Landweber exact formal group laws and smooth cohomology theories

A lgebraic & G eometric T opology 9 (2009) 1751–1 790 1751 Land weber exa ct f ormal gr oup laws and smoot h cohomology theories U LRICH B UNKE T HOMA S S CHICK I NGO S CHR ¨ ODER M ORITZ W IETHA UP The main aim of this paper is the construction of a smooth (sometimes called differential) extension d MU of the c ohomo logy theory com plex cobo rdism MU , using cycles for d MU ( M ) which are essentially pro per maps W → M with a ﬁxed U -structur e and U -co nnection on the (stable) normal b undle of W → M . Crucial is that this model allows the construction of a p roduct structur e and of pushdown map s for this smooth extension of MU , which have all the expected proper ties. Moreover , we show , using the L andweber exact functor principle, that ˆ R ( M ) : = d MU ( M ) ⊗ MU ∗ R deﬁnes a multiplicative smoo th e xtension of R ( M ) : = MU ( M ) ⊗ MU ∗ R whenever R is a Landweber exact MU ∗ -modu le. An example for this construc- tion is a new w ay to deﬁne a multiplicative smooth K-theory . 1 Introd uction Smooth (also called diff erentiabl e) ext ensions of gener alized cohomology theories recent ly became an intensi ve ly studied mathematical topic with many appl ications rangin g from arithmetic geometry to string theory . Found ational contrib utio ns are [ CS85 ], [ B ry93 ] (in the case o f ordinary c ohomology ) and [ HS05 ]. The latter paper gi ves amon g many other resul ts a gener al constr uction of smooth ex tensions in homo- top y theore tic terms. Fo r cohomology theories based on geometr ic or analytic cycles there are often alternati ve m odels. This appli es in particular to ordina ry cohomology whose smooth extensio n has v arious dif ferent realiza tions ([ CS85 ], [ Gaj97 ], [ Bry93 ], [ DL05 ], [ HS05 ], [ BKS ]). T he papers [ S S ] or [ BS09 ] show that all these realizat ions are isomorph ic. An example of a cycle model of a smooth extensio n of a generalized cohomology theory is the m odel of smooth K -theory intr oduced in [ BS07 ], se e also [ Fre00 ], [ FH00 ]. Published: 26 September 200 9 DOI: 10.2 140/agt. 2009.9.1 751 1752 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up The present paper co ntrib utes geometric models of smooth extension s of cobord ism theori es, where t he ca se of complex cobordism the ory MU is of part icular importa nce. In [ BS09 ] we obtain general results about uniquen ess of smooth ext ensions which in particul ar appl y to smooth K -th eory and s mooth complex cobordis m theory ˆ MU . In detail, any two smooth exte nsions of comple x cobordi sm theory or complex K - theory which admit an int egratio n along R : S 1 × M → M are isomorphic by a unique isomorph ism compati ble with R . In case of mul tiplicati ve e xtensio ns the isomorphis m is au tomatically multiplic ativ e. Note that the extensio n ˆ MU const ructed in t he prese nt paper has an inte gration and is multipl icati ve. W e e xpect that o ur model ˆ MU of th e smooth e xtensio n of MU is uniqu ely isomorp hic to the one gi ven by [ HS05 ]. So far th is fact can not immediatel y be deduced from the abov e uniqueness result since for the latter model the functor ial prop erties of the inte gration map ha ve not been de velo ped yet in suf ﬁcient detail. Ho wev er , for the uniqu eness of the ev en part we do not need the integratio n. Therefore in ev en de grees our exte nsion ˆ MU is uniquely isomorphic to the model in [ HS05 ]. An advant age of geometric or analytic models is that th ey allo w th e introd uction of additi onal st ructures like products, smooth orientat ions an d integra tion ma ps with good proper ties. The se addi tional propertie s are fundame ntal for applic ations. In [ HS05 , 4.10] methods for integ rating smooth cohomology classes w ere discussed , b ut furthe r work will be required in order to turn these ideas into construc tions w ith good functo rial proper ties. In the case of smooth ordinary co homology the p roduct a nd the integratio n ha ve been co nsidered in vario us places (see e.g. [ CS85 ], [ DL05 ], [ Bry93 ]) (he re smooth orient ations are ordinary orientati ons). The case of smooth K -the ory , discuss ed in detail in [ BS07 ], sho ws that in partic ular the theory of orienta tions an d integ ration is consid erably more complicate d for general ized cohomology theories. In th e present paper we construct a multi plicati ve exten sion of the compl ex cobo rdism cohomol ogy theory MU . Furthermor e, we introdu ce the notion of a smooth MU - orient ation and de v elop the co rrespond ing theory o f inte gratio n. The sa me idea s could be appl ied with minor modiﬁcation s to other cobordism theories . For an MU ∗ -module R one can try to deﬁne a new cohomolo gy theory R ∗ ( X ) : = MU ∗ ( X ) ⊗ MU ∗ R for ﬁnite CW -complex es X . By Land weber’ s famous re sult [ Lan76 ] this constructio n wo rks and gi ves a multiplica tiv e complex oriented cohomology theory pro vided R is a ring ov er MU ∗ which is in add ition Landweber e xact . In T heorem 2.5 we ob serve that by the same idea one can de ﬁne a multipli cati ve smooth extens ion A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1753 ˆ R ( X ) : = ˆ MU ( X ) ⊗ MU ∗ R of R . It immediately follo ws that this smooth ext ension admits an inte gration for smooth ly MU -oriented proper submersio ns. In this way we co nsiderabl y enlarg e the class of exampl es of gene ralized cohomology theori es which admit mult iplicati ve exten sions and integrat ion maps. The construct ion can e.g. be app lied to Landweber exact e lliptic co homology th eories [ LRS95 ], [ F ra92 ] and comple x K -theory 1 . In Section 2 we re view the mai n result of Landweber [ Lan76 ] an d the deﬁnition of a smooth ext ension of a generalize d cohomology theory . W e state the main result asserti ng the exis tence of a multiplic ativ e smooth extens ion of MU w ith orientation s and integ ration. Then w e realize the idea sketche d abo ve and construct a m ultipli cati ve smooth ex tension for eve ry Landweber exact formal group law . In Section 3 w e rev iew the stan dard const ructions of cobordi sm theorie s using homo- top y theory on the one hand, and cycles on the other . Furthermore, we re vie w the notion of a genu s. In Section 4 we construct o ur mod el of th e mu ltiplicati ve smoot h e xtension of co mplex cobord ism. Furth ermore, we introdu ce the notion of a smooth MU -or ientation and constr uct the integ ration map. Thomas Schick was partially funded by the Courant R esearch Center “High er order structures in Mathematics” with in the Ge rman initiave of excellence. In go Schr ¨ oder an d Mo ritz Wiethaup were partially funded by DFG GK 535 “Groups and Geometry” . 