q is not a ˆ Q -dgmo dule map but it ca n b e a djusted as describ ed in the following lemma. Lemma 5.3 Ther e exists a ˆ Q -dgmo dule map χ : s − r ˆ Q → ˆ Q and a ˆ R -dgmo dule homotopy h : ss − r ˆ Q → ˆ Q fr om χ t o ˆ φ ˆ φ ! such that β h = β ′ h = 0 . In p articular ( β , β ′ ) χ = ( β , β ′ ) ˆ φ ˆ φ ! . 32 Pro of. Since n ≥ 2 m + 3 , r = n − m ≥ m + 1, s o we know that H dim( V ) and so H ≥ r − 1 ( Q ⊕ A P L ( T )) = 0 = H ≥ r − 1 ( ˆ Q ). Thus a lift H exists . Our desired h : ss − r ˆ Q → ˆ Q is the homotopy corres po nding to H . 5.4 A dgmo dule mo del of j ∗ ⊕ sρ . Recall from Lemma 4.7 that the map of mapping cones j ∗ ⊕ sρ : A P L ( W ) ⊕ ι sA P L ( W , B ) → A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) is a CDGA mo del of k ∗ : A P L ( B ) → A P L ( ∂ T ). O ur a im in this subs e ction is to give another mo del (as dgmo dules) of that map. W e c onsider Diagrams (20 ) and (21 ) with the maps ˆ φ ! and γ ′ from Lemma 5.1 as well as the ˆ Q -dgmo dule map χ : s − r Q → ˆ Q and the ˆ R -dgmo dule homo topy h : χ ≃ ˆ φ ˆ φ from Lemma 5.3. Recall the notation from Lemma 3.13 . 33 Lemma 5.4 The map ˆ φ ⊕ h id: ˆ R ⊕ ˆ φ ! ss − r ˆ Q → ˆ Q ⊕ χ ss − r ˆ Q is a morphi sm of ˆ R -dgmo dules making the fol lowing diagr am c ommutative in the c ate gory of ˆ R -dgmo dules R ⊕ φ ! ss − r Q φ ⊕ id   ˆ R ⊕ ˆ φ ! ss − r ˆ Q α ⊕ ss − r β ≃ o o α ′ ⊕ sγ ′ ≃ / / ˆ φ ⊕ h id   A P L ( W ) ⊕ ι sA P L ( W , B ) j ∗ ⊕ sρ   Q ⊕ φφ ! ss − r Q ˆ Q ⊕ χ ss − r ˆ Q β ⊕ ss − r β ≃ o o β ′ ⊕ sργ ′ ≃ / / A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) (22) Mor e over t he horizontal maps in this diagr am ar e quasi-isomorphi sms and the b ottom r ow c onsists of ˆ Q -dgmo dule maps. Pro of. Since Lemma 5.3 says that β h = β ′ h = 0, the dia g ram in the statement of the lemma is commutativ e. W e s ee that the horizo n tal maps o f the diagr am are quasi- isomorphisms by Lemma 3.13. The fact that the b ottom line of the diagra m is of ˆ Q -dgmo dules follows from the construction of β , β ′ and γ ′ . 5.5 Extending dgmo dule structure to CDGA struct ure F or this subsectio n we supp ose ﬁxed Dia g rams (20) and (21) with the maps γ ′ and ˆ φ ! constructed in Lemma 5.1 and χ and h from Lemma 5.3. Here we use CDGA squa r es. At ﬁrs t sight this may seem clumsy but it keeps track of the ˆ R and ˆ Q mo dule structures in a c o nv enient wa y . Notation concerning squares is describ ed in Section 3.1 . Lemma 5.5 Assume that r is even. Ther e exists a u nique CDGA stru ct ur e on ˆ Q ⊕ χ ss − r ˆ Q (r esp e ctively, Q ⊕ ss − r Q ) extending its ˆ Q -dgmo dule (r esp e ct ively, Q - dgmo dule) structure . Mor e over we c an ﬁnd CDGA isomorphisms ˆ e and e , such that e ( z ) = ss − r 1 , ˆ e ( z ) = ( ˆ q , ss − r 1) for some q ∈ ˆ Q r − 1 , e | Q = id , ˆ e | ˆ Q = id , and which make the fol lowing diagr am c ommu te ˆ Q ⊗ Λ( z ) ˆ e / / β ⊗ id   ˆ Q ⊕ χ ss − r ˆ Q β ⊕ ss − r β   Q ⊗ Λ( z ) e / / Q ⊕ ss − r Q wher e | z | = r − 1 and dz = 0 . 34 Pro of. Recall that φφ ! = 0 for dimensio n re a sons. The ma p e is determined b y the conditions that e ( z ) = ss − r 1, e | Q = id and the fact that it is a Q - mo dule map. It is clearly a n isomorphism. Since χ ( ss − r 1) is a co cycle in ˆ Q and since H ≥ r ( ˆ Q ) = 0, there ex ists ˆ q ∈ ˆ Q r − 1 such that d ( ˆ q ) = − χ ( ss − r 1). F or degree reasons β ( ˆ q ) = 0. W e see also tha t ( ˆ q , ss − r 1) is a co cy c le in ˆ Q ⊕ χ ss − r ˆ Q . Then ˆ e is determined by the formula ˆ e ( z ) = ( ˆ q , ss − r 1) and is an isomorphism. Clear ly the diagr a m commutes. The CDGA structures are unique since all pro ducts ar e determined by ( ss − r (1)) 2 and the ˆ Q and Q mo dule str uctures and ss − r (1) must square to 0 since it is in o dd dimension. Lemma 5.6 F or r even, let D denote the c ommut ative squar e D = ˆ R ˆ φ / /  _   ˆ Q  _   ˆ R ⊕ ˆ φ ! ss − r ˆ Q ˆ φ ⊕ h id / / ˆ Q ⊕ χ ss − r ˆ Q. and let ( ˆ Q ⊗ Λ z , dz = 0) b e the r elative Sul livan algebr a fr om L emma 5.5 to gether with t he CDGA isomorphism ˆ e : ˆ Q ⊗ Λ z → ˆ Q ⊕ χ ss − r ˆ Q . (i) Ther e exists a CDGA squar e D ′ = ˆ R ˆ φ / /   ˆ Q  _   ˆ A ψ / / ˆ Q ⊗ Λ z and a we ak e quivalenc e b etwe en ˆ R -dgmo dule squar es Θ =  id id θ 3 ( ˆ e ) − 1  : D → D ′ . (ii) Su pp ose C = C 1 / /   C 2   C 3 / / C 4 is another CDGA squar e and ther e is a map Θ ′ =  θ ′ 1 θ ′ 2 θ ′ 3 θ ′ 4  : D → C such that θ ′ 1 and θ ′ 2 ar e CGDA maps, θ ′ 3 is a ˆ R -dgmo dule map and θ ′ 4 is a ˆ Q -dgmo dule maps wher e C 3 (r esp e ctively, C 4 ) has b e en given the ˆ R -dgmo dule (re sp e ctively, ˆ Q -dgmo dule) induc e d by θ ′ 1 (r esp e ctively, θ ′ 2 ). If H i ( C 3 ) = H i ( C 4 ) = 0 fo r 35 i ≥ 2 r − 3 and C 3 → C 4 is a ﬁbr ation, then t her e exists a CDGA map θ 3 such that Θ =  θ ′ 1 θ ′ 2 θ 3 θ ′ 4 ˆ e  : D ′ → C is a map of CDGA squar es m aking the fol lowing diagr am c ommut e. D Θ / / Θ ′   D ′ Θ ~ ~ } } } } } } } } C Pro of. In view of Lemma 5.5 to construct D ′ and Θ , and pr ov e part (i), it is enough to c o nstruct a r elative Sulliv an model ˆ R → ˆ A (th us also giving ˆ A a ˆ R -dgmo dule str ucture), a CDGA map ψ : ˆ A → ˆ Q ⊗ Λ( z ) and an ˆ R -dgmo dule map θ 3 : ˆ R ⊕ ˆ φ ! ss − r ˆ Q → ˆ A so that the following diagr am commutes ˆ R ⊕ ˆ φ ! ss − r ˆ Q θ 3 / / ˆ e − 1 ( ˆ φ ⊕ h id ) & & M M M M M M M M M M ˆ A ψ   ˆ Q ⊗ Λ z and θ 3 is a n equiv a le nce. Since ˆ φ is a cellular coﬁbration of CDGA s, ˆ Q = ˆ R ⊗ Λ U and so ss − r ˆ Q = ss − r ˆ R ⊗ Λ U . T he r e is an isomor phism o f ˆ R -dgmo dules ρ : ss − r ˆ R ⊗ Λ U → ˆ R ⊗ ss − r Λ U given by ρ ( ss − r α ⊗ β ) = ( − 1) (1 − r ) | α | α ⊗ ss − r β where α ∈ ˆ R and β ∈ Λ U . Thus ss − r ˆ Q ∼ = ˆ R ⊗ ss − r Λ U as ˆ R -dgmo dules and so ˆ R ⊕ ˆ φ ! ss − r ˆ Q ∼ = ˆ R ⊕ ˆ R ⊗ ss − r Λ U . Setting U 0 = ss − r Λ U , ˆ R ⊕ ˆ φ ! ss − r ˆ Q is isomorphic a s an ˆ R -dgmo dule to ˆ R ⊗ ( Q ⊕ U 0 ). The inclusion Q ⊕ U 0 → Λ U 0 induces an inclusion R ⊗ ( Q ⊕ U 0 ) → R ⊗ Λ U 0 . There is a unique diﬀerential on R ⊗ Λ U 0 satisfying the Liebniz law such tha t the inclusio n f 0 : ˆ R ⊕ ˆ φ ! ss − r ˆ Q ∼ = R ⊗ ( Q ⊕ U 0 ) → ˆ R ⊗ Λ U 0 is an ˆ R -dgmo dule map. Clearly then we get a commuting diagram ˆ R $ $ I I I I I I I I I I I / / ˆ R ⊕ ˆ φ ! ss − r ˆ Q f 0   ˆ R ⊗ Λ U 0 . Moreov er any ˆ R -dgmo dule ma p from ˆ R ⊕ ˆ φ ! ss − r ˆ Q into a CDGA extends uniquely to a CDGA map out o f ˆ R ⊗ Λ U 0 . In par ticular we get a C DGA ma p g 0 : ˆ R ⊗ 36 Λ U 0 → ˆ Q ⊕ χ ss − r ˆ Q a nd a c omm utative diagr a m ˆ R ⊕ ˆ φ ! ss − r ˆ Q f 0 / / ˆ φ ⊕ h id ' ' O O O O O O O O O O O ˆ R ⊗ Λ ( U 0 ) g 0   ˆ Q ⊕ χ ss − r ˆ Q. The elements of the cokernel of f 0 are elements in Λ U 0 of pro duct length at least tw o . So they are in degree at least 2 r − 2. W e add a minimal set of generator s U 1 of degree at least 2 r − 3 to kill the cohomolo gy of the cokernel of f 0 . W e then get a CDGA extension h 1 : ˆ R ⊗ Λ U 0 → ˆ R ⊗ Λ U 0 ⊗ Λ U 1 and a n ˆ R -dgmo dule map f 1 = h 1 f 0 : ˆ R ⊕ ˆ φ ! ss − r ˆ Q → ˆ R ⊗ Λ U 0 ⊗ Λ U 1 . W e have assumed that n ≥ 2 m + 3 a nd r = n − m , so 2 r − 2 = n − m + r − 2 ≥ m + r + 1 > m + r − 1 . Recalling that ˆ Q mo dels A P L ( V ) a nd ˆ Q ⊕ χ ss − r ˆ Q mo dels A P L ( ∂ T ) we know that ˆ Q ⊕ χ ss − r ˆ Q has tr ivial co ho mology in degr ees grea ter than n − 1 = m + r − 1 and so a lso in degr ees gr eater than or equal to 2 r − 2 wher e the b oundaries of U 1 lie. Ther efore the map g 0 can be extended ov er these new ele men ts to a CDGA map g 1 : ˆ R ⊗ Λ U 0 ⊗ Λ U 1 → ˆ Q ⊕ χ ss − r ˆ Q . Of co urse f 1 has a new cokernel but since H 0 ( ˆ R ) = Q (o f cour se the ho- mology is 0 in neg ative degrees) a nd U 1 was chosen minimally the cohomo l- ogy o f this cokernel is still in degrees greater than 2 r − 3. So again w e ca n add gener a tors to kill the cohomolo gy o f the co kernel. W e contin ue this pr o- cess countably many times to get our CDGA, ˆ A = ˆ R ⊗ Λ ( ⊕ i ≥ 0 U i ) and our map θ 3 = f ∞ : ˆ R ⊕ ˆ φ ! ss − r ˆ Q → ˆ A . Since each element of the c o homology of the cokernel of θ 3 was killed at the next sta ge, H ( θ 3 ) is surjective. Since 2 r − 2 > n , H ≥ 2 r − 2 ( ˆ R ⊕ ˆ φ ! ss − r ˆ Q ) = 0 and w e hav e not killed anything in H ( ˆ R ⊕ ˆ φ ! ss − r ˆ Q ); s o H ( θ 3 ) is injective. Thus θ 3 is a qua si-isomorphis m o f ˆ R - dgmo dules. The orig inal map g 0 extends to a CDGA map g ∞ : ˆ A → ˆ Q ⊕ χ ss − r ˆ Q since H ≥ 2 r − 2 ( ˆ Q ⊕ χ ss − r ˆ Q ) = 0 and so at ea ch stage all obstr uctions are triv- ial for degr ee reasons. So we can set ψ = ˆ e − 1 g ∞ and we hav e completed the construction of D ′ and Θ . W e now pro ceed to construct the map Θ and prove part (ii). An y ˆ Q - dgmo dule map out of ˆ Q ⊕ χ ss − r ˆ Q is automatically a CDGA map since ( ss − r 1) · ( ss − r 1) = 0 b ecause ss − r 1 is of o dd degr ee. Thus θ ′ 4 and θ ′ 4 ˆ e are CDGA maps. So w e only ha ve to co ns truct a CDGA map θ 3 : ˆ A → C 3 extending θ ′ 3 and making the following diagram commute ˆ A ˆ eψ / / θ 3   ˆ Q ⊕ χ ss − r ˆ Q θ ′ 4   C 3 / / C 4 . 