2 The Land weber construction and s mooth extensions 2.1 The Landwe ber construction 2.1.1 Let X 7→ MU ∗ ( X ) denote the multipli cati ve cohomo logy theory (de ﬁned on the catego ry of CW -comple xes) called complex cobordism. W e ﬁx an is omorphism MU ∗ ( CP ∞ ) ∼ = MU ∗ [[ x ]] . The K ¨ unneth formula then giv es MU ∗ ( CP ∞ × CP ∞ ) ∼ = MU ∗ [[ x , y ]] . The tensor product of line bu ndles induces an H -space structure µ : CP ∞ × CP ∞ → CP ∞ . Under the abov e id entiﬁcation s the map µ ∗ : MU ∗ [[ z ]] → MU ∗ [[ x , y ]] is determin ed by the element f ( x , y ) : = µ ∗ ( z ) ∈ MU ∗ [[ x , y ]] . 1 It is an interesting p roblem to u nderstand explicitly the relation with the mode l [ BS07 ]. Note that we abstractly know that t he smooth extensions are isomorphic by [ BS09 ]. A lgebraic & G eo metric T opology 9 (2009) 1754 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up By a result of Quillen [ Qui69 ] the pair ( MU ∗ , f ) is a u niv ersal formal group law . This means that, giv en a commutati ve ring R and a formal group law g ∈ R [[ x , y ]] , there exi sts a unique ring homomor phism θ : MU ∗ → R such that θ ∗ ( f ) = g . 2.1.2 Let R be a commutati ve ring ov er MU ∗ . Then one can ask if the functor X 7→ MU ∗ ( X ) ⊗ MU ∗ R is a cohomolog y theory on the categ ory of ﬁnite CW -comple xes. The result of Landwebe r [ Lan76 ] determines necessary and suf ﬁcient conditions. A ring which satisﬁes these condi tions is called Landweber exa ct. 2.1.3 Actually , Landweber sho ws a stronger resu lt which is crucial for the presen t paper . For any space or spectr um X the homolog y MU ∗ ( X ) h as the structu re of a comodul e over the coalgeb ra M U ∗ MU i n MU ∗ -modules . By d uality , if X is ﬁnite, then M U ∗ ( X ) ∼ = MU ∗ ( S ( X )) also has a comodul e structure , where S ( X ) denotes the Alexa nder -Spanier dual (see [ Ada74 ]) of X . Theor em 2.1 (Landweber [ Lan76 ] ) Let M be a ﬁnitely presente d MU ∗ -module which h as the st ructure of a comodule ov er MU ∗ MU , an d consi der a Landweb er exact formal group law ( R , g ) so that in particula r R is a ring over MU ∗ . Then for all i ≥ 1 we ha ve Tor MU ∗ i ( M , R ) = 0 . 2.2 Smooth cohomology theories 2.2.1 In the present subsect ion B denote s a compact smooth manifold. Let N be a Z -grade d vect or space ove r R . W e conside r a generaliz ed cohomology theor y h with a natural transfo rmation of cohomology theori es c : h ( B ) → H ( B , N ) , where H ( B , N ) is ordinary coho mology with c oef ﬁcients in N . The n atural u niv ersal e xample is gi ven by N : = h ∗ ⊗ R , where c is the canoni cal transfor mation. Let Ω ( B , N ) : = Ω ( B ) ⊗ R N , where Ω ( B ) d enotes the smooth real dif ferentia l forms on B . Note that this deﬁnition only coinc ides with the corres ponding deﬁnition of N -v alued forms in [ BS09 ] if N is degree-wise ﬁnite-dimensi onal. By dR : Ω d = 0 ( B , N ) → H ( B , N ) w e denote the de Rham map which as sociates to a closed form the corres ponding cohomology clas s. T o a pair ( h , c ) we assoc iate the notion of a smooth ext ension ˆ h . Note that manifolds in the presen t paper may ha ve bounda ries. Deﬁnition 2.2 A smooth ext ension of the pair ( h , c ) is a functor B → ˆ h ( B ) from the category of compact smooth manifolds to Z -grade d g roups together with natural transfo rmations (1) R : ˆ h ( B ) → Ω d = 0 ( B , N ) (curv ature) A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1755 (2) I : ˆ h ( B ) → h ( B ) (for get smooth data) (3) a : Ω ( B , N ) / im ( d ) → ˆ h ( B ) (action of forms) . These transfo rmations are required to satisfy the follo wing axioms: (1) The follo wing diagram commutes ˆ h ( B ) R   I / / h ( B ) c   Ω d = 0 ( B , N ) dR / / H ( B , N ) . (2) (2–1) R ◦ a = d . (3) a is of deg ree 1 . (4) The sequence (2–2) h ( B ) c → Ω ( B , N ) / im ( d ) a → ˆ h ( B ) I → h ( B ) → 0 . is exact . 2.2.2 If h is a multiplic ativ e cohomolog y theory , th en one can consid er a Z -graded ring R over R and a multiplicati ve transformati on c : h ( B ) → H ( B , R ) . In th is case we also talk about a multipli cati ve smooth exten sion ˆ h of ( h , c ) . Deﬁnition 2.3 A smooth extens ion ˆ h of ( h , c ) is called multiplicat iv e, if ˆ h togeth er with the transf ormations R , I , a is a smooth exte nsion of ( h , c ) , and in addition (1) ˆ h is a funct or to Z -grade d rings, (2) R and I are multiplic ativ e, (3) a ( ω ) ∪ x = a ( ω ∧ R ( x )) for x ∈ ˆ h ( B ) and ω ∈ Ω ( B , R ) / im ( d ) . 2.2.3 The ﬁrst goal o f t he present pa per is the co nstructio n of a multiplicat iv e smooth ext ension of the pair ( MU , c ) , where c : MU ∗ ( B ) → MU ∗ ( B ) ⊗ Z R ∼ = H ∗ ( B , MU R ) is the canonica l natural transformati on (see 3.4.7 ). The follo wing theorem is a special case of Theorem 4.21 which giv es a cons truction of multiplicati ve smooth extensio ns of more genera l pairs ( MU , h ) . Theor em 2.4 The pair ( MU , c ) admits a multipli cati ve smooth exten sion. The existen ce of a smooth exte nsion also follows from [ HS05 ], but there, no ring structu re is constru cted. A lgebraic & G eo metric T opology 9 (2009) 1756 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 2.2.4 In the present paper w e consider smooth extens ions of generalize d coho mology theori es deﬁned on the c ategor y of compact manifold s. The reason lies in the f act that we w ant to ap ply the Landwebe r exact fu nctor theorem. If R is a ge neralized comple x orient ed co homology theory s atisfying the wedge axiom to w hich the L andweber exact functo r theorem applies, then for ﬁnite C W -compl exe s X R ∗ ( X ) ∼ = MU ∗ ( X ) ⊗ MU ∗ R . In gener al this equality does not exte nd to inﬁnite CW - complex es since the tensor produ ct on the right-ha nd side does not necessaril y commute with inﬁnite products. If o ne omits th e co mpactness conditio n in the D eﬁnition s 2.2 an d 2.3 , th en one obtains the axioms for smooth and multiplicati ve smoo th e xtensio ns deﬁne d on the categor y of al l manifolds. If the coef ﬁcients groups R is degree- wise ﬁnitely ge nerated (see the corres ponding remark in 2.2.1 ), then we obtain the same notion as in [ BS09 ] Our const ruction of the smooth e xtensio n of the comple x cobordism theory does not depen d on any compactn ess assu mption so that there is al so a correspon ding version of Theorem 2.4 furnishing a multiplicati ve smoo th extensi on of ( MU , c ) deﬁned on the cate gory of all smooth m anifol ds. 2.2.5 W e also introduc e the notion of a smooth M U -orientatio n (Deﬁniti on 4.2 7 ) of a proper submersi on p : W → B a nd deﬁne a push-forw ard p ! : ˆ MU ( W ) → ˆ MU ( B ) whi ch reﬁnes the in tegrat ion map p ! : MU ( W ) → MU ( B ) (Deﬁnition 4.34 ). In Subsection 4.4 we sho w that integrat ion is compatible w ith the structure maps a , R , I of the smooth ext ension, functori al, compa tible with pull-back and the product. W e refer to this subsec tion and Theorem 2.7 for further details . I ntegra tion maps play a fundamental role in app lications of generalized co homology theor ies. This is th e cas e e.g. in the conte xt of T -duality , where we hope to e ventua lly gene ralize our in vesti gatons [ BS05 ] to a setti ng in smooth cohomol ogy . 2.3 Smooth extensions for Land weber exact formal gr oup laws 2.3.1 If ( R , g ) is a Landweber exact formal group law , then we let R ∗ ( X ) : = MU ∗ ( X ) ⊗ MU ∗ R denote the associated cohomolog y theory on ﬁnite CW -comple xes. W e consid er the pair ( R , c R ) , where c R : R → R ⊗ Z R = : R R is the canonical map. Theor em 2.5 If ( R , g ) is a Landweber exact formal group law , then ( R , c R ) has a multiplic ati ve smooth ex tension ˆ R , gi ven by ˆ R ( B ) = ˆ MU ( B ) ⊗ MU ∗ R . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1757 Pro of W e start with Theorem 2.4 which states that ( MU , c ) has a multiplicat iv e smooth ext ension. Since Ω k ( ∗ ) = 0 for k 6 = 0 , Ω 0 ( ∗ ) ∼ = R , and MU odd = 0 , the natural map ˆ MU ev ( ∗ ) → MU ev ( ∗ ) i s an isomorphis m. Hence ˆ MU ev ( ∗ ) ∼ = MU ∗ , and for a compact manifold B the group ˆ MU ( B ) is an M U ∗ -module. W e set ˆ R ( B ) : = ˆ MU ∗ ( B ) ⊗ MU ∗ R and deﬁne th e stru cture maps R , I , a by tensoring th e corr espondin g structu re maps fo r ˆ MU . Here we identi fy R ∗ ( B ) ∼ = MU ∗ ( B ) ⊗ MU ∗ R and Ω ( B , R R ) ∼ = Ω ( B , MU R ) ⊗ MU ∗ R . The only non- triv ial point to show is tha t the sequence R ( B ) c R → Ω ( B , R R ) / im ( d ) a → ˆ R ( B ) I → R ( B ) → 0 is ex act. Let us reformulat e this as the exact ness of (2–3) 0 → Ω ( B , R R ) / c R ( R ( B )) → ˆ R ( B ) → R ( B ) → 0 . W e start from the exac t sequence 0 → Ω ( B , MU R ) / c ( MU ∗ ( B )) → ˆ MU ( B ) → MU ∗ ( B ) → 0 . T ensori ng by R gi ves Tor MU ∗ 1 ( MU ∗ ( B ) , R ) → ( Ω ( B , MU R ) / c ( MU ∗ ( B ))) ⊗ MU ∗ R → ˆ MU ( B ) ⊗ MU ∗ R → MU ∗ ( B ) ⊗ MU ∗ R → 0 . Since the tenso r produc t is right exact we ha ve ( Ω ( B , MU R ) / c ( MU ∗ ( B ))) ⊗ MU ∗ R ∼ = Ω ( B , R R ) / c R ( R ( B )) . W e conclude the ex actness of ( 2–3 ) from Landweber’ s Theorem 2.1 which states that Tor MU ∗ 1 ( MU ∗ ( B ) , R ) ∼ = 0 . 2.3.2 Let p : V → A be a prope r submers ion which is smoothly MU -orient ed (s ee 4.27 ) by o p . Recall that ˆ R ( V ) = ˆ MU ( V ) ⊗ MU ∗ R . Deﬁnition 2.6 W e deﬁne the pus h-forwar d map p ! : ˆ R ( V ) → ˆ R ( A ) by p ! ( x ⊗ z ) : = p ! ( x ) ⊗ z . W e must sho w that the push-forwar d is w ell deﬁned . Let u ∈ MU ( ∗ ) ∼ = ˆ MU ev ( ∗ ) . W e must sho w that p ! ( x ∪ u ) ⊗ z = p ! x ⊗ uz . This indeed foll ows from th e special case of the project ion formula Lemma 4.39 , p ! ( x ∪ u ) = p ! ( x ) ∪ u . The smooth MU -orientatio n o p of the proper submersion p gi ves rise to a form A ( o p ) ∈ Ω ( V , R R ) w hich we describe in detail in D eﬁnition 4.29 . The nex t theorem states that the natu ral and expec ted prop erties of a push-f orward hold true. A lgebraic & G eo metric T opology 9 (2009) 1758 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up Theor em 2.7 The follo wing diagram commutes: Ω ( V , R R ) / im ( d ) R V / A A ( o p ) ∧ ...   a / / ˆ R ( V ) p !   I / / R * * R ∗ ( V ) p !   Ω ( V , R R ) R V / A A ( o p ) ∧ ...   Ω ( A , R R ) / im ( d ) a / / ˆ R ( A ) I / / R 4 4 R ∗ ( A ) Ω ( A , R R ) Furthermor e, we ha ve the project ion formula p ! ( p ∗ x ∪ y ) = x ∪ p ! y , x ∈ ˆ R ( A ) , y ∈ ˆ R ( V ) . The push- forward is compatib le with pull-bac ks, i.e. for a Cartesian diagram W q   F / / V p   B f / / A we ha ve q ! ◦ F ∗ = f ∗ ◦ p ! : ˆ R ( V ) → ˆ R ( B ) , where q is smooth ly MU -orien ted by f ∗ o p . If : C → V is a second proper submersion with smooth MU -orien tation o r , then the composi tion s : = p ◦ r has the composed orientation o s : = o p ◦ o r (see 4.32 ), and we ha ve s ! = p ! ◦ r ! : ˆ R ( C ) → ˆ R ( A ) . Pro of This follo ws immediately by tensorin g with id R the corresp onding res ults of the push-fo rward fo r t he exten sion of ( MU , c ) . These are all prov en in Section 4.4 . Cor ollary 2.8 Let ( R 1 , g 1 ) and ( R 2 , g 2 ) be two Landweber exac t formal group laws with corre sponding cohomolog y theories R i ( B ) : = MU ( B ) ⊗ MU ∗ R i . L et φ : R 1 → R 2 be a natural transfo rmation of MU -modules . Then φ lifts to a natura l transformatio n of smoo th coho mology theories as in [ BS09 , Deﬁnit ion 1.5] or [ BS 07 , Deﬁniti on 1.3], ˆ φ ( B ) : = id ˆ MU ( B ) ⊗ φ . In particu lar , we ha ve a (multipli cati ve) smooth complex orientation ˆ MU ( B ) → ˆ K ( B ) from smooth comple x cobordism to smooth K -theory . Here, we use again that ˆ K ( B ) is uniquely determined as a multiplicat iv e extensio n of K -theory [ BS09 ]. A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1759 3 Normal G -structures and cobordi sm theories 3.1 Repr esentatives of the stable normal b undle 3.1.1 In the prese nt paper we construc t geometric models of smooth ext ensions of cobord ism c ohomology theories associat ed to the familie s G ( n ) of classic al groups like U ( n ) , SO ( n ) , Spin ( n ) , or Spin c ( n ) . W e use the notation MG ( B ) and are in particul ar interes ted in the case where B is a smooth m anifol d. A cycle for MG n ( B ) is a proper smooth map W → B with a n ormal G -structure suc h that dim( B ) − dim( W ) = n . The relatio ns are gi ven by bordis ms. Cycles fo r the s mooth ext ension will h av e in a ddition a geometric normal G -structure. In order to make a precis e deﬁnitio n we introd uce a rather concr ete vers ion of the notion of the stabl e normal b undle. 