37 Clearly the ˆ R -dgmo dule ma p θ ′ 3 extends uniquely to a CDGA map h 0 : ˆ R ⊗ Λ U 0 → C 3 making the following diagram commute ˆ R ⊗ Λ U 0 h 0   g 0 / / ˆ Q ⊕ χ ss − r ˆ Q θ 4   C 3 / / C 4 . Since H i ( C 3 ) = 0 for i > 2 r − 2 , we can further extend to a CDGA ma p h ′ 1 : ˆ R ⊗ Λ U 0 ⊗ Λ U 1 → C 3 . W e show that the following CDGA diagra m ˆ R ⊗ Λ U 0 ⊗ Λ U 1 h ′ 1   g 1 / / ˆ Q ⊕ χ ss − r ˆ Q θ 4   C 3 / / C 4 commutes up to CDGA homotopy . Indeed it alr e ady commutes on ˆ R ⊗ Λ U 0 and hence the tw o wa ys of going ar o und diﬀer by a n element of [Λ U 1 , C 4 ] and this group is 0 since H i ( C 4 ) = 0 for i ≥ 2 r − 3 a nd U 1 has no elements in degrees ≤ 2 r − 3. Because C 3 → C 4 is a surjection we can r e pla ce h ′ 1 by a map h 1 : ˆ R ⊗ Λ U 0 ⊗ Λ U 1 → C 3 making the last diagram commute ex a ctly . Using this same metho d at each stage and taking the direct limit we get our desir ed CDGA map θ 3 : ˆ A → C 3 making the diagra m commute. The approa ch taken in the las t lemma gives a hint of how to appro ach the problem of e x tending an ˆ R -dgmo dule structure to a CDGA structure when there are no dimension restrictions. At each stag e one would have to choose which r e pr esentativ es of the co kernel to kill. Now we descr ib e a CDGA mo del of Diagra m (14) Lemma 5.7 Assume t hat r is even. With t he semi-t r ivial CDGA on t he map- ping c ones, the squar e E = R φ / /  _   Q  _   R ⊕ φ ! ss − r Q φ ⊕ id / / Q ⊕ φφ ! ss − r Q. is a c ommuting squar e of CDGAs. Also ther e exists a CDGA squar e E ′ = R φ / /  _   Q  _   A κ / / Q ⊗ Λ( z ) . 38 such that κ is a ﬁbr ation and t her e exists a CDGA quasi-isomorphism ξ : R ⊕ φ ! ss − r Q → A such that Θ 1 =  id id ξ e − 1  : E → E ′ is a we ak e quivalenc e of CDGA squar es. Pro of. That E is a CDGA square follows using Pro p os ition 3.18. T o cons truct E ′ and Θ 1 we fac to r the CDGA ma p φ ⊕ i d : R ⊕ φ ! ss − r Q → Q ⊕ φφ ! ss − r Q as an acyclic coﬁbra tio n ξ : R ⊕ φ ! ss − r Q → A followed by a ﬁbr ation κ : A → Q ⊕ φφ ! ss − r Q . The lemma fo llows easily . In the next lemma D a nd D ′ are the diagr ams of Lemma 5.6 , E and E ′ are the diagra ms of the prev ious lemma and F , F ′ and Θ 6 are the diag r ams a nd map from Lemma 4.7. Lemma 5.8 Assume that r is even. L et ρ b e t he m ap fr om 4.4, γ ′ fr om Se ction 5.2, ˆ e fr om L emma 5.5 and Θ 1 fr om L emma 5.7. Set µ ′ = ( i ∗ + 0)( β ′ ⊕ sργ ′ ) ˆ e : ˆ Q ⊗ Λ z → A P L ( ∂ T ) which is a quasi-isomorph ism. Ther e exist CDGA maps ζ : ˆ A ≃ → A and ζ ′ : ˆ A ≃ → A P L ( B ) such that if we set Θ 2 =  α β ζ β ⊗ id  and Θ 3 =  α ′ β ′ ζ ′ µ ′  then we have a chain of qu asi-isomorph isms of CDGA s qu ar es E Θ 1 → E ′ Θ 2 ← D ′ Θ 3 → F . Pro of. Set Θ 4 =  α β α ⊕ s s − r β β ⊕ h ss − r β  : D → E and Θ 5 =  α ′ β ′ α ′ ⊕ sγ ′ β ′ ⊕ sργ ′  : D → F ′ . By Lemmas 4.7, 5.4 and 5.7 w e hav e a weak equiv alences of ˆ R -dgmo dule squares Θ 4 Θ 1 : D → E ′ and Θ 6 Θ 5 : D → F . Also the maps be tw een the ob jects in the right hand co lumn of each square are ˆ Q -dgmo dule ma ps. W e ca n apply Lemma 5.6 to these maps and g et the string of quasi-is omorphisms of s quares as stated in the lemma. 6 Mo del of the pro jectivization of a complex bundle In this section we supp ose that f : V → W is a smo oth embedding of c lo sed manifolds of co dimension 2 k . W e will assume that k > 1. W e also s uppo s e that 39 the normal bundle ν of the embedding has s ome ﬁxed structure of a complex vector bundle, ν : C k → E π → V . Let T b e a co mpact tubular neighborho o d of V in W . By the T ubular Neigh- bo rho o d Theo rem, we ca n identify T with the disk bundle D ν and ∂ T with its sphere bundle S ν in s uc h a way that the zero section of D ν corr esp o nds to the inclusion σ : V → T . W e ﬁx such an identiﬁcation. W e als o supp ose we ha ve b een given some CDGA mo del Q of A P L ( T ) a nd a common mo del ˆ Q Q ˆ Q β ′ / / β o o A P L ( T ) of Q and A P L ( V ) such that ( β , β ′ ) is sur jectiv e and β and β ′ are quasi-iso rmorphisms as in Sectio n 5.1 . Clearly σ ∗ β ′ : ˆ Q → A P L V is also a quas i- isomorphism. By Lemma 3.6 such a common mo del can b e co nstructed from any CDGA mo del Q o f A P L ( T ). The aim of this s ection is to descr ib e the pro jective bundle P ν ass o ciated to ν and give a CDGA model for this pro jective bundle. In Sectio n 6 .1 we re view the deﬁnition o f the pro jectiv e bundle a nd of Cher n classes a nd prove the triviality of a certain line bundle (Lemma 6.1). Next in Sectoin 6.2 we consider the pullback ov e r a po in t of the sphere bundle and of its pro jectivization. Using orientation infor mation we show that the models o f these pullbacks a nd the mo del of the sphere bundle P ν can b e chosen in a compatible wa y (Lemmas 6.4 and 6.6). T he n in Section 6.3 we co ns truct a mo del o f the pro jectivization of the sphere bundle and us e the re sults from Section 6.2 to show this mo del is compatible with the mo del o f the b ounda ry of the nor ma l bundle (Lemma 6.8 and P rop osition 6 .9). 6.1 The pro jective bundle a nd the Chern classes Next we r ecall the deﬁnition of the Che r n classe s of a co mplex bundle using the asso ciated pro jective bundle as describ ed in [2 , IV.20]. Consider the universal complex line bundle γ 1 ov er C P ( ∞ ). C → E γ 1 → C P ( ∞ ) where C P ( ∞ ) = { l : l is a C line in C ∞ } and E γ 1 = { ( l , v ): l ∈ C P ( ∞ ) , v ∈ l } . This co mplex line bundle can also b e viewed as an o r iented real vector bundle of rank 2. Therefore we have an a sso ciated E ule r cla ss e ( γ 1 ) and we set a ∞ : = − e ( γ 1 ) ∈ H 2 ( C P ( ∞ ) , Z ) 40 This is our prefer r ed gener ator of the cohomolog y algebra H ∗ ( C P ( ∞ ) , Z ). Note that if j : S 2 = C P (1 ) → C P ( ∞ ) is the o b vious inclusion then j ∗ ( a ∞ ) ∈ H 2 ( S 2 ; Z ) is the or ien tation class corr esp onding to the orientation coming fr om its complex structure. T o any complex vector bundle ν we ca n asso cia te its pr oje ctive bund le which is deﬁned as follows (see [2, page 269]). Set E 0 = E \ { zero section } and consider the bundle ν 0 : C k \ { 0 } → E 0 π 0 → V Then C ∗ = C \ { 0 } acts on each ﬁbre in E 0 by co mplex mult iplication a nd we can de ﬁne the orbit space P ν = E 0 / C ∗ . In o ther words, P ν = { ( v , ℓ ) : v ∈ V , ℓ is a complex line in the ﬁbre E v := π − 1 ( v ) } and we hav e the pro jective bundle C P ( k − 1) inc → P ν π ′ → V Denote by q : E 0 → P ν the quotient map and consider the co mmutative diagram E 0 q / / π 0 A A A A A A A A P ν π ′ ~ ~ | | | | | | | | V . (23) Since the inclusio n ∂ T ∼ = S ν ֒ → E 0 is a homo topy equiv alence, we also us e q to denote the comp osition ∂ T ∼ = S ν / / E 0 q / / P ν . W e now come to the deﬁnition of the Chern clas ses c i ( ν ) ∈ H 2 i ( V ; Z ) in terms of the pr o jective bundle. The pullback o f ν along π ′ is a C k − bundle π ′∗ ( ν ) ov e r P ν containing a tauto logical line bundle deﬁned by λ = { ( v , ℓ, x ) : v ∈ V , ℓ is a line in E v , x ∈ ℓ } (24) The complex line bundle λ is classiﬁed b y some map α : P ν → C P ( ∞ ) ≃ B U (1 ), that is λ ∼ = α ∗ ( γ 1 ). W e deﬁne the c anonic al class of ν as the cohomo logy class a = α ∗ ( a ∞ ) ∈ H 2 ( P ν , Z ) . (25) Since the restric tio n o f that class to each ﬁbr e C P ( k − 1) of π ′ is a genera tor of the co homology (beca use the pullback of λ to that ﬁbre is the universal line bundle o ver C P ( k − 1)), the Leray-Hirsch Theo rem gives an isomorphism of H ∗ ( V , Z ) − alg e bras H ∗ ( P ν , Z ) ∼ = H ∗ ( V , Z )[ a ] / k X i =0 c i ( ν ) a k − i ! (26) 41 where c 0 ( ν ) = 1 and by deﬁnition the c i ( ν ) ∈ H 2 i ( V , Z ) for i > 0 are the Cher n classes. Notice a str aightforw ard calculation shows that a = − c 1 ( λ ) . (27) Lemma 6.1 q ∗ ( a ) = 0 . Pro of. Consider the fo llowing diagra m of vector bundles q ∗ λ   / / λ ! ! D D D D D D D D D   / / π ′∗ ( ν ) / /   pullback ν   E 0 q / / P ν π ′ / / α ' ' P P P P P P P P P P P P P V C P ( ∞ ) = B U (1) The cohomolo gy cla ss q ∗ ( a ) is the clas sifying class of the line bundle q ∗ ( λ ). W e will prov e that this bundle is trivia l, so that q ∗ ( a ) = 0. By (2 4) we have q ∗ λ = { ( e, v , ℓ, x ) : e ∈ E 0 , v ∈ V , v = π 0 ( e ) , ℓ is a line in E v , x ∈ ℓ } ∼ = { ( e, x ) : e ∈ E 0 , x ∈ C .e ⊂ E π 0 ( e ) } Therefore we hav e the trivializ ation E 0 × C ∼ = → q ∗ λ , ( e, z ) 7→ ( e, e.z ) 6.2 Orien t ations As in Section 4.1 we supp ose ∗ ∈ V is so me ﬁxed p oint. Co nsider the map η : S 2 k − 1 → C P ( k − 1) obtained as the restriction of the map q : S ν → P ν to the ﬁbres o ver our p oint ∗ ∈ V . The a im of this subsection is to give an explicit CDGA mo del of the map η . First we deﬁne some more cohomolog y class e s. Recall the c la ss a ∈ H 2 ( P ν ) deﬁned in (2 5). W e also denote by a ∈ H 2 ( C P ( k − 1)) its res triction to the 42 ﬁbre. Notice that the r estriction of λ to C P ( k − 1) is the tautolog ical line bundle γ k − 1 and hence by Equation (2 7) and the naturality of the Chern class es a = − c 1 ( γ k − 1 ) . (28) Equiv alently we can take Equation (28) to b e the deﬁnition of a ∈ H 2 ( C P ( k − 1)). The ﬁbre D 2 k of the disk bundle D ν over our po int ∗ ∈ V has a cano nical orientation given b y its complex structure D 2 k ⊂ C k . This de ter mines the cohomolog y c la ss u D 2 k ∈ H 2 k ( D 2 k , S 2 k − 1 ; Z ) (29) of Equation (15) which through the connecting homomo rphism δ : H 2 k − 1 ( S 2 k − 1 ; Z ) ∼ = / / H 2 k ( D 2 k , S 2 k − 1 ; Z ) determines an orientation class u S ∈ H 2 k − 1 ( S 2 k − 1 , Z ) . (30) Next we extend η to some map ˜ η : D 2 k → C P ( k ) a nd describ e C P ( k ) as a suitable pus hout. F or z = ( z 1 , . . . , z k ) ∈ C k we set || z || 2 =  Σ k i =1 | z i | 2  1 / 2 . Consider the disc and the sphere D 2 k = { z ∈ C k : || z || 2 ≤ 1 } S 2 k − 1 = { z ∈ C k : || z || 2 = 1 } . Deﬁne a ma p ˜ η : D 2 k → C P ( k ) , z 7→ [ z 1 : . . . : z k : 1 − || z || 2 ] Then η = ˜ η | S 2 k − 1 where C P ( k − 1) is considered as the hype rplane with la st co ordinate 0 . W e need to replac e the map η by a n eq uiv alent map that is a co ﬁbration. This is done us ing the standard mapping cy linder construction. W e will describ e it explicitly so that it is ea sy to see how it is co mpa tible with ˜ η . Deﬁne the contraction ρ : D 2 k → D 2 k , z 7→ z / 2 . Set X = { ˜ η ( z ): z ∈ D 2 k , 1 / 2 ≤ || z || 2 ≤ 1 } ⊂ C P ( k ) and deno te by l : S 2 k − 1 → D 2 k , ˜ l : X → C P ( k ) and s : C P ( k − 1) → X t he obvious inclusions. Then the c o mpo site ˜ h = ˜ η ρ : D 2 k → C P ( k ) is a homeomorphis m onto its imag e and a coﬁbratio n. Also we can think of X as a tubular neighbourho o d of C P ( k − 1) in C P ( k ) although we will not use this fact. W e summarize a few facts ab out these maps in the following lemma. Its pro of is straightforward. 43 Lemma 6.2 (i) The re striction of ˜ h to S 2 k − 1 induc es a c oﬁbr ation h : S 2 k − 1 → X (ii) Ther e is a pushout S 2 k − 1 h / / l   X ˜ l   D 2 k ˜ h / / C P ( k ) (iii) The inclusion map s : C P ( k − 1) → X is a homotopy e quivalenc e. (iv) The fol lowing diagr am c ommutes up to homotopy S 2 k − 1 h / / η & & L L L L L L L L L L X C P ( k − 1) . s O O By abuse o f notation w e also use a ∈ H 2 ( X ) to denote the pre ima ge thro ugh the isomorphism s ∗ of the element a ∈ H 2 ( C P ( k − 1)) deﬁned in (28). Lemma 6.3 Consider the Sul livan algebr a (Λ( x, z ); dx = 0 , dz = x k ) with | x | = 2 and | z | = 2 k − 1 . Ther e exist s a homotopy c ommutative diagr am of CDGA Λ( x, z ) g 0   pro j / / Λ( z ) f 0   A P L ( X ) A P L ( h ) / / A P L ( S 2 k − 1 ) . such that [ g 0 ( x )] = a (deﬁne d in (28)) and [ f 0 ( z )] = − u S (deﬁne d in (30)). Pro of. Since k > 1, the restriction map ι : H 2 ( X, S 2 k − 1 ) → H 2 ( X ) is an isomorphism and so the class a lifts to a unique cla ss a 0 ∈ H 2 ( X, S 2 k − 1 ). Denote by ˜ a ∈ H 2 ( C P ( k )) the opp osite o f the ﬁr st Chern c la ss o f the ta u- tological line bundle γ k ov er C P ( k ). Since γ k − 1 is the r estriction of γ k ov er C P ( k − 1), Eq uation (28) and the naturality o f the Cher n class es implies that ˜ l ∗ (˜ a ) = a . In vie w of the pushout of Lemma 6.2(ii), ˜ a lifts to some class ˜ a 0 ∈ H 2 ( C P ( k ) , D 2 k ) such that ( ˜ l , l ) ∗ (˜ a 0 ) = a 0 . 44 Let α 0 ∈ A 2 P L ( X, S 2 k − 1 ) ∩ ker d b e a representative of a 0 and α b e its image in A 2 P L ( X ) ∩ ker d . Since a k = 0 in H 2 ( X ), there exis ts ζ ∈ A 2 k − 1 P L ( X ) such that dζ = α k . Deﬁne g 0 : (Λ ( x, z ); dz = x k ) → A P L ( X ) by g 0 ( x ) = α and g 0 ( z ) = ζ . Since h ∗ ( α ) = 0, h ∗ ( ζ ) is a co cycle in A P L ( S 2 k − 1 ) and s o we ca n deﬁne f 0 : Λ ( z ) → A P L ( S 2 k − 1 ) , z 7→ h ∗ ( ζ ) . This deﬁnition makes the diagra m of the lemma commutativ e. W e pro ceed to prov e that [ h ∗ ( ζ )] = − u S . This will imply that [ f 0 ( z )] = − u S and thus complete the pro of of the lemma. Let δ and δ ′ denote the c o homology connecting ho mo morphisms of the pa ir s ( D 2 k , S 2 k − 1 ) and ( X , S 2 k − 1 ) res pectively . Consider the following diagr am in cohomolog y H 2 k − 1 ( S 2 k − 1 ; Z ) δ ′ + + δ $ $ H 2 k ( X ; Z ) H 2 k ( X, S 2 k − 1 ; Z ) ι o o H 2 k ( D 2 k ; Z ) H 2 k ( C P ( k ); Z ) ˜ h ∗ o o ˜ l ∗ O O H 2 k ( C P ( k ) , D 2 k ; Z ) ˜ ι o o ( ˜ l,l ) ∗ O O H 2 k ( D 2 k , S 2 k − 1 ; Z ) O O H 2 k ( C P ( k ) , X ; Z ) . ˜ ι ′ O O ( ˜ h,h ) ∗ o o A diagram chase at the chain level gives us the formula ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 δ = − ˜ ι (( ˜ l , l ) ∗ ) − 1 δ ′ , (31) the minus sign cor resp onding to the fact that in the May er-Vietoris sequence there is a minus sign on one of the maps. By the deﬁnition o f the pushout o f Lemma 6 .2(ii) a nd the co ns truction of ζ we have δ ′ ([ h ∗ ζ ]) = [ α k ] = a k 0 ∈ H 2 k ( X, S 2 k − 1 ) Since ( ˜ l , l ) ∗ and ˜ ι a re multiplicative we get ˜ ι (( ˜ l , l ) ∗ ) − 1 δ ′ [ h ∗ ζ ] = ˜ a k ∈ H 2 k ( C P ( k )) (32) On the other hand since ˜ a = − c 1 ( λ ), [22, page 17 0] says that the orientation of C P ( k ) induced by its complex structure corr esp onds to the class ˜ a k ∈ H 2 k ( C P ( k )) . 45 Since ˜ h preser ves the given orientations, we hav e ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 u D = ˜ a k , and hence ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 δ ( u S ) = ˜ a k . (33) Equations (31), (32) and (33) imply that ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 δ ( u S ) = − ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 δ [ h ∗ ζ ] Thu s since ˜ ι ′ (( ˜ h, h ) ∗ ) − 1 δ : H 2 k − 1 → H 2 k C P ( n ) is a n isomo rphism we deduce that [ h ∗ ζ ] = − u S This completes the pro of of the lemma. Lemma 6.4 L et | x | = 2 and | z | = 2 k − 1 . Supp ose we ar e given a CDGA diagr am (Λ( x, z ); dz = x k ) g   pro j / / Λ( z ) f   A P L ( C P ( k − 1)) A P L ( η ) / / A P L ( S 2 k − 1 ) . such that [ g ( x )] = a and [ f ( z )] = − u S . Then the diagr am is homotopy c ommu - tative and f and g ar e qu asi-isomorph isms. Pro of. Observe that since H 2 k − 1 ( C P ( k − 1)) = 0 the homoto p y class of the map g is deter mined b y the fact that [ g ( x )] = a . This equation also implies that g is a quasi-isomor phism. Similarly the eq uation [ f ( z )] = − u S determines the homotopy cla ss o f the map and implies that it is a qua si-isomorphis m. The lemma then follows easily from Lemma 6.2 (iii) and (iv) and Lemma 6.3 . Lemma 6.5 Consider the map µ ′ : ˆ Q ⊗ Λ( z ) → A P L ( ∂ T ) fr om L emma 5.8. (i) Consider t he c onne cting homomorphism δ : H 2 k − 1 ( ∂ T ) → H 2 k ( T , ∂ T ) and t he Thom class θ fr om Se ction 4.1. Then δ [ µ ′ ( z )] = − θ. (ii) L et inc : S 2 k − 1 → ∂ T b e the inclusion. Then inc ∗ [ µ ′ ( z )] = − u S . 46 Pro of. By the deﬁnition of µ ′ in Lemma 5.8, the following diagr am commutes ˆ Q ⊗ Λ( z ) µ ′ % % L L L L L L L L L L L L L L L L L L L L L L L L L ˆ e / / ˆ Q ⊕ ss − 2 k ˆ Q β ′ ⊕ sργ ′   A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) i ∗ +0   A P L ( ∂ T ) . By the constr uction of ˆ e in Lemma 5.5 there exists ˆ q ∈ ˆ Q 2 k − 1 such that ˆ e ( z ) = ( ˆ q , ss − 2 k 1). By Lemma 5.1 ργ ′ ( ss 2 k (1)) = θ and we all see that ( β ′ ⊕ sργ ′ )( ˆ q , ss − 2 k 1) = ( β ′ ( ˆ q ) , sθ ) , so µ ′ ( z ) = i ∗ ( β ′ ( ˆ q )) . (34) Since z is a co cycle, ( β ′ ( ˆ q ) , sθ ) ∈ A P L ( T ) ⊕ ι ′ sA P L ( T , ∂ T ) is a lso a co cycle which implies that in A P L ( T ) d ( β ′ ( ˆ q )) = − ι ′ ( θ ) . The deﬁnition of the connecting homomor phism then implies that δ [ i ∗ β ′ ( ˆ q )] = − [ θ ] = − θ Combining this formula with Equa tion (34) proves (i). (ii) This is a consequence of (i), of the fact that u D 2 k = inc ∗ ( θ ) (see Equation (17) a nd of the natura lit y of the connecting homomorphis m in the following diagram H 2 k − 1 ( ∂ T ) δ / / inc ∗   H 2 k ( T , ∂ T ) inc ∗   H 2 k − 1 ( S r − 1 ) ∼ = δ / / H 2 k ( D 2 k , S 2 k − 1 ) . Lemma 6.6 The inclusion of the p oint ∗ ∈ V determines an augmentation on ˆ Q u sing t he c omp osition ˆ Q σ ∗ β ′ → A P L ( V ) → A P L ( ∗ ) = Q 47 In turn the augmentation determines the pr oje ct ion map pro j: ˆ Q ⊗ Λ( z ) → Λ( z ) . Supp ose f : (Λ( z ); dz = 0) → A P L ( S 2 k − 1 ) is any CDGA map s uch that [ f ( z )] = − u S . Then the fol lowing diagr am c ommu tes u p t o CDGA homotopy ˆ Q ⊗ Λ( z ) pro j / / µ ′   Λ( z ) f   A P L ( ∂ T ) inc ∗ / / A P L ( S 2 k − 1 ) Pro of. In this pr o of we denote by ∗ an augmentation ma p followed by a unit map. F or ﬁxed domain and ra ng e this comp osition is unique up to homoto p y since all of our CDGAs are homolog ically connected. Since pro j | ˆ Q = ∗ , f pro j | ˆ Q = ∗ . Also µ ′ | ˆ Q factors up to homotopy through A P L ( V ) so inc ∗ µ ′ | ˆ Q ≃ ∗ . Because dz = 0 in ˆ Q ⊗ Λ( z ) (s e e Lemma 5.5) the diag ram commutes if and only if inc ∗ [ µ ′ ( z )] = [ f pro j( z )] in H A P L ( S 2 n − 1 ). Lemma 6.5 says that inc ∗ [ µ ′ ( z )] = − u S and by assumption [ f ( z )] = − u S . Th us the diagram homotopy commutes as required. 