3.1.2 Let X be a spa ce or manifold . For k ∈ N we deno te by R k X the (tot al space of the) triv ial real vector bu ndle X × R k → X . Let f : A → B be a smooth map between manifold s. Deﬁnition 3.1 A representat iv e of the stable normal bun dle of f is a real vecto r bundl e N → A togeth er with an exact seque nce 0 → T A ( df , α ) − − − → f ∗ TB ⊕ R k A → N → 0 , where we ﬁx only the homoto py class of the project ion to N . There is a natural notion of an isomorphis m of representati ves of stable normal b undles. For an i nteger l ∈ N it is evide nt ho w to deﬁne t he l -fold s tabilizati on N ( l ) : = N ⊕ R l A as representati ve of the stable normal b undle with correspond ing short exact se quence. 3.1.3 Let q : C → B be a smooth map which is trans versal to f . Then we ha ve a Cartesian diagra m C × B A Q − − − − → A   y F   y f C q − − − − → B of manifol ds. If 0 → T A ( df , α ) − − − → f ∗ TB ⊕ R k A u − → N → 0 repres ents the stable normal b undle of f , th en we deﬁne the pull-back representat iv e of the stable normal b undle of F by 0 → T ( C × B A ) ( dF ,β ) − − − → F ∗ TC ⊕ R k C × B A γ − → Q ∗ N → 0 , with β : = Q ∗ α ◦ dQ an d γ : = Q ∗ u ◦ ( F ∗ dq ⊕ id R k C × B A ) . Note that Q ∗ ( N ( l )) ∼ = ( Q ∗ N )( l ) . A lgebraic & G eo metric T opology 9 (2009) 1760 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 3.1.4 W e now discus s the stable normal bu ndle of a composit ion. Let g : B → C a smooth map and (3–1) 0 → TB s ← ( dg ,β ) − − − → g ∗ TC ⊕ R l B v − → M → 0 be a repre sentati ve of the stable normal b undle of g . T hen we deﬁne 0 → T A ( d ( gf ) ,γ ) − − − − − → ( gf ) ∗ TC ⊕ R l A ⊕ R k A w − → N ⊕ f ∗ M → 0 as the associated representa tiv e of the stable normal b undle of g ◦ f . Here γ : = ( f ∗ β ◦ df , α ) and w : = ( u ◦ ( f ∗ s ⊕ id R k ) , f ∗ v ◦ pr ( gf ) ∗ TC ⊕ 0 ⊕ R k ) , where s is the split indica ted in ( 3–1 ). This split is unique up to homotop y (since the space of such split s is con ve x) so that the homoto py class of w is well deﬁned. 3.2 G -structur es and connections on the stable normal bun dle 3.2.1 Let G be a Lie group with a homomorph ism G → GL ( n , R ) and consider an n -dimensional real vector bu ndle ξ → X . Deﬁnition 3.2 A G -struct ure on ξ is a pair ( P , φ ) of a G -princ ipal b undle P → X and an isomor phism of vect or b undles φ : P × G R n ∼ → ξ . Deﬁnition 3.3 A geometric G -struct ure on ξ is a triple ( P , φ, ∇ ) , where ( P , φ ) is a G -struct ure and ∇ is a conn ection on P . Note that the tri vial bun dle R n X has a canon ical G -structur e with P = X × G → X . 3.2.2 In order to deﬁne a cobord ism theory we consider a sequence of groups G ( n ) , n ∈ M for an inﬁni te sub monoid M of ( N ≥ 0 , + ) which ﬁ t into a chain o f commuta tiv e diagra ms G ( n )   / / GL ( n , R )   G ( n + k ) / / GL ( n + k , R ) . T ypically , M = N or M = 2 N . This is in particul ar used in order to deﬁne stabili zation. In order to deﬁne the multipli cati ve struct ure we require in additio n G ( n ) × G ( m )   / / GL ( n , R ) × GL ( m , R )   G ( n + m ) / / GL ( n + m , R ) . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1761 Examples are O ( n ) , SO ( n ) , Spin ( n ) , or Spin c ( n ) . In the presen t paper we are in particu lar interested in the comple x cobordi sm theory MU . In this case we hav e M = 2 N and we s et G (2 n ) = U ( n ) . By ab use of n otation we wil l use th e symbol G to denote such a fa mily of grou ps, and by MG the corr espondin g cobordis m theory . 3.2.3 Let f : A → B be a smooth map between manifold. Deﬁnition 3.4 A representati ve of a normal G -struct ure on f is gi ven by a pair ( N , P , φ ) , where N is a repres entati ve of the stable normal bund le, and ( P , φ ) is a G ( n ) -struct ure on N , where n : = dim( N ) , n ∈ M . For notation al con veni ence, we write N instead of the short exa ct sequence with quotie nt N which is also contain ed in the data of a represent ativ e of the stable normal b undle. Deﬁnition 3.5 A repre sentati ve of a geometri c normal G -struct ure on f is gi ven by a quadru ple ( N , P , φ, ∇ ) , where N is a repres entati ve of the stabl e normal bu ndle of f , and ( P , φ, ∇ ) is a geomet ric G ( n ) -struct ure on N , where n : = dim( N ) , n ∈ M . There are natural n otions of isomorphisms of represent ativ es of normal G -structures or geomet ric normal G -structure s. In the follo wing we discuss the opera tions "sta- bilizat ion", "pull-back", and "co mposition" on the lev el o f rep resentati ves of nor mal G -struc ture and geometric normal G -structures. 3.2.4 Let ( N , P , φ ) be a rep resentati ve of a normal G -structur e on f : A → B and consid er l ∈ M . T he stabiliz ation N ( l ) is N ⊕ R l A . It has a canonic al G ( n ) × G ( l ) - structu re with underl ying principal b undle P × G ( l ) → A . W e ge t a G ( n + l ) -struc ture with the under lying principa l bun dle P ( l ) : = ( P × G ( l )) × G ( n ) × G ( l ) G ( l + n ) . Deﬁnition 3.6 W e deﬁne the stabilizati on of ( N , P , φ ) by ( N , P , φ )( l ) : = ( N ( l ) , P ( l ) , φ ( l )) . Let ( N , P , φ, ∇ ) is a represen tati ve o f a geometric normal G -structure, then the con- nectio n ∇ i nduces a connection ∇ ( l ) on P ( l ) . Deﬁnition 3.7 W e deﬁne the stabilizat ion of ( N , P , φ, ∇ ) by ( N , P , φ, ∇ )( l ) : = ( N ( l ) , P ( l ) , φ ( l ) , ∇ ( l )) . A lgebraic & G eo metric T opology 9 (2009) 1762 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 3.2.5 W e no w consider the pull- back and use the notat ion introduced in 3.1.3 . If ( P , φ ) is a G ( n ) -structure on N , then ( Q ∗ P , Q ∗ φ ) is a G ( n ) -structur e on Q ∗ N . Deﬁnition 3.8 W e deﬁne the pull-back of a normal G -struct ure by q ∗ ( N , P , φ ) : = ( Q ∗ N , Q ∗ P , Q ∗ φ ) . Deﬁnition 3.9 W e deﬁne the pull-back of a geometric normal G -struct ure by q ∗ ( N , P , φ, ∇ ) : = ( Q ∗ N , Q ∗ P , Q ∗ φ, Q ∗ ∇ ) . 