6.3 A mo del of the pro jectiv e and the sphere bundles Recall from the star t of Sec tion 6 that V ֒ → W has co dimension 2 k and s o this is also the real ra nk of the nor mal bundle ν , which also has a complex structure of rank k . Recall a lso the maps σ , β and β ′ from the start of Section 6. W e de s crib e the Sulliv an mo dels that we will prove ar e mo dels for the pr o- jective bundle P ν . Set γ 0 = 1 and for 1 ≤ i ≤ k let γ i ∈ Q 2 i ∩ ker d be representatives of the images of the Chern classe s c i ( ν ) ∈ H 2 i ( V , Z ) → H 2 i ( V , Q ) ( σβ ′ ) ∗ ∼ = H 2 i ( ˆ Q ) β ∗ ∼ = H 2 i ( Q ) . Since β is a surjective quas i-isomorphism there exists ˆ γ i ∈ ˆ Q 2 i ∩ ker d such that β ( ˆ γ i ) = γ i . Also w e can take ˆ γ 0 = 1. Let | x | = 2 a nd | z | = 2 k − 1 a nd de ﬁne relative Sulliv an mo dels ( Q ⊗ Λ( x, z ); D x = 0 , Dz = Σ k i =0 γ i x k − i ) (35) and s imila rly ( ˆ Q ⊗ Λ( x, z ); ˆ D x = 0 , ˆ D z = Σ k i =0 ˆ γ i x k − i ) (36) These mo dels a re motiv ated by Equa tion (26) for the Cher n cla sses. F or the next lemma re c all that σ : V → T corresp onds to the inclusio n of the zero section. In addition reca ll the clas s es a ∈ H 2 ( P ν ) deﬁned in (25) and its restriction to C P ( k − 1) also denoted by a ∈ H 2 ( C P ( k − 1)). 48 Lemma 6.7 Consider the pr oje ctive bun d le asso ciate d to ν C P ( k − 1) inc / / P ν π ′ / / V . L et ˆ D b e the diﬀer ential given in Equation (36). Supp ose g : (Λ( x, z ); dz = x k ) → A P L ( C P ( k − 1)) is any map such that [ g ( x )] = a . Then t her e exist s a quasi- isomorphi sm θ ′ : ( ˆ Q ⊗ Λ( x, z ); ˆ D ) → A P L ( P ν ) making the fol lowing diagr am c ommute up to homotopy ˆ Q σ ∗ β ′   / / ( ˆ Q ⊗ Λ( x, z ) , ˆ D ) θ ′ ≃   pro j / / (Λ( x, z ) , dz = x k ) g   A P L ( V ) π ′∗ / / A P L ( P ν ) inc ∗ / / A P L ( C P ( k − 1)) . Pro of. As observed at the star t o f the section σ ∗ β ′ is a quas i-isomorphism. Deﬁne θ ′ | ˆ Q ⊗ Λ( x ) so tha t θ ′ | ˆ Q = ( σ π ′ ) ∗ β ′ and θ ′ ( x ) is any repr esentativ e o f the ima g e of a under the map H ∗ ( P ν , Z ) → H ∗ ( P ν , Q ) ∼ = H ( A P L ( P ν )) Then we know fro m the deﬁnition of the Chern cla sses (s e e Section 6.1) that Σ k i =0 c i ( ν ) a k − i = 0 in H ∗ ( P ν , Z ) and so [Σ k i =0 π ′ ∗ σ ∗ β ′ ( ˆ γ i )( θ ′ ( x )) k − i ] = 0 in H ( A P L ( P ν )). Thus a n extension ov er z of θ ′ | ˆ Q ⊗ Λ( x ) exists. Let θ ′ be any such extension. It is clear us ing Equatio n (26) that θ ′ is a q uasi-isomor phism and makes the left hand squa re of our diag ram c o mm ute. Since (inc ∗ θ ′ ) | ˆ Q = 0 the right hand square commutes when restricted to ˆ Q . Since g ( x ) repre s ent s inc ∗ ( a ) we see that the right hand square c o mm utes up to homotopy when restricted to Λ( x ). Thus it commutes up to homotopy when restricted to ˆ Q ⊗ Λ ( x ). W e know that [Λ( z ) , A P L C P ( k − 1)] = H 2 k − 1 ( C P ( k − 1)) = 0. This implies that, up to homotopy , ther e is a unique extension of the map ˆ Q ⊗ Λ ( x ) → A P L ( C P ( k − 1)) ov er ˆ Q ⊗ Λ( x, z ). Thu s the right ha nd s quare in the diag ram c o mm utes up to homotopy a nd the lemma has been prov en. In this se c tion θ ′ yielded a mo del of A P L ( P ν ). Lemma 6.8 will show that the tw o a re compatible. This lemma co n trols the automo rphism of the mo del o f A P L ( ∂ T ). If Q is our mo del of A P L ( T ) then the mo del of A P L ( ∂ T ) is Q ⊗ Λ z . An automorphism φ ∈ [ A P L ( ∂ T ) , A P L ( ∂ T )] can be considered as a n element of [ Q ⊗ Λ z , Q ⊗ Λ z ] ∼ = [ Q, Q ⊗ Λ z ] × [Λ z , Q ⊗ Λ z ]. W e control the ﬁrst fa c to r by working in the categ ory of Q - dgmo dules. Another wa y to think of this is that we work with o b jects tog ether with maps fr om Q . T o handle the seco nd factor we can observe that for dimensio n reaso ns [Λ z , Q ⊗ Λ z ] ∼ = [Λ z , Λ z ] and in turn we have the ge neral isomor phism [Λ z , Λ z ] ∼ = Hom(H 2k − 1 ∂ T , H 2k − 1 ∂ T). So w e only hav e to con trol a single element in homology and this is the r eason we have 49 bee n keeping tr ack of orientation cla sses. In the last section µ ′ gav e us a mo del of A P L ( ∂ T ). Recall β from the start of the s ection, µ ′ from Lemma 5.8 , θ ′ from Lemma 6.7, q : ∂ T → P ν deﬁned in Section 6.1 and the CDGAs Q ⊗ Λ( x, z ) and ˆ Q ⊗ Λ( x, z ) deﬁned in equations (35) and (36). Lemma 6.8 Assume t hat 2 k ≥ dim( V ) + 2 . The fol lowing diagr am c ommutes up to homotopy Q ⊗ Λ( x, z ) pro j / / Q ⊗ Λ( z ) ˆ Q ⊗ Λ( x, z ) β ⊗ id O O pro j / / θ ′   ˆ Q ⊗ Λ( z ) β ⊗ id O O µ ′   A P L ( P ν ) q ∗ / / A P L ( ∂ T ) (37) Pro of. The top squa r e clearly commu tes. Since w e will b e using diﬀerent inclu- sions, to av oid confusion for the r e st of the pro of set I = inc ∗ : A P L ( ∂ T ) → A P L ( S 2 k − 1 ). T o av oid to o muc h clutter in the equations we will also sometimes write I instead of I ∗ for induced maps. Re c all the cla s ses a ∈ H 2 ( P ν ) deﬁned in (25) and also its r e striction to C P ( k − 1) also denoted b y a ∈ H 2 ( C P ( k − 1)). Let g : Λ( x, z ) → A P L ( C P ( k − 1 )) be any map such that [ g ( x )] = a and f : Λ( z ) → A P L ( S 2 k − 1 ) b e any map such that [ f ( z )] = − u S . Note that these equations determine g and f up to homotopy . T o see that the b ottom square of (37) commutes consider the following cub e: ˆ Q ⊗ Λ( x, z ) θ ′   pro j / / pro j ( ( Q Q Q Q Q Q Q Q Q Q Q Q ˆ Q ⊗ Λ( z ) µ ′   pro j ' ' N N N N N N N N N N N Λ( x, z ) g   pro j / / Λ( z ) f   A P L ( P ν ) q ∗ / / inc ∗ ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q A P L ( ∂ T ) I ' ' N N N N N N N N N N N A P L ( C P ( k − 1)) A P L ( η ) / / A P L ( S 2 k − 1 ) . The back face is the one we w is h to show is homo to p y commutativ e. The top and b o ttom faces clearly commute. The fro n t, rig h t and left faces are homotopy 50 commutativ e by Lemmas 6.4, 6 .6 and 6 .7. So we g et that I µ ′ (pro j) ≃ I q ∗ θ ′ . Next c onsider the coaction sequence ([23, Chapter I Section 3 Prop ositio n 4 ]) asso ciated to the map from the coﬁbr a tion sequence of CDGA Λ( s − 1 z ) → ˆ Q ⊗ Λ( x ) → ˆ Q ⊗ Λ( x, z ) → Λ( z ) int o A P L ( ∂ T ) → A P L ( S 2 k − 1 ). W e get a commutativ e dia gram o f sets [Λ( z ) , A P L ( ∂ T )] I ∗   p ∗ / / [ ˆ Q ⊗ Λ( x, z ) , A P L ( ∂ T )] I ∗   inc ∗ / / [ ˆ Q ⊗ Λ( x ) , A P L ( ∂ T )] I ∗   [Λ( z ) , A P L ( S 2 k − 1 )] p ∗ / / [ ˆ Q ⊗ Λ( x, z ) , A P L ( S 2 k − 1 )] inc ∗ / / [ ˆ Q ⊗ Λ( x ) , A P L ( S 2 k − 1 )] By [23, Chapter I Section 3 Prop ositio n 4 ’] the rows are exa ct in the sense that, if inc ∗ f = inc ∗ g then there exists α ∈ [Λ( z ) , ] such that α · f = g , where · denotes the action o f the group [Λ( z ) , ] on [ ˆ Q ⊗ Λ( x, z ) , ]. Note that o ur coﬁbration sequence is a mo del of a ﬁbration sequence S 2 k − 1 Q = K ( Q , 2 k − 1) → P ν Q → V Q × K ( Q , 2 ) → K ( Q , 2 k ) and at the space level the co action sequence is the ma pping class sequence. (see [27, Chapter 2]). F or any coﬁbratio n s equence A → B → C → Σ A a nd group ob ject G in a pointed mo del categor y the coaction · : [Σ A, G ] × [ C , G ] → [ C, G ] from the coﬁbratio n s e q uence is compatible with the group action φ : [ C , G ] × [ C, G ] → [ C, G ] induced by th e multiplication on G . In particular for an y α ∈ [Σ A, G ] a nd f , g ∈ [ C, G ]: α · φ ( f , g ) = φ ( α · f , g ) . (38) Consider the diagr am ˆ Q β ′   θ ′ | ˆ Q ' ' N N N N N N N N N N N N N N N N N N N N N N N N N N A P L ( T ) σ ∗ / / i ∗   A P L ( V ) ( π ′ ) ∗ / / ( π | ∂ T ) ∗ x x r r r r r r r r r r A P L ( P ν ) ( q | ∂ T ) ∗ t t h h h h h h h h h h h h h h h h h h h A P L ( ∂ T ) The top triangle commutes up to ho motopy by Lemma 6.7 . The b ottom left triangle commutes up to homotopy since σ : V → T is the zero se c tion of π and 51 the b ottom rig ht triangle commutes since it is A P L of Diagra m (2 3). Th us Diagram (3 7) commutes up to homotopy when restr icted to ˆ Q . Since θ ′ ( x ) = a a nd by Lemma 6.1 q ∗ ( a ) = 0, Diagr am (37) restricted to Λ ( x ) commutes and so restricted to ˆ Q ⊗ Λ( x ) commut es up to homoto py . Thus a s homotopy classes inc ∗ µ ′ ( proj ) = inc ∗ q ∗ θ ′ , a nd there e x ists α ∈ [Λ( z ) , A P L ( ∂ T )] such that α · µ ′ pro j = q ∗ θ ′ . Since the action is natural in the se c ond v ar iable I α · I µ ′ pro j = I q ∗ θ ′ . As o bserved a bove I µ ′ ( proj ) = I q ∗ θ ′ , so I α · I µ ′ ( proj ) = I µ ′ ( proj ) and so I α · 0 = I α · (( I µ ′ ( proj ))( I µ ′ ( proj )) − 1 ) = ( I α · I µ ′ ( proj ))( I µ ′ ( proj )) − 1 = ( I µ ′ ( proj ))( I µ ′ ( proj )) − 1 = 0 with the ﬁrs t equality following from E quation (38 ) since A P L ( S 2 k − 1 ) is a group ob ject in CDGA. This implies that p ∗ ( I α ) = 0. Since H 2 k ( ˆ Q ) = 0 b y lo