3.2.6 W e no w discuss the compositi on. Continuing with the notation of 3.1.4 we consid er A f → B g → C and represe ntati ves of normal G -structu res ( N , P , φ ) and ( M , Q , ψ ) on f and g . The sum N ⊕ f ∗ M has a natura l G ( n ) × G ( m ) -struc ture with underlyi ng G ( n ) × G ( m ) -b undle P × A f ∗ Q , and there fore a G ( n + m ) -structu re with underly ing bund le R : = ( P × A f ∗ Q ) × G ( n ) × G ( m ) G ( n + m ) with isomorp hism ρ : R × GL ( n + m ) R n + m ∼ = N ⊕ f ∗ M . Deﬁnition 3 .10 W e deﬁne the composition of repres entati ves of normal G -struct ures by ( M , Q , ψ ) ◦ ( N , P , φ ) : = ( N ⊕ f ∗ M , R , ρ ) . If ∇ P and ∇ Q are connection s on P and Q , th en we get an induced con nection ∇ R on R . Deﬁnition 3.11 W e deﬁne the compositi on of representa tiv es of geometric normal G -struct ures by ( M , Q , ψ , ∇ ) ◦ ( N , P , φ, ∇ ) : = ( N ⊕ f ∗ M , R , ρ, ∇ R ) . 3.2.7 The follo w ing assertion s are obviou s. Lemma 3.12 (1) On the lev el of repres entati ves of normal G -struct ures or geomet- ric normal G -struct ures, pul l-back and composition commut e with stabili zation. (2) On the le vel of represent ativ es o f normal G -struct ures or geometri c normal G -struct ures, pull-bac k and composition are functorial . (3) On the le vel of represent ativ es o f normal G -struct ures or geometri c normal G -struct ures, pull-bac k and composition commute with each other . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1763 3.3 A cycle model f or MG 3.3.1 Let us ﬁ x a family of groups G and M as in 3.2.2 . It determines a multiplicati ve cohomol ogy theory which i s repres ented by a Thom spectrum MG . The map G ( n ) → GL ( n , R ) induces a map of classifying spaces BG ( n ) → BGL ( n , R ) . Let ξ n → BG ( n ) denote the pull-back of the uni vers al R n -b undle. Then for n ∈ M we deﬁne MG n : = BG ( n ) ξ n , where for a vecto r bund le ξ → X we write X ξ for its Thom space. The family of space s MG n , n ≥ 0 , ﬁts into a spec trum with struc ture maps Σ d MG n ∼ = BG ( n ) ξ n ⊕ R d BG ( n ) → BG ( n + d ) ξ n + 1 ∼ = MG n + d , n , n + d ∈ M where we use the canon ical Cartesian diagra m ξ n ⊕ R d BG ( n ) − − − − → ξ n + d   y   y BG ( n ) − − − − → BG ( n + 1) . The ring struct ure is induce d by MG n ∧ MG m ∼ = BG ( n ) ξ n ∧ BG ( m ) ξ m ∼ = ( BG ( n ) × B G ( m )) ξ n ⊞ ξ m → BG ( n + m ) ξ n + m ∼ = MG n + m , for n , m ∈ M , using the can onical C artesia n diagram ξ n ⊞ ξ m − − − − → ξ n + m   y   y BG ( n ) × BG ( m ) − − − − → BG ( n + m ) . For l / ∈ M we set MG l : = Σ l − d MG d , where d ≤ l is maximal with d ∈ M . T he corres ponding structure maps and multiplic ation maps are giv en as suspens ions of the maps desc ribed abov e. If A is a manifold (or more generally a ﬁnite CW -comple x), then the homotop y theoretic deﬁnitio n of the cobord ism cohomolog y group is MG n ( A ) : = lim k [ Σ k A + , MG n + k ] , where the limit is tak en over the stab ilization maps [ Σ k A + , MG n + k ] → [ ΣΣ k A + , Σ MG n + k ] → [ Σ k + 1 A + , MG n + k + 1 ] , and A + is the union of A and a disjoin t base point. T emporaril y we use the bold-f ace notati on of the homotop y theoreti c deﬁnitio n of the cobord ism cohomology theory . For d etails we ref er to [ Swi02 ] or [ Sto68 ]. A lgebraic & G eo metric T opology 9 (2009) 1764 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 3.3.2 W e now prese nt a cycle model of the G -cobord ism theory . Let A be a smooth manifold . Deﬁnition 3.13 A precycle ( p , ν ) of degree n ∈ Z ov er A consis ts of a smooth map p : W → A from a smoot h manifold W of dimension dim( W ) = dim( A ) − n , and a repres entati ve ν of a normal G -struct ure on p (see 3.4 ). A c ycle o f de gree n ∈ Z ov er A is a prec ycle ( p , ν ) of de gree n , where p is proper . There is a natura l notion of an isomorph ism of precy cles. 3.3.3 Let c : = ( p , ν ) be a precycle ov er A and q : B → A be transvers e to p . Deﬁnition 3.14 W e deﬁne the pull-b ack q ∗ c : = ( q ∗ p , q ∗ ν ) , a precy cle over B . The pull-b ack is functo rial by Lemma 3.12 . 3.3.4 W e no w consid er precycle s c = ( p , ν ) ov er A and d = ( q , µ ) ov er C with underl ying maps p : B → A and q : A → C . Deﬁnition 3.15 W e deﬁne the composit ion d ◦ c : = ( q ◦ p , µ ◦ ν ) using 3.10 . The composi tion d ◦ c is a prec ycle ov er C . The composition is associati ve and compatib le with pull-ba ck. 3.3.5 Let c : = ( p , ν ) , p : W → A , and d : = ( q , µ ) , q : V → B be precyc les ove r A and B . Then we can form the diagram W × V Q   / / V q   W p   W × B o o P   r / / B A A × B s o o . Deﬁnition 3 .16 W e deﬁne th e product of the prec ycles c and d to be the precyc le c × d : = s ∗ c ◦ r ∗ d ov er A × B . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1765 Note that there is an equi v alent deﬁniti on based on on the diagra m W × V   / / W p   V q   A × V o o   / / A B A × B o o . It follo ws from the functoriali ty of the composition and its compati bility with the pull-b ack that the product of precy cles is assoc iati ve. 3.3.6 W e consid er a precy cle b : = (( f , p ) , ν ) ov er R × A . Deﬁnition 3.17 The precy cle b is called a bordism datum if f is transv erse to { 0 } ∈ R and p |{ f ≥ 0 } is proper . W e deﬁne the precycl e ∂ b : = i ∗ b , w here i : A → R × A , i ( a ) : = (0 , a ) . 3.3.7 Let c = ( p , ν ) be a prec ycle and l ∈ N . Deﬁnition 3.18 W e deﬁne the l -fold stabiliz ation o f the precycle c by c ( l ) : = ( p , ν ( l )) (see 3.6 ). 3.3.8 W e no w come to the geometric picture of the cobordis m theory MG . W e consid er a smooth manifold A and let ZMG ( A ) denote the semigroup of isomorphism classe s of cycle s over A w ith respect to disjoi nt u nion. Recall that a relation ∼ on a semigrou p is compatib le with the semigroup structure if a ∼ b implies that a + c ∼ b + c for all c . Deﬁnition 3.19 Let “ ∼ ” be the minimal equi valen ce relation which is compati ble with the semigro up structure and satisﬁes: (1) If b is a bordism datum, then ∂ b ∼ 0 . (2) If l ∈ N , then c ( l ) ∼ c . W e let MG ( A ) : = ZMG ( A ) / ∼ denote the quotie nt semigroup. 