oking at mo dels we get that π 2 k − 1 ( S 2 k − 1 ) ⊗ Q → π 2 k − 1 ( P ν ) ⊗ Q is injective. Thus p ∗ : [Λ( z ) , A P L ( S 2 k − 1 )] → [ ˆ Q ⊗ Λ( x, z ) , A P L ( S 2 k − 1 )] is injective, and so I α = 0. Since Q ⊗ Λ( z ) → Λ( z ) with dz = 0 mo dels A P L ( ∂ T ) → A P L ( S 2 k − 1 ) a nd H 2 k − 1 ( Q ) = 0, I induce s an isomor phism on H 2 k − 1 and so I ∗ : [Λ( z ) , A P L ( ∂ T )] → [Λ( z ) , A P L ( S 2 k − 1 )] is a n isomorphism. Thus α = 0 and so µ ′ pro j = q ∗ θ ′ as ho- motopy classes. Prop ositio n 6.9 L et ν : C k → E → V b e a c omplex ve ctor bun d le of r ank k such that 2 k ≥ dim V + 2 and Q b e a CDGA we akly e quivalent t o A P L ( V ) . Set γ 0 = 1 , and for 1 ≤ i ≤ k − 1 let γ i ∈ Q 2 i ∩ ker d b e c o cycle r epr esentatives of the Chern classes c i ( ξ ) ∈ H 2 i ( V ; Q ) . Using the same notation as at t he b e ginning of Se ction 6 the diagr am ( Q ⊗ Λ z , D z = 0) ( Q ⊗ Λ( x, z ); D x = 0 , D z = P k i =0 γ i x k − i ) φ o o Q S 3 f f M M M M M M M M M M M M '  4 4 j j j j j j j j j j j j j j j j j j j with φ | Q = inc , φ ( x ) = 0 , φ ( z ) = z , | x | = 2 and | z | = 2 k − 1 is a CDGA mo del of the diagr am E 0 q / / π 0 A A A A A A A A P ν π ′ ~ ~ | | | | | | | | V . Pro of. Observe tha t Lemma 3.9 also works in the case that f and f ′ are identit y maps. Thus the prop osition follows from Lemma 6 .8 b y applying Lemma 3.9 t wice. 52 Notice that since 2 k ≥ dim V + 2, γ k = 0 , and so D z = Σ k − 1 i =0 γ i x k − i . Note that in this section we never used the hypo thes is that V is a manifold; only paracompa ctness is needed to deﬁne the classifying map of the tautological bundle λ and we also had some dimension restrictions . 7 The mo del of the blo w-u p F or all this s ection w e ha ve the following a s sumptions and nota tion. Let f : V ֒ → W b e an embedding of c onnected closed manifo lds of co dimension 2 k with a ﬁxed complex structure on its normal bundle ν . Set n = dim( W ) and m = dim( V ) as befo re and r = n − m = 2 k . Assume that n ≥ 2 m + 3 and that H 1 ( f ) is injective. Let φ : R → Q be a CDGA-mo del of the embedding f such that R ≥ n +1 = 0 and Q ≥ m +2 = 0. Supp ose we hav e ﬁxed some shriek map φ ! : s − 2 k Q → R of R - dgmo dules. Re c a ll that all this can b e done using Pr op o sition 4.5. Set γ 0 = 1 and for 1 ≤ i ≤ k − 1 let γ i ∈ Q 2 i ∩ ker d b e re presentativ e s of the Chern classes c i ( ν ) ∈ H 2 i ( V ; Z ). W e will a lso use the notation of Diag ram (39 ) of Section 7.1 below. In this sectio n we pull together what we have done up to now and give o ur mo del of the blow-up. In Section 7.1 we reca ll a deﬁnition o f the blow-up f W and of the map ˜ π : f W → W which are suitable fo r studying homotopy theoretic questions. In 7.2 we deﬁne a CDGA mo del B ( R, Q ) which dep ends on the mo del φ : R → Q of the embedding f : V → W , the shr iek ma p φ ! : s − 2 k Q → R and the representatives γ i of the C he r n classes . In Section 7.3 we prove the equiv alence of t wo diag r ams of CDGAs (Lemma 7.3). The ﬁrst is constructed from our mo dels R and Q , φ , φ ! and the γ i and the seco nd c o mes from taking A P L of Diagram (39 ). In Section 7.4 we ta ke pullbacks derived fro m the equiv a lent dia - grams of Lemma 7.3 a nd show that they are equiv a le nt to B ( R, Q ) (Lemma 7.4) and A P L ( f W )(Lemma 7.5). Putting thes e tog e ther we prov e o ur main theorem (Theorem 7.6) that B ( R , Q ) is a mo del for A P L ( f W ). Under a cer tain nilp otence condition this also implies that the rational ho motopy type of the blow-up a long f : V → W depends only on the rational homotopy class of the map f and the Chern classes of the normal bundle of V in W (Cor ollary 7.7). 7.1 The homotop y t yp e of the blo w-up Consider the following cubica l dia gram with a triangle added to the b otto m face. As in Section 5, T is a tubular neig h b orho o d of V in W and B = W \ T . The pro jective bundle P ν was deﬁned in Section 6.1 a s were the pro jection maps q , π 0 and π ′ . The maps i , k , l and π were ﬁrst seen in Section 4.1. So we hav e come acro ss all the maps in the diagra m except ˜ π . The space f W is deﬁned a s 53 the pushout of the top face. ∂ T k / / q ! ! C C C C C C C C B   @ @ @ @ @ @ @ P ν / / π ′   f W ˜ π   ∂ T  _ i   k / / π 0 ! ! D D D D D D D D B l A A A A A A A A T π / / V f / / W (39) W e know that when we re pla ce f π by j the o utside b ottom qua dr ilateral is a pushout (see Sectio n 4.1). In fac t the b ottom face o f the cub e is als o a pus hout since π is a de fo rmation r etraction. How ever the map b etw een these pushouts induced by π is a homoto py equiv alence and not a homeomorphism. Deﬁnition 7.1 The blo w-up of W a long V is the pushout f W and the map ˜ π : f W → W is t he map induc e d by π ′ b etwe en pushouts c omprising t he top and b ottom fac es of t he ab ove cub e. This deﬁnition of blow-up is equiv a len t to those of [1 1] a nd [21 , 7.1 ]. 7.2 Description of the mo del B ( R, Q ) for t he blow-u p W e now c onstruct a CDGA, B ( R, Q ), which will b e a CDGA mo del o f the blo w- up f W of W along V . W e als o deﬁne a morphism ι ( R , Q ): R → B ( R, Q ) that will mo del the pro jectio n ˜ π : f W → W . Let x and z b e ge ner ators such that | x | = 2 and | z | = 2 k − 1 and denote by Λ + ( x, z ) the aug men tation ideal of the free g raded co mm utative algebra Λ( x, z ). The CDGA B ( R, Q ) is of the form: B ( R, Q ) =  R ⊕ Q ⊗ Λ + ( x, z ) , D  . The graded commutativ e algebr a str ucture on B ( R , Q ) is induced b y the m ultiplications on R and Q ⊗ Λ + ( x, z ), and by the R -mo dule structure on the free Q -mo dule Q ⊗ Λ + ( x, z ) induced by the alg ebra map φ : R → Q . More explicitly for r ∈ R , q ∈ Q and w ∈ Λ + ( x, z ), r · ( q ⊗ w ) = ( φ ( r ) · q ) ⊗ w, ( q ⊗ w ) · r = ( − 1) ( | w | + | q | ) | r | ( φ ( r ) · q ) ⊗ w . 54 Let d R and d Q denote the diﬀerentials o n R and Q r e spec tiv ely , r ∈ R and q ∈ Q . The diﬀerential D on B ( R, Q ) is determined by the Leibnitz law and the formulas D ( r ) = d R r D ( q ⊗ x ) = d Q q ⊗ x D ( q ⊗ z ) = d Q q ⊗ z + ( − 1) | q | φ ! ( s − 2 k q ) + k − 1 X i =0 ( q · γ i ) ⊗ x k − i ! There is a n obvious inclusion morphism ι ( R, Q ) : R → B ( R , Q ) . Notice that B ( R , Q ) actually dep ends not only on R and Q but a lso on the γ i , φ and φ ! . These are implicit and not included in the nota tio n. Lemma 7.2 B ( R, Q ) is a CDGA and ι ( R, Q ) : R → B ( R, Q ) is a CDGA mor- phism. Pro of. F or q , q ′ ∈ Q we hav e ( q ⊗ z ) · ( q ′ ⊗ z ) = 0, ther efore we need to check that the Leibnitz law applied to D (( q ⊗ z ) · ( q ′ ⊗ z )) gives zero. This follows b ecause, for degr ee r e asons, φ ! φ = 0. T o chec k the rest of the deﬁnition o f CDGA is straightforward. 7.3 Tw o equiv alent diagrams Lemma 7.3 Consider the m ap e fr om L emma 5. 5 and the r elative Sul livan algebr a ( Q ⊗ Λ( x, z ) , D ) fr om Equation (35). The CDGA diagr am Q inc / / Q ⊗ Λ( x, z ) e pro j / / Q ⊕ ss − r Q R ⊕ φ ! ss − r Q φ ⊕ id o o is we akly e quivalent to the diagr am A P L ( V ) ( π ′ ) ∗ / / A P L ( P ν ) q ∗ / / A P L ( ∂ T ) A P L ( B ) k ∗ o o Pro of. W e b egin by ﬁxing a common mo del ˆ φ : ˆ R → ˆ Q as in Sec tio n 5.1 . All the notation is given in the le mmas we refer to. Lemma 5 .8 g ives us a commutative diagram of CDGAs 55 Q ⊕ ss − r Q e − 1   R ⊕ φ ! ss − r Q φ ⊕ id o o ξ   Q ⊗ Λ( z ) A κ o o ˆ Q ⊗ Λ( z ) β ⊗ id O O µ ′   ˆ A ζ O O ζ ′   ψ o o A P L ( ∂ T ) A P L ( B ) k ∗ o o (40) with a ll vertical a rrows being weak eq uiv alences. Cho ose representatives ˆ γ i ∈ ˆ Q of c i ( ν ) s uch that β ( ˆ γ i ) = γ i . This ca n b e done since β is an acyc lic ﬁbration. Next Le mma s 6.7 and 6.8 imply tha t we hav e a ho motopy co mm utative diagram of CDGA Q =   inc / / Q ⊗ Λ( x, z ) =   e pro j / / Q ⊕ φφ ! ss − r Q e − 1   Q inc / / Q ⊗ Λ( x, z ) pro j / / Q ⊗ Λ( z ) ˆ Q β O O inc / / σ ∗ β ′   ˆ Q ⊗ Λ( x, z ) β ⊗ id O O pro j / / θ ′   ˆ Q ⊗ Λ( z ) µ ′   β ⊗ id O O A P L ( V ) ( π ′ ) ∗ / / A P L ( P ν ) q ∗ / / A P L ( ∂ T ) (41) with all vertical arrows b eing weak equiv alences. Note that Q ⊕ φφ ! ss − r Q = Q ⊕ ss − r Q since Q ≥ m +2 = 0 en tails φφ ! = 0, so we can g lue Diagram (40 ) to the rig ht of Diag r am (41) and a pply Lemma 3.9 three times to g et the de s ired result. 7.4 Tw o pairs of pullbac ks and t he pro of the the main theorem Lemma 7.4 In CDGA, t he pul lb ack of Q inc   R ⊕ φ ! ss − r Q φ ⊕ id / / Q ⊕ ss − r Q (42) 56 is R and the pul lb ack of Q ⊗ Λ( x, z ) e pro j   R ⊕ φ ! ss − r Q φ ⊕ id / / Q ⊕ ss − r Q (43) is B ( R, Q ) . The map b etwe en the pul lb acks induc e d by the inclusion Q → Q ⊗ Λ( x, z ) is ι ( R, Q ) : R → B ( R, Q ) . F or b oth of the diagr ams ab ove the map fr om the homotopy pul lb ack to the pul lb ack is a we ak e quivalenc e. Thus t he map b etwe en the homotopy pul lb acks induc e d by t he inclusion Q → Q ⊗ Λ ( x, z ) is e quivalent to ι ( R, Q ) : R → B ( R , Q ) . Pro of. Consider the ma ps g : B ( R , Q ) → R ⊕ φ ! ss − 2 k Q determined by the equations g ( r, 0) = ( r, 0) g (0 , q ⊗ z ) = (0 , ( − 1) | q | ss − 2 k q ) g (0 , q ⊗ x i ⊗ z ǫ ) = (0 , 0) if i > 0 and g ′ : B ( R, Q ) → Q ⊗ Λ( x, z ) ( r , q ⊗ x i ⊗ z ǫ ) 7→ φ ( r ) + q ⊗ x i ⊗ z ǫ Since ( φ ⊕ id) g = e (pro j) g ′ , g and g ′ determine a map h : B ( R, Q ) → pullbac k of Diagram (4 3) . Because the forgetful functor from CDGA to graded mo dules co mm utes with taking pullbacks it is easy to check that h is an isomorphism. Similarly R is the pullback of Diagram 42 and ι ( R, Q ) : R → B ( R, Q ) is the induced map b etw een the pullbacks. Next we will show that these pullbacks are indeed homotopy pullbacks. The short exact sequence of diﬀerential gr a ded mo dules 0 → R → Q ⊕ ( R ⊕ φ ! ss − r Q ) → Q ⊕ ss − r Q → 0 gives rise to a May er-Vietor is long exact sequence o n homolog y . This maps into the cor resp onding long ex act sequence fo r the homoto p y pullback. Th us the map fro m R to the ho mo topy pullback o f Diagram 4 2 is an eq uiv alence by the ﬁve lemma. The map fro m B ( R, Q ) into the homotopy pullback of Diagram 43 is s imilarly a weak equiv alence . The fact that the induced map b et ween the homotopy pullbac ks is equiv alent to ι ( R, Q ) : R → B ( R , Q ) follows by naturality . 57 Lemma 7.5 R e c al ling the cub e (39), the homotopy pul lb ack of A P L ( V ) ( π i ) ∗   A P L ( B ) k ∗ / / A P L ( ∂ T ) (44) is qu asi-isomorph ic to A P L ( W ) and the homotopy pul lb ack of A P L ( P ν ) q ∗   A P L ( B ) k ∗ / / A P L ( ∂ T ) (45) is quasi-isomorphic to A P L ( f W ) . The map b et we en the homotopy pul lb acks in- duc e d by π ′ ∗ : A P L ( V ) → A P L ( P ν ) is we akly e quivalent to ˜ π ∗ : A P L ( W ) → A P L ( f W ) . Pro of. Recall (39) the following cube in which the b ottom and the top faces are push-outs : ∂ T / / q ! ! C C C C C C C C B   @ @ @ @ @ @ @ P ν / / π ′   f W ˜ π   ∂ T / / π 0 ! ! D D D D D D D D B A A A A A A A A V / / W Let U b e the homotopy pullback o f (4 4) and f : A P L ( W ) → U b e the induced map. W e hav e maps b etw een Mayer-Vietoris sequences H ∗− 1 ( A P L ( ∂ T )) / / =   H ∗ ( A P L ( W )) H ( f )   / / H ∗ ( A P L ( V )) ⊕ H ∗ ( A P L ( B )) =   / / H ∗ ( A P L ( ∂ T )) =   H ∗− 1 ( A P L ( ∂ T )) / / H ∗ ( U ) / / H ∗ ( A P L ( V )) ⊕ H ∗ ( A P L ( B )) / / H ∗ ( A P L ( ∂ T )) , hence f is a weak equiv ale nc e by the ﬁve lemma. Similarly A P L ( ˜ W ) is weakly equiv alent to the homotopy pullback of (45). The fact that the induced map is weakly e quiv alent to ˜ π ∗ follows by naturality . 58 7.5 Pro of of main theorem Here is the main result of the pap er. Theorem 7. 6 L et f : V ֒ → W b e an emb e dding of c onne cte d close d manifold s of c o dimension 2 k with a ﬁxe d c omplex stru ct ur e on its n ormal bund le ν . Set n = dim( W ) and m = dim( V ) as b efor e and r = n − m = 2 k . Assume that n ≥ 2 m + 3 and that H 1 ( f ) is inje ctive. L et φ : R → Q b e a CDGA-mo del of the emb e dding f such that R ≥ n +1 = 0 and Q ≥ m +2 = 0 . L et φ ! : s − 2 k Q → R b e a shriek map of R -dgmo dules. Set γ 0 = 1 and for 1 ≤ i ≤ k − 1 let γ i ∈ Q 2 i ∩ ker d b e r epr esentatives of the Chern classes c i ( ν ) ∈ H 2 i ( V ; Z ) . The CDGA B ( R, Q ) =  R ⊕ Q ⊗ Λ + ( x, z ) , D  deﬁne d in S e ction 7.2 is a CDGA mo del of A P L ( f W ) wher e f W is the blow-up of W along V . Also ι ( R , Q ): R ֒ → B ( R , Q ) is a CDGA m o del of A P L ( ˜ π ): A P L ( W ) → A P L ( f W ) . Pro of. The theor em follows dir ectly fr o m Lemmas 3.8, 7 .3 , 7.4 a nd 7.5 . Note that if n ≥ 2 m + 3 and H 1 ( f ) is injectiv e, by Pro p os ition 4.5 any mo del of f can b e repla ced by one satisfying the hypo theses o f the theorem. Corollary 7.7 With the hyp otheses of The or em 7.6, if we assume that V , W and the blow-up f W ar e nilp otent sp ac es then the r ational homotopy typ e of f W is determine d by the r ational homotopy class of f and by the r ational Chern classes c i ( ν ) ∈ H 2 i ( V ; Q ) of t he normal bund le. Pro of. This is an immediate application of Sulliv an’s theor y [26] to Theo rem 7.6 . In particular (see [3 , Section 9]) since V and W ar e nilpo tent if f , g : V → W are r ationally ho mo topic then A P L ( f ) a nd A P L ( g ) hav e the same mo dels R → Q . Since the Chern cla s ses are als o the s ame, B ( R , Q ) with mo del A P L ( f W ) along b oth embedding s . Since f W is nilp otent A P L ( f W ) deter mines the r ational homotopy type of f W . W e hav e a homotopy pusho ut P µ   / / f W   V / / W and s o by the V an Kamp en theo rem π 1 ( f W ) → π 1 ( W ) is an is omorphism. So if W is nilp otent then π 1 ( f W ) = π 1 ( W ) is a nilp otent group but the action of π 1 59 on the ho mo topy g roups of the universal cov er of f W may not b e nilp otent. In certain cases Rao [24] ha s determined when a homotopy pushout is nilpo ten t, how ever it seems diﬃcult to see if they apply in our s itua tion. Ho wev e r no tice that the nilp otence co ndition in the co rollary is a utomatically satisﬁed if V is nilpo ten t a nd W is simply connected b ecause then f W is also simply co nnec ted. Hence we hav e prov ed our ﬁrst theo rem fro m the intro duction: Theorem 7. 8 L et f : V → W b e an emb e dding of smo oth close d ori entable manifolds su ch that W is simpl y c onne ct e d and V is nilp otent and oriente d. Supp ose t hat the normal bund le ν is e quipp e d with the stru ctur e of a c omplex ve ctor bun d le and assume that dim W ≥ 2 dim V + 3 . Then the r ational homo- topy typ e of the blow-up of W along V , f W c an b e explicitly determine d fr om the r ational homotopy typ e of f and fr om t he Chern classes c i ( ν ) ∈ H 2 i ( V ; Q ) . Even without the nilp otence co ndition we ca n still determine a mo del for A P L ( f W ). Corollary 7.9 With the hyp otheses in the ﬁrst p ar agr aph of Se ction 7, a CDGA mo del of A P L ( f W ) is determine d by any m o del A P L ( f ) and by the r ational Chern classes c i ( ν ) ∈ H 2 i ( V ; Q ) of t he normal bund le. As we will s ee in the next s ection this is enough to determine H ∗ ( f W ) (Theore m 8.6). 8 Applications In this s ection we a pply our mo del of the blow-up to three situations. Fir st, after s ome prelimina ries in Section 8.1 we describ e in Section 8 .2 the mo del o f the blow-up of C P ( n ) along a subma nifo ld. This is somewha t simpler than our general mo del since here the shriek map is more easily des c rib ed. Secondly , Section 8.3 exa mines the mo del of the example of McDuﬀ of the blo w-up of C P ( n ) along the Ko dair a -Thurston manifold. W e recover directly the Bab enko- T aimanov r esult that this g C P ( n ) ha s a nontrivial Mas sey pr o duct a nd is th us nonformal. Finally in Section 8.4 we describ e the cohomo logy algebra of certa in blow-ups. Some other s p ecia l ca ses have be e n studied by Gitler [10]. In Section 8.5 w e calculate the ra tional homotopy type of blow-ups o f C P (5) alo ng C P (1). It turns o ut that there are inﬁnitely many ratio na lly inequiv alent ones. 8.1 Preliminaries on symplectic manifolds A symple ctic form o n a 2 n -dimensional manifold M is a nondegener ate clo sed diﬀerential 2 -form ω . Thus ω n is a volume for m. The pair ( M , ω ) is ca lle d a symple ctic manifold (see [21]). The form ω is calle d inte gr al if it b elong s to the image H 2 ( M , Z ) → H 2 ( M , R ) ∼ = H 2 (Ω ∗ ( M )) where Ω ∗ ( M ) is the de Rham complex of diﬀerential forms. 60 It is well k nown that to any s ymplectic form ω we can asso ciate an almost complex structure o n the tange n t bundle of M . This almos t complex structure in turn induces a pr eferred orientation on M and hence a generator u M ∈ H 2 n ( M , Z ). Deﬁnition 8.1 Deﬁne the r e al numb er l M by the Equation [ ω n ] = l M · u M ∈ H 2 n ( M ; R ) . Since ω n is a volume form and u M is induced from the almost complex structure asso ciated to ω we know that l M is a p ositive rea l num b er. If ω is integral then l M is a po sitive integer. W e ca nnot a lwa ys cho ose a n integral ω so that l M = 1. F or example if [ ω ] ∈ H 2 ( S 2 × S 2 , Z ) then [ ω ] 2 ∈ H 4 ( S 2 × S 2 , Z ) is alwa ys divisible by 2. The num b er l M will app ear in the formula when we blow-up o f C P ( n ) a long M . An example of a sy mplectic manifold is given b y the complex pro jective space C P ( n ) equipp ed with the 2-for m ω 0 asso ciated to the F ubini-Study metric ([21, Example 4.21 ] or [11, pag e 31]). It is classic a l (se e for ex a mple [11, pages 30-3 2 and 1 44-15 0]) that [ ω 0 ] = − c 1 ( γ 1 ) ∈ H 2 ( C P ( n ); Z ) In particular ω 0 is integral and l C P ( n ) = 1. 