3.3.9 Let 0 deno te th e c ycle of degree n gi ven by the empty man ifold. The following Lemma will be usefu l in calcul ations. Lemma 3.20 Let c be a cycle which is equi v alent to 0 . T hen there exists a bordism datum b and l ≥ 0 such that c ( l ) ∼ = ∂ b . W e lea ve the proof to the intereste d reader . A lgebraic & G eo metric T opology 9 (2009) 1766 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 3.3.10 W e no w describe the func toriality , the product , or ientation s, an d the integrati on on the le vel of cyc les. (1) F unctorialit y Let f : B → A be a smooth m ap and x ∈ MG ( A ) . W e can repres ent x by a cycle c = ( p , ν ) su ch that p and f are transv erse. Then f ∗ x is repres ented by f ∗ c . (2) P r oduct Let c and d be cycle s for x ∈ MG n ( A ) and y ∈ MG m ( B ) . Then x × y ∈ MG n + m ( A × B ) is represen ted by the c ycle c × d (see 3.16 ). W e ge t the interio r product using the pull-back along the diagona l. (3) Integration Let d be a cycle over A with underly ing map q : V → A . In this situati on we hav e an integra tion map q ! : M G ( V ) → MG ( A ) . If x ∈ MG ( V ) is repres ented by the cycle c , then q ! ( x ) is repres ented by the cyc le d ◦ c (see 3.15 ). (4) S uspension Let i : ∗ → S 1 denote the embedding of a point. For each d ∈ M , the tri vial bun dle R d ∗ repres ents the stable normal b undle which of course has a canon ical G ( d ) -structure. In this way i is the underlyin g map of a cyc le { i } ∈ ZMG 1 ( S 1 ) which represent s a class [ i ] ∈ MG 1 ( S 1 ) . For a manifold A we deﬁne MG k ( A ) → MG k + 1 ( S 1 × A ) , x 7→ { i } × x , which on the le vel o f cycles is represe nted by c 7→ { i } × c . This transf ormation is essent ially the suspen sion morphism (not an isomorp hism, since we neith er use reduce d cohomology nor the suspensio n of A ). 3.3.11 In order to sho w that the operat ions deﬁned abo ve on the cyc le le vel descend throug h the equi valenc e relati on ∼ the follo wing observ ations are usefu l. Let b = (( f , p ) , µ ) be a bordism datum ov er A w ith under lying map ( f , p ) : W → R × A . Assume that q : B → A is transver se to p and p |{ f = 0 } . Then we can form the bordism datum ( id R × q ) ∗ b ov er B which will be denoted by q ∗ b . Note that q ∗ ∂ b ∼ = ∂ q ∗ b . Let e be a c ycle ov er B . Then we ca n form b × e which we ca n interpret as a b ordism datum ov er B × A . Note that ∂ ( b × e ) ∼ = ∂ b × e . Let d be a cycle with underl ying map A → B . Let pr : R × B → B be the projection . Then we can form the bord ism datum pr ∗ d ◦ b o ver B . Note that ∂ ( pr ∗ d ◦ b ) ∼ = d ◦ ∂ b . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1767 Finally , if c is a cycle ov er W , then we ca n form the bordi sm datum b ◦ c o ver B , and ha ve ∂ ( b ◦ c ) ∼ = ∂ b ◦ c . 3.3.12 W e no w hav e a geometric and a homotopy theoretic picture of the G -cobordism theory which we disti nguish at the moment by using roman and bold-f ace letters. Pro position 3.21 There is an i somorphism of ring-v alued functor s MG ( A ) ∼ = MG ( A ) on compact manifolds. This isomorphism preserv es t he product and is compatibl e with push-f orward. Pro of This follo ws from the Pontryag in-Thom constr uction. Since this construc tion for cobordism cohomolog y (as oppo sed to homology) seems not to be so well kno wn let us shortl y indicate the main ideas . For concre teness let us consider the case of comple x cobordism MU and ev en 2 n . W e ha ve MU 2 n ( A ) ∼ = colim i [ Σ 2 i A , MU 2 n + 2 i ] . Let h : Σ 2 i A → MU 2 n + 2 i repres ent s ome class in MU 2 n ( A ) . Recall that MU 2 n + 2 i = BU ( n + i ) ξ n + i is the Thom space the uni versa l b undle ξ n + i → BU ( n + i ) . The latter is itself the colimit of Thom spaces BU ( n + i ) ξ n + i ∼ = colim k Gr n + i ( C n + i + k ) ξ n + i of tau tological b undles ξ n + i ov er th e Grassmann ians Gr n + i ( C n + i + k ) of ( n + i ) - dimensio nal subspac es in C n + i + k . W e can assume that h fact ors over some Thom space Gr n + i ( C n + i + k ) ξ n + i , and that the induced map S 2 i × A p → Σ 2 i A f → Gr n + i ( C n + i + k ) ξ n + i is smooth and transver se to the zero section of ξ n + i , where p is the canonical projec- tion. The preimage of the zero section is a subman ifold W ⊂ S 2 i × A of codimension 2 n + 2 i . W e let f : W → A be ind uced by the pro jection. W e use the sta ndard e m- beddin g S 2 i → R 2 i + 2 in order to trivia lize the bundle TS 2 i ⊕ R S 2 i ∼ = S 2 i × R 2 i + 2 . The embeddi ng W ֒ → S 2 i × A thus indu ces naturall y an embeddin g TW → T ( A × S i ) | W ∼ = f ∗ T A ⊕ TS 2 i | W → f ∗ T A ⊕ R 2 i + 2 W . Moreo ver , the dif ferentia l of h identiﬁes the no rmal bu ndle N : = f ∗ T A ⊕ R 2 i + 2 W / TW with the pull-back h ∗ | W ξ n + i ⊕ C W , w hich has a canonical compl ex structure. In this way we get the normal b undle sequence 0 → TW → f ∗ T A ⊕ R 2 i + 2 M → N → 0 and t he U -structure ν = ( N , P , φ ) on N . Note t hat f : W → A is pro per so that we get a cyc le ( f , ν ) of deg ree n . O ne no w proceed s as in the case of bordi sm homolo gy and A lgebraic & G eo metric T opology 9 (2009) 1768 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up sho ws that the class [ f , ν ] ∈ MU 2 n ( A ) only depends on the class [ h ] ∈ MU 2 n ( A ) . In this way we get a map MU 2 n ( A ) → MU 2 n ( A ) . Con versely one starts with a cycle ( f , ν ) of deg ree n . One observes that up to stabi- lizatio n and homotop y the normal b undle sequence 0 → TW → f ∗ T A ⊕ R k W → N → 0 comes from an embeddin g of i : W ֒ → S k − 1 × A such that f = pr A ◦ i . Then we let W → BU ( n + ( n + k ) / 2) be a class ifying m ap o f N (nec essarily , dim R N = n + k is eve n). It giv es rise to a map of Thom spaces W N → BU ( n + i ) ξ ( n + k ) / 2 . W e ﬁnally precomp ose with the clutchin g m ap Σ k A → W N in order to get a map h : Σ k A → MU n + k . One checks that this constr uction giv es the in vers e map MU n ( A ) → MU n ( A ) . A furthe r standa rd argu ment checks that these maps are compatib le with the abelian group and ring struct ures and the push-fo rward. In vie w of Proposition 3.21 we can drop the bold-fac e notation for the homotopy theore tic cobordis m. 3.4 Po wer series and genera 3.4.1 The basic