8.2 Blo w-ups of CP(n) Let ( M , ω ) b e a symplectic manifold. By a symple ctic emb e dding f : ( M , ω ) → ( C P ( n ) , ω 0 ) (46) we mean a smo o th embedding such that f ∗ ( ω 0 ) = ω . It can b e prov ed that f resp ects the almost complex structures a sso ciated to ω a nd ω 0 , therefo r e the normal bundle o f f admits a natural complex struc tur e and we ca n consider the blow-up g C P ( n ) of C P ( n ) alo ng M . Notice also that, since ω 0 is in tegra l, if a sy mplectic manifold ( M , ω ) symplectica lly embeds in ( C P ( n ) , ω 0 ) then ω is int egra l. Mor eov er the co n verse is almos t true thanks to a theorem of Tischler and Gromov [12], [29]: if M is a manifold equipp ed with a n integral symplectic form ω then, for n la rge enough, there e x ists a symplectic em b edding as in Equation (46). The ﬁr s t application of o ur mo del is to the blow-up of C P ( n ) along a sym- plectically embedded submanifold. T o s implify the description we will use: Lemma 8.2 L et Q b e a CDGA-mo del of some close d symple ctic manifold of di- mension 2 m with symple ct ic form ω ∈ Q 2 ∩ ker d and supp ose that Q ≥ 2 m +2 = 0 . Consider the  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dule str u ctur e on Q induc e d by multiplic a- tion by ω . Then ther e exists a sub-  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dule I ⊂ Q such that Q = I ⊕ Λ( ω ) / ( ω m +1 ) as  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dules. Mor e over if Q 2 m = Q · ω m then this sub-  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dule is un ique. 61 Pro of. T ake a co mplemen tary vector subspace S of Q · ω m ⊕ dQ 2 m − 1 in Q 2 m and set I (2 m ) = S ⊕ Q 2 m +1 . W e know I (2 m ) is a s ub-  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dule of Q since Q ≥ 2 m +2 = 0 . Suppose that for some k ≤ m a nd for eac h j > k we hav e alrea dy deﬁned a diﬀerential submo dule I (2 j ) ⊂ Q such that Q ≥ 2 j = Q { ω j , ω j +1 , . . . , ω m } ⊕ I (2 j ) . W e deﬁne I (2 k ) as follows. Consider the mo rphism λ k : Q 2 k · ω → Q 2 k +2 pr → Q 2 k +2 /I (2 k +2) ∼ = Q · ω k +1 . Set I (2 k ) = ker ( λ k ) ⊕ Q 2 k +1 ⊕ I (2 k +2) . Since α ∈ ker λ k implies that α · ω ∈ I (2 k +2) , it is straig h tforward to c heck that I (2 k ) is a s ub-  Λ( ω ) / ( ω m +1 ) , 0  - dgmo dule o f Q such that Q ≥ 2 j = Q { ω j , ω j +1 , . . . , ω m } ⊕ I (2 j ) . Finally I = I (0) is the desired submo dule of Q . If Q = I ⊕ Λ( ω ) / ( ω m +1 ) a s  Λ( ω ) / ( ω m +1 ) , 0  -dgmo dules then ker λ j m ust be included in I . Since Q ≥ 2 m +2 = 0, we m ust have that I (2 m ) = S ⊕ Q 2 m +2 and then we can show b y induction that I ≥ 2 k = I (2 k ) = ker( λ k ) ⊕ Q 2 k +1 ⊕ I 2 k +2 for all k < m . This implies that I is c o mpletely deter mined by the choice of S . Therefore when Q 2 m = Q · ω m the ideal I is unique. Let ( M , ω ) b e a symplectic manifold o f dimension 2 m , f : M → C P ( n ) b e a symplectic em b edding and ν be the normal bundle of the embedding . Let ( Q, d ) b e a CDGA-mo del of M suc h that Q ≥ 2 m +2 = 0. Let ω ∈ Q 2 ∩ ker d b e a representative of the sy mplectic form and γ i ∈ Q 2 i ∩ ker d b e representatives of the Chern c la sses c i ( ν ) ∈ H 2 i ( M ) of the normal bundle of M in C P ( n ). Then R = (Λ( a ) / ( a n +1 ) , 0 ) is a CDGA mo del of C P ( n ) with [ a ] = [ ω 0 ] ∈ H 2 ( C P ( n )) and a n represents the orientation class o f C P ( n ). A mo del of the embedding f : M ⊂ C P ( n ) is given by the map φ : R → Q deﬁned by φ ( a j ) = ω j which is indeed a CDGA morphism since Q ≥ 2 m +2 = 0. This induces an R -dgmo dule structure on Q . Let I be a diﬀerential submo dule o f Q complementary to the image of φ which e x ists by Lemma 8 .2 . Let 2 k = 2 n − 2 m , the co dimension of M ins ide C P ( n ). Lemma 8.3 The map φ ! : s − 2 k Q → R deﬁne d by φ ! ( ω j ) = l M a j + k and φ ! ( I ) = 0 is a shriek map, wher e l M is given by ω m = l M u M for the orientation class u M determine d by the almost c omplex str u ctur e. (Se e D eﬁnition 8.1.) Pro of. This follows directly from the deﬁnition of a shriek map (Deﬁnition 4.2 ). F rom Section 7.2 we get B (Λ( a ) / ( a n +1 ) , Q ) =  Λ( a ) ( a n +1 ) ⊕ Q ⊗ Λ + ( x, z ) , D  (47) 62 a CDGA with | x | = 2, | z | = 2 k − 1. T he alg ebra structur e extends the algebra structure o n Q and the Λ( a ) / ( a n +1 ) − mo dule structure and the diﬀerential is determined by the Leibnitz law and the following equatio ns D ( a ) = 0 D ( q ⊗ x j ) = dq ⊗ x j D ( q ⊗ z ) = dq ⊗ z + ( − 1) | q | φ ! ( s − 2 k q ) + k − 1 X i =0 q γ i ⊗ x k − i ! Notice that taking the pro duct of a with an element of Q gives multiplication by ω . Note that for the nex t theor em our standa rd dimension hypothesis w ould be 2 n ≥ 4 m + 3 or n ≥ 2 n + 3 / 2 but since n is an integer this is equiv alent to n ≥ 2 m + 2. Theorem 8. 4 L et ( M , ω ) b e a symple ctic manifold of dimension 2 m and sup- p ose we have b e en given a symple ctic emb e dding f : ( M , ω ) → ( C P ( n ) , ω 0 ) . L et Q b e a CDGA mo del of M su ch that Q is c onne ct e d and Q ≥ 2 m +1 = 0 . L et ω ∈ Q 2 ∩ ker d b e a r epr esentative of the symple ctic form. If n ≥ 2 m + 2 t hen the CDGA B (Λ( a ) / ( a n +1 ) , Q ) of Equation (47) is a mo del of the blow-up g C P ( n ) . Pro of. Since R = Λ( a ) / ( a n +1 ) and Q ≥ 2 n +1 = 0, φ deﬁned ab ove is the only homo- topy class of a CDGA-morphism such that φ ( a ) = ω . Therefore it is a mo del of the embedding f . The mor phism φ ! is clea rly a shriek map. Ther e fo re a ll the hypotheses of Theor em 7.6 are fulﬁlled a nd the theorem follows. 8.3 McDuﬀ ’s example Next we lo ok at the mo del of the example of McDuﬀ ([20] or [21, Exerc is e 6.55 ]) that we now review. W e start with the Ko dair a-Thurston manifold ([28] or [30, Example I I.2.1] ) which is a closed symplectic nilmanifold V o f dimension 4 and is deﬁned as the pro duct of the circle S 1 with the o rbit space R 3 / Γ where Γ is the unifor m lattice of the in tegra l upp er triangula r 3 × 3 − ma trices in the Heisenberg group. A CDGA-mo del o f that manifold is given by the following exterior algebra on four gener ators of degree 1 (see [30, example I I.1.7 (2)]): ( Q, d ) = (Λ( u 1 , y 1 , v 1 , t 1 ) , du = dy = dt = 0 , dv = uy ) . The symplec tic form is represented in this mo del by ω = u v + y t . By the symplectic em b edding theorem of Tischler and Gromov ([29], [12, 3.4.2 ]) for n ≥ 5 there exists a s y mplectic e m b edding f : V ֒ → C P ( n ) such that f ∗ ( a ) = [ ω ]. T o fulﬁll the h y po theses of Theor em 8.4 we will supp ose that n ≥ 6 . Then m = 2 and k = n − 2 ≥ 4. In [20, p 271] w e s ee that the Chern classes 63 of V a re trivial and ([22, Theo rem 14.10]) the Chern classes of C P ( n ) are c ( C P ( n )) = P n i =0 c i ( C P ( n )) = (1 + a ) n +1 where a ∈ H 2 ( C P ( n ) , Z ) r e pr esents the for m ω 0 describ ed in Section 8.1. Therefore the Chern classes o f the no rmal bundle ν of the embedding ar e given by the Equation c ( ν ) · c ( V ) = f ∗ ( c ( C P ( n ))) which yields c ( ν ) = 1 + ( n + 1) ω + n ( n + 1) 2 ω 2 . The morphism φ ! : s 4 − 2 n Q → Λ( a ) / ( a n +1 ) is characterized by Lemma 8 .3 and satisiﬁes, φ ! ( s 4 − 2 n 1) = 2 a n − 2 , φ ! ( s 4 − 2 n uv ) = φ ! ( s 4 − 2 n y t ) = a n − 1 , φ ! ( s 4 − 2 n uv y t ) = a n , and φ ! ( s 4 − 2 n ζ ) = 0 for any other monomia l ζ in Q = Λ( u, y , v , t ). Using these data in the deﬁnition of the CDGA B = B (Λ ( a ) / ( a n +1 ) , Q ) =  Λ( a ) / ( a n +1 ) ⊕ Λ( u, y , v , t ⊗ Λ + ( x, z ) , D  ab ov e gives a mo del of the McDuﬀ example. F rom this CDGA-mo del of the McDuﬀ example we r ecov er the Ba b enko- T aimanov [1] result. Theorem 8. 5 Ther e exist non-trivial Massey pr o ducts in g C P ( n ) . Pro of. In B we hav e D ( v ⊗ x 2 ) = ( u ⊗ x ) · ( y ⊗ x ) a nd ( y ⊗ x ) · ( y ⊗ x ) = 0, so D (0) = ( y ⊗ x ) · ( y ⊗ x ). Thus v y ⊗ x 3 is in the triple Massey pro duct h [ u ⊗ x ] , [ y ⊗ x ] , [ y ⊗ x ] i . Also [ vy ⊗ x 3 ] is not in the ideal of H ∗ ( g C P ( n )) generated b y [ u ⊗ x ] and [ y ⊗ x ] s o 0 6∈ h [ u ⊗ x ] , [ y ⊗ x ] , [ y ⊗ x ] i . In this example the Massey pro duct h u, y , y i in V became h [ u ⊗ x ] , [ y ⊗ x ] , [ y ⊗ x ] i in g C P ( n ). This is the gener al wa y in which Massey pr o ducts pro pa gate to the blow-up. More g enerally we show in [1 9 ] that any o bstruction to forma lit y in any manifold propaga tes to an obstructio n in its blow-up in C P ( n ). 8.4 The cohomology algebra of a blo w-up In the nex t theore m f ! is the classical co homological shriek map (see Se c tio n 4.2). Also the algebra str ucture o n H ∗ ( W ) ⊕ H ∗ ( V ) ⊗ Λ + ( x ) is determined by the formula on basic tensors ( w, v ⊗ x i ) · ( w ′ , v ′ ⊗ x j ) = ( ww ′ , w v ′ ⊗ x j + ( − 1) | w ′ || v | w ′ v ⊗ x i + v v ′ ⊗ x i + j ) where w, w ′ ∈ H ∗ ( W ) a nd v , v ′ ∈ H ∗ ( V ). Theorem 8. 6 L et f : V → W b e an emb e dding of close d oriente d manifolds of c o dimension 2 k with a ﬁxe d c omplex stru ctur e on the normal bund le ν . Assu me that dim W ≥ 2 dim V + 3 . L et c i ( ν ) ∈ H 2 i ( V ) b e the Chern classes. L et I b e the ide al in H ∗ ( W ) ⊕ H ∗ ( V ) ⊗ Λ + ( x ) gener ate d by the set { f ! ( v ) + Σ k − 1 i =0 v · c i ( ν ) ⊗ x k − i : v ∈ H ∗ ( V ) } . 