datum for a multiplica tiv e smooth extensio n of a generalize d co ho- mology theory h is a pair ( h , c ) , where c : h → HR is a natural transformati on from h into the ordinary cohomo logy w ith coef ﬁcients in a graded ring R over R . The transfo rmation c induc es in particular a homomo rphism of coef ﬁcients h ∗ → R ∗ . Our constr uction of smooth extensio ns of cobordism theori es is base d on a description of c in terms of char acteristic numbers of stable normal b undles. A ring homomorphi sms c : MG ∗ → R ∗ is called a G -genus. One can class ify SO and U -genera in terms of formal power series (see [ HBJ92 ] and 3.22 ). G enera for other cobord ism th eories c an be deri ved from transformati ons like MSpin → MSO . Since the details in the real and comple x case diff er slight ly , in the present paper we restri ct to our main example G : = MU , i.e. M = 2 N ≥ 0 , G (2 n ) = U ( n ) . It is easy to modify the constru ctions for other cases like MSpin c , MSO or Spin c . 3.4.2 Let R be a commutativ e Z -grad ed algeb ra ove r R with 1 ∈ R 0 . By R [[ z ]] we denote the graded ring of formal power series, where z has degr ee 2 . Let φ ∈ R [[ z ]] 0 be a po wer serie s of the f orm 1 + φ 1 z + φ 2 z 2 + . . . (n ote that deg( φ i ) = − 2 i ). T o such a po wer series we associate a genus r φ : MU ∗ → R ∗ as in [ MS , Section 19]. A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1769 Theor em 3.22 ([ HBJ92 ]) The corresp ondence φ → r φ gi ves a bije ction between the set R [[ z ]] 0 and R -v alued U -gener a. In the follo wing we describe the associat ed na tural transfo rmation r φ : MU ( A ) → H ( A , R ) of cohomology theories on the le vel of cycles, follo wing the procedure as descri bed in [ MS ]. 3.4.3 W e deﬁne the power series K φ ∈ R [[ σ 1 , σ 2 , . . . , ]] 0 (where σ i has degree 2 i ) such that K φ ( σ 1 , σ 2 , . . . ) = ∞ Y i = 1 φ ( z i ) holds if we replace σ i by the elementary symmetric functio ns σ i ( z 1 , . . . ) . 3.4.4 Note that G (2 k ) = U ( k ) (see 3.2.2 ). Let N → W be an n -dimensiona l real vec tor b undle for n ev en with a G ( n ) -structure ( P , φ ) . Then we ha ve Chern cl asses c j ( N ) : = c j ( P ) ∈ H 2 j ( W , R ) . Deﬁnition 3.23 W e deﬁne the charac teristic class φ ( N ) : = K φ ( c 1 ( N ) , c 2 ( N ) , . . . ) ∈ H 0 ( A , R ) . The follo w ing propertie s are well-kno wn (see [ HBJ92 ]). Lemma 3.24 (1) Let R k A ha ve the tri vial G ( k ) -struct ure. Then we hav e φ ( R k ) A = 1 for all k ≥ 0 . (2) If M is a second b undle w ith a G ( m ) -struct ure, and N ⊕ M has the induced G ( n + m ) -struct ure, then we ha ve φ ( N ⊕ M ) = φ ( N ) ∪ φ ( M ) . (3) If f : B → A is a continuou s map, then we ha ve f ∗ φ ( N ) = φ ( f ∗ N ) , if we equip f ∗ N with the induc ed G ( n ) -struct ure. 3.4.5 Consider a cycle c = ( p , ν ) ∈ ZMU ( A ) of degree n with underlying map p : W → A and normal U -structu re ν = ( N , P , φ ) . Then p is a proper map which is orient ed for the ordinary co homology theo ry HR . In particular , we hav e an integra tion p ! : H ∗ ( W , R ) → H ∗ + n ( A , R ) . Deﬁnition 3.25 W e deﬁne ˜ r φ ( c ) : = p ! ( φ ( N )) ∈ H n ( A , R ) . A lgebraic & G eo metric T opology 9 (2009) 1770 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 3.4.6 The follo wing Lemma implies half o f T heorem 3.22 . What is m issing is th e ar gument that ev ery R -val ued U -genus comes from a formal po wer series. Lemma 3.26 The map ˜ r φ descen ds through ∼ and induces a natural transfo rmation r φ : MU ( A ) → H ( A , R ) of ring- val ued functors. Pro of Using the ﬁrst and secon d property in 3.24 one checks that ˜ r φ ( c ) = ˜ r φ ( c ( l )) . Assume that b = (( f , q ) , µ ) with underlying map ( f , q ) : W → R × A and µ = ( M , Q , λ ) is a bord ism datum. Then we get the Cartesian diagram V i / / p   W ( f , q )   { 0 } × A / / R × A Set N : = i ∗ M . Theref ore p ! ( φ ( N )) = p ! ( φ ( i ∗ M )) = p ! ( i ∗ φ ( M )) = 0 by the bor dism in var iance of the pus h-forward in ordin ary cohomology and the t hird prope rty of 3.24 . Thus the trans formation r φ is well deﬁned . It is natu ral since for f : B → A whi ch is transve rse to p we ha ve a C artesia n diagra m F ∗ N   / / N   f ∗ V q   F / / V p   B f / / A , the b undle F ∗ N repres ents the stable normal b undle of q , and q ! ( φ ( F ∗ N )) = q ! ( F ∗ φ ( N )) = f ∗ p ! ( φ ( N )) by the proje ction formula. This implies that f ∗ ˜ r φ ( c ) = ˜ r φ ( f ∗ c ) . W e claim th at th e tra nsformation i s als o mu ltiplicati ve. T o this end we consider a cycle d = ( q , µ ) with underlyi ng map q : V → B and normal G -structu re µ = ( M , Q , λ ) . Then the unde rlying prop er map of c × d ∈ ZMU ( A × B ) is p × q : W × V → A × B , and the b undle N ⊞ M represents its normal G -structure. W e thus hav e ( p × q ) ! ( φ ( N ⊞ M )) = ( p × q ) ! ( φ ( N ) × φ ( M )) = p ! ( φ ( N )) × q ! ( φ ( M )) . This implies ˜ r φ ( c × d ) = ˜ r φ ( c ) × ˜ r φ ( d ) . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1771 3.4.7 The most i mportant example for the present paper is giv en by the ring M U R : = MU ∗ ⊗ Z R . T he MU ∗ -module MU R is Landweber exact. Hence, for a compact manifold or ﬁnite CW -complex A we hav e H ∗ ( A , MU R ) ∼ = MU ∗ ( A ) ⊗ MU ∗ MU R and therefo re a canonical natural trans formation r : MU ∗ ( A ) → H ∗ ( A , MU R ) , x 7→ x ⊗ 1 . This transfor mation is a genus r = r φ for a certain po w er series φ ∈ MU R [[ x ]] 0 . W e refer to [ HBJ92 ] for furthe r details on φ . 4 The smooth extension of M U 4.1 Characteristic forms 4.1.1 Let φ ∈ R [[ z ]] 0 be as in 3.4.2 and G be the f amily of gr oups 3.2 .2 a ssociated t o U ( n ) , n ≥ 0 .. W e ﬁrst lift th e construct ion of the cha racteristic class φ ( N ) ∈ H 0 ( A , R ) of ve ctor bundl es N → A with G ( n ) -struct ure to the form lev el. Let ( P , ψ , ∇ N ) be a geometric G ( n ) -struc ture on N → A . By R ∇ N ∈ Ω 2 ( A , End ( N )) we deno te the curv ature of the connectio n ∇ N . The ﬁ ber -wise polynomia l bu ndle morphism det : End ( N ) → R A ext ends to det : Ω ev ( A , End ( N )) → Ω ev ( A ) . As usual we deﬁne the Chern forms c i ( ∇ N ) ∈ Ω 2 i ( A ) by 1 + c 1 ( ∇ N ) + c 2 ( ∇ N ) + · · · = det (1 + 1 2 π i R ∇ N ) . Deﬁnition 4.1 If N → A is a real v ector bundl e with a geometric G ( n ) -struct ure, then we deﬁne φ ( ∇ N ) : = K φ ( c 1 ( ∇ N ) , c 2 ( ∇ N ) , . . . ) ∈ Ω 0 ( A , R ) . 