64 Then we have an isomorphism of algebr as H ∗ ( f W ) ∼ = ( H ∗ ( W ) ⊕ H ∗ ( V ) ⊗ Λ + ( x )) /I . Pro of. Recall the mo del of A P L ( f W ) o btained in Theorem 7.6: B = B ( R, Q ) =  R ⊕ Q ⊗ Λ + ( x, z ) , D  W e deﬁne an increa sing ﬁltration on tha t CDGA by F 0 B = R ⊕ Q ⊗ Λ + ( x ) F p B = B for p ≥ 1 This ﬁltr a tion is compatible with the CDGA structure. The E 1 − term of the asso ciated s pectr al s equence is E 0 , ∗ 1 = H ( R ) ⊕ H ( Q ) ⊗ Λ + ( x ) E 1 , ∗ 1 = H ( Q ) ⊗ Λ( x ) · z E p, ∗ 1 = 0 for p 6 = 0 , 1 and the d 1 − diﬀerential is non trivial only on E 1 , ∗ 1 , b eing there d 1 ([ q ] ⊗ x r ⊗ z ) = ( − 1) | q | f ! ([ q ]) + k − 1 X i =0 [ q ] · c i ( ν ) ⊗ x r + k − i ! hence d 1 ([ q ] ⊗ x r ⊗ z ) = ± [ q ] ⊗ x r + k + (terms with lower p ow ers o f x ) . Thu s d 1 is injective and therefore E 2 = H ( E 1 , d 1 ) = H ( R ) ⊕ H ( Q ) ⊗ (Λ + ( x )) im d 1 with im d 1 = { f ! ( v ) + k − 1 X i =0 v · c i ( ν ) ⊗ x k − i : v ∈ H ∗ ( V ) } ! The E 2 − term is concentrated in column p = 0 , so the s pectr al s equence col- lapses at E 2 where E 2 = E 0 , ∗ as alge bras. Therefore there is no p ossibility for extensions and H ∗ ( f W ) ∼ = E ∞ = E 2 as algebra s . Theorem 8.6 determines the algebra structure of the c ohomology of the blow-up under the “stable” condition: dim( W ) ≥ 2 dim( V ) + 3. This result is complementary to a theorem of Gitler [1 0, Theorem 3.11] that determines the cohomolog y algebr a H ∗ ( f W ) under the hypo thesis that f ∗ : H ∗ ( W ) → H ∗ ( V ) is surjective. 65 8.5 A Simple Example As no ted in Section 8.1 the F ubini-Study metric ω 0 ∈ H 2 ( C P ( n )) is an integral symplectic for m. Th us for any l ∈ Z with l > 0, l ω 0 ∈ H 2 ( C P ( n )) is also one. So by [29] ther e exists a symplectic embedding f l : C P (1 ) → C P (5) such that f ∗ l ( ω 0 ) = l ω ′ 0 , wher e ω 0 denotes the F ubini-Sudy metric in H 2 ( C P (5)) and ω ′ 0 denotes it in H 2 ( C P (1)). Identifying a with ω 0 ∈ H 2 ( C P (5)) and a ′ with ω 0 ∈ H 2 ( C P (1)), we g et iso mo rphisms H ∗ ( C P (5)) ∼ = Λ( a ) / ( a 6 ) and H ∗ ( C P (1)) ∼ = Λ( a ′ ) / (( a ′ ) 2 ). Also f ∗ l ( a ) = l a ′ and f ! l : s − 8 Λ( a ′ ) / (( a ′ ) 2 ) → Λ( a ) / ( a 6 ) is given b y the formulas φ ! ( s − 8 1) = la 4 φ ! ( s − 8 a ′ ) = a 5 . Next we ca lculate c 1 ( ν ), the ﬁrst Chern class o f the norma l bundle. Using the following three formulas f ∗ ( c ( C P (5)) = 1 + 6 f ∗ ( a ) = 1 + 6 l a ′ c ( C P (1 )) = 1 + 2 a ′ and c ( ν ) c ( C P (1)) = f ∗ ( c ( C P (5))) ∈ H ∗ ( C P (1)) , a s imple calculation gives us that c ( ν ) = 1 + (6 l − 2) a ′ and hence c 1 ( ν ) = (6 l − 2) a ′ . Theorem 8.6 tells us that the cohomolog y of the blow-up g C P l (5) of C P (5) along f l is H ∗ ( g C P l (5)) ∼ = (Λ( a ) / ( a 6 ) ⊕ (Λ( a ′ ) / (( a ′ ) 2 ) ⊗ Λ + ( x ))) /  l a 4 + (6 l − 2) a ′ x 3 + x 4  (48) Since f l ( a ) = l a ′ , a · (1 ⊗ x ) = l a ′ ⊗ x so we can write ax = l a ′ x or 1 l ax = a ′ x . Thu s a 2 x = l 2 ( a ′ ) 2 x = 0 , and also l a 4 + (6 l − 2) a ′ x 3 + x 4 = 1 l ( l 2 a 4 + (6 l − 2) ax 3 + l x 4 ). So it is stra ig ht forward to check that we a lso get an iso morphism H ∗ ( g C P l (5)) ∼ = Λ( a, x ) /  a 6 , a 2 x, l 2 a 4 + (6 l − 2 ) ax 3 + l x 4  (49) with the isomorphis m in (49 ) r e alized by mapping a to a and x to x on the right hand side of (48). 66 Prop ositio n 8.7 If g C P l (5) and g C P r (5) ar e r ational ly homotopy e quivalent then l = r . Pro of. W e hav e just seen tha t H ∗ ( g C P l (5)) ∼ = Λ( a, x ) /  a 6 , a 2 x, l 2 a 4 + (6 l − 2 ) ax 3 + l x 4  and H ∗ ( g C P r (5)) ∼ = Λ( b, y ) /  b 6 , b 2 y , r 2 b 4 + (6 r − 2) by 3 + r y 4  Assume that g C P l (5) and g C P r (5) ar e rationally homoto p y equiv alent. So there is an isomorphism g : H ∗ ( g C P l (5)) → H ∗ ( g C P r (5)) . W rite g ( a ) = α 1 b + β 1 y and g ( x ) = α 2 b + β 2 y . W e will ﬁrst show that β 1 = α 2 = 0. W e know that a 2 x = 0 in H ∗ ( g C P l (5)), so g ( a 2 x ) = 0 . This implies that α 2 1 α 2 = 0, β 2 1 β 2 = 0 and 2 α 1 β 1 β 2 + α 2 β 2 1 = 0. Since g is an isomorphism it is stra ightforw a rd to chec k that the only p ossibility is β 1 = 0 = α 2 . W e know that l 2 a 4 + (6 l − 2) ax 3 + lx 4 = 0 ∈ H ∗ ( g C P l (5)). So lo ok ing at the co eﬃcients o f b 4 , by 3 and y 4 in g ( l 2 a 4 + (6 l − 2 ) ax 3 + l x 4 ) = 0 we get that for some ﬁxed δ ∈ Q l 2 α 4 1 = δ r 2 so δ = l 2 r 2 α 4 1 (50) (6 l − 2) α 1 β 3 2 = δ (6 r − 2) (51) and l β 4 2 = δ r. (52) Observe that none of l , r , δ , β 2 and α 1 can b e 0 so we can divide by them at will. Equations (50) and (52) imply that l r =  β 2 α 1  4 (53) Subbing (50) into (51 ), ta king four th p ow ers a nd gather ing α 1 to one side we get (6 l − 2) 4  β 2 α 1  12 = l 8 r 8 (6 r − 2) 4 67 so using (53) we get (6 l − 2) 4 l 5 = (6 r − 2) 4 r 5 (54) Let h ( x ) = (6 x − 2) 4 x 5 . Calc ulus tells us that h ( x ) is decrea sing x ≥ 2, also h (1) 6 = h (2), h (1) 6 = h (3 ) and h (1) > h (4). T ogether these imply that h ( x ) takes distinct v alues on each p ositive integer. Thus we must have r = l as requir ed. Since all of the g C P l (5) are not rationally homotop y eq uiv alent they are not integrally homo topy eq uiv alent a nd hence not diﬀeomorphic. Using the H ∗ ( C P (5); Z ) module structure on H ∗ ( g C P l (5); Z ) gives another w ay to see that all of the g C P l (5) are not integrally homotopy equiv a lent. If we lo o k at blow-ups o f C P (1) in C P (4) we would have diﬃculty proving the las t theorem bec ause tw o of the relations in the coho mology a lg ebra w ould b e in the s a me degree. This leads to some interesting alg ebraic equatio ns which s e em diﬃcult to so lve. Howev er it can s till b e shown using the mo dule structure that the blow-ups have diﬀerent int egra l homo top y type. This lea ds us to the following question w hich seems more likely to hav e a p ositive answer if the co dimension is large . Question: Let ( M , ω ) b e an integral sy mplec tic manifold, l a p ositive integer and f l : M → C P ( n ) b e an embedding such that f ∗ l ( ω 0 ) = l ω . Let g C P l ( n ) b e the blowup along f l . Are all of the g C P l ( n ) rationally inequiv alent? References [1] I.K. Bab enko and U.A. T aimanov , On the existenc e of nonformal simply c onne cte d symple ctic manifold , Russ ia n Math. Sur veys 53 :4 (1998 ), 1082- 1083. [2] R. Bott and L.W. T u , Diﬀer ential forms in algebr aic top olo gy , Gr aduate T ext in Mathema tics, vol. 8 2, Spr ing er-V er lag, 1982 . [3] A.K. Bous ﬁeld and V.K.A.M. Gugenhei m , On PL de Rham the ory and r ational homotopy typ e , Me moirs of the A.M.S. 179 , (1 976). [4] G.E. Bredon , T op olo gy and ge ometry , Graduate T e x ts in Ma thematics, vol. 139, Springer-V erlag, 1 995. [5] P . Del igne, P . Griﬃ ths, J. Morgan and D. Sulliv an , R e al homotopy the ory of K¨ ahler manifolds , Inv entiones Math. 29 (19 7 5), 245 –274. [6] W.G. Dwy e r and J. Spalinski , Homotopy the ories and mo del c ate gories , in Handb o ok of algebra ic to po logy , edited by I.M. Ja mes, Elsevie r Science B.V., 1995 , pp. 73-1 26. 68 [7] Y. F ´ elix, S. Halp erin and J.-C. Thom as , Gor enstein sp ac es , Adv a nces in Math 71 (1988 ), 92–11 2. [8] Y. F´ eli x, S. Halp erin and J.-C. Thomas , Diﬀer ent ial gr ade d alge- br as in t op olo gy , in Handb o ok of algebr aic top olo gy , edited by I.M. James, Elsevier Science B.V., 199 5, pp. 82 9-865. [9] Y. F´ eli x, S. Halp erin and J.-C. Thomas , R ational homotopy the ory , Graduate T exts in Mathema tics , vol. 205, Springer-V erlag, 200 1. [10] S. Gi tler , The c ohomolo gy of blow-ups , Bol. So c. Mat. Mexicana 37 (1 982), 167-1 75. [11] P . Griﬃths and J. Harris , Principles of algebr aic ge ometry , Pur e and Applied Mathematics, John Wiley & Sons, 19 7 8. [12] M. Gromov , Partial diﬀer ential re lations , Springer, Berlin–Heidelbe r g, 1986. [13] S. 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Swi tzer Al gebr aic top olo gy—homotopy and homolo gy , Die Grundlehren der mathematischen Wissenschaften, Band 2 12. Spring er- V erlag , New Y ork-Heidelb erg, 1 975. [28] W. P . Th urston , Some simple examples of c omp act symple ctic manifolds , Pro c. Amer. Math. So c 55 (1976 ), pp. 467- 4 68. [29] D. Tisc hler , Close d 2 − forms and an emb e dding the or em for symple ctic manifolds , J. Diﬀ. Geom. 12 (19 7 7), 229 –235. [30] A. T ralle and J. Oprea , S ymple ctic manifolds with no K¨ ahler struct ur e , Lecture Notes in Math., vol. 166 1 , Springer- V erlag , 1 9 97. Addresses : Pascal Lambrec hts Univ ersit´ e d’Artois Univ ersit´ e de Louv a in Institut Math´ ema tique Chemin du Cyclotron, 2 B-1348 L o uv ain-la-Neuve, BELGIUM e-mail : lambrec hts@ma th.ucl.ac.be Donald Sta nley Univ ersity of Regina e-mail : stanley @math. uregina.ca 70

The rational homotopy type of a blow-up in the stable case

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