4.1.2 The propertie s stat ed in Lemma 3.24 lift to the form le vel by well-kno w n proper ties of the Chern-W eil calculus. Lemma 4.2 (1) Let k ≥ 0 and R k A ha ve the trivial G ( k ) -struct ure with the tri vial conne ction. Then we ha ve φ ( ∇ R k A ) = 1 . (2) If M → A is a second bund le with a geometri c G ( m ) -struct ure and assume that N ⊕ M has the induced geometric G ( n + m ) -struct ure, then we ha ve φ ( ∇ N ⊕ M ) = φ ( ∇ N ) ∧ φ ( ∇ M ) . (3) Assume that f : B → A is a smooth map. Then we ha ve f ∗ φ ( ∇ N ) = φ ( ∇ f ∗ N ) , if we equip f ∗ N with the induc ed geometric G ( n ) -struct ure. A lgebraic & G eo metric T opology 9 (2009) 1772 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 4.1.3 Deﬁnition 4.3 A geometric prec ycle over A is a pair ( p , ν ) of a smooth m ap p : V → A and a geometric normal G -struct ure ν (see 3.5 ). A geometric precy cle is a cycle if p is prope r . Usually we will denote geometric precycles by ˜ c , where c d enotes the underlying prec ycle. Since a principal bun dle a lways admits connection s, ev ery precycle can be reﬁned to a geometric prec ycle. If ν = ( N , P , φ, ∇ ) , then we will write ∇ ν : = ∇ . 4.1.4 Let Ω −∞ ( A ) : = C −∞ ( A , Λ ∗ T ∗ A ) denote the di ffer ential forms with dist ribu - tional coef ﬁcients. W e identify this space with the topological dual of C ∞ c ( A , Λ n −∗ T ∗ A ⊗ Λ A ) , w here Λ A → A is the real orien tation b undle and n = d im( A ) . For this identiﬁ- cation we use cup product and inte gration of an n -form w ith v alues in the orient ation b undle ove r A . Finally , w e deﬁne Ω −∞ ( A , R ) : = Ω −∞ ( A ) ⊗ R R using the algebraic tensor produc t. A morphism of complex es induc ing an isomorphism in cohomol ogy is called a quasi- isomorph ism. It is well-kno wn (see [ dR84 ], or do this ex ercise using Lemma 4.1 1 ) that the inclus ion Ω ( A ) ֒ → Ω −∞ ( A ) is a quasi-is omorphism. H ence, Ω ( A , R ) ֒ → Ω −∞ ( A , R ) is a qua si-isomorph ism, too. 4.1.5 Let p : V → A be a pro per smooth orient ed map. The or ientation of p gi ves an isomorph ism p ∗ Λ A ∼ → Λ V . W e then deﬁne the push-forw ard p ! : Ω −∞ ( V ) → Ω −∞ ( A ) of de gree dim( A ) − dim( V ) by the formul a < p ! ω , σ > = < ω , p ∗ σ > , ω ∈ Ω −∞ ( V ) , σ ∈ Ω ( A , Λ A ) holds true. By tensoring with the identity of R w e get the map p ! : Ω −∞ ( V , R ) → Ω −∞ ( A , R ) . Stokes’ theorem implies p ! ◦ d = d ◦ p ! . W e get an induced map in cohomolo gy such that the follo wing diag ram commutes : (4–1) H ∗ ( Ω −∞ ( V , R )) deRham − − − − → ∼ = H ∗ ( V , R )   y p !   y p ! H ∗ ( Ω −∞ ( A , R )) deRham − − − − → ∼ = H ∗ ( A , R ) . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1773 4.1.6 Let ˜ c = ( p , ν ) be a geometri c cycl e of de gree n . Deﬁnition 4.4 W e deﬁne T ( ˜ c ) : = p ! ( φ ( ∇ ν )) ∈ Ω n −∞ ( A , R ) . This form is close d, and by ( 4– 1 ) we h av e t he following equality in de Rham cohomol- ogy: (4–2) [ T ( ˜ c )] = p ! ( φ ( N )) = ˜ r φ ( c ) . 4.1.7 W e now consider a bordism datum b = (( f , q ) , µ ) ov er a manifold A with ( f , q ) : W → R × A . W e b uild the composition q ! ◦ χ { f ≥ 0 } : Ω k ( W ) → Ω k + l −∞ ( A ) , where l : = dim( A ) − dim( W ) , a nd χ U is the multi plication operation with the ch arac- teristi c functio n of the subset U . Stoke s’ theore m implies in this case that (4–3) d ◦ q ! ◦ χ { f ≥ 0 } − q ! ◦ χ { f ≥ 0 } ◦ d = ( q 0 ) ! ◦ i ∗ , where q 0 : W 0 → A is deﬁned by the Cartesian diagram W 0 i → W q 0 ↓ q ↓ A a 7→ (0 , a ) → R × A , i.e. q 0 is the underlyi ng map of ∂ b . Deﬁnition 4.5 Let ˜ b : = (( f , q ) , ˜ ν ) be a geomet ric reﬁnement of b . W e deﬁne T ( ˜ b ) : = q ! ◦ χ { f ≥ 0 } ( φ ( ∇ ˜ ν )) ∈ Ω −∞ ( A ) . Equation ( 4–3 ) sho ws that (4–4) dT ( ˜ b ) = T ( ∂ ˜ b ) . 4.2 The smooth extension of MU 4.2.1 In the presen t subsection we construct the smooth exte nsion associated to the pair ( MU , r φ ) , where φ ∈ R [[ z ]] 0 is as in 3.4.2 , and r φ is the associa ted natural transfo rmation MU ( A ) → H ( A , R ) . Recall the n otions o f a cycle and a geometric cycle from 3.13 and 4.3 . The c ycles for the smoot h extensio n ˆ MU of MU will be called smooth cy cles. A lgebraic & G eo metric T opology 9 (2009) 1774 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up Deﬁnition 4.6 A smooth cycle of degree n is a pair ˆ c : = ( ˜ c , α ) , where ˜ c is a geometric cyc le of degree n , and α ∈ Ω n − 1 −∞ ( A , R ) / im ( d ) is such that T ( ˜ c ) − d α : = Ω ( ˆ c ) ∈ Ω n ( A , R ) . The point here is that T ( ˜ c ) − d α is a smooth representati ve of the cohomolo gy cl ass repres ented by T ( ˜ c ) . The latter is in general a singular form. T o be ex plicit note that in the deﬁnitio n abo ve im ( d ) : = im ( d : Ω n − 2 −∞ ( A , R ) → Ω n − 1 −∞ ( A , R )) , i.e. we allo w dif ferential s of forms with distr ibu tion coef ﬁcients. 4.2.2 There is an evid ent notion of an isomorph ism of smooth cyc les. W e form the graded semigroup Z ˆ MU ( A ) of isomorphism classes o f smooth c ycles such th at the sum is gi ven by ( ˜ c , α ) + ( ˜ c ′ , α ′ ) = ( ˜ c + ˜ c ′ , α + α ′ ) , where, as in the non-g eometric case, ˜ c + ˜ c ′ is gi ven by the disjo int union. 4.2.3 The smooth cobordism group ˆ MU ( A ) will be deﬁned as the quotient of Z ˆ MU ( A ) by an equi valenc e relation generated by stabilizatio n and bordi sm. Deﬁnition 4.7 Let “ ∼ ” be the minimal equi v alence relation on Z ˆ MU ( A ) which is compatib le with the semigroup structu re (see 3.3.8 ) and such that (1) For l ∈ M we hav e ( ˜ c , α ) ∼ ( ˜ c ( l ) , α ) , w here ˜ c ( l ) is the l -fold stabiliz ation deﬁned by ( p , ν )( l ) : = ( p , ν ( l )) (see 3.7 ). (2) For a geo metric bordism datum ˜ b we ha ve ( ∂ ˜ b , T ( ˜ b )) ∼ 0 . W e deﬁne ˆ MU n ( A ) : = Z ˆ MU n ( A ) / ∼ as the semigroup of equiv alence classe s of smooth cyc les of degree n . W e will write [ ˜ c , α ] for the equi valenc e class of ( ˜ c , α ) . 4.2.4 Lemma 4.8 ˆ MU n ( A ) is a grou p. Pro of Let [ ˜ c , α ] ∈ ˆ MU ( A ) . It suf ﬁces t o show that it ad mits an in verse . Since MU ( A ) is a group there exists a cycle c ′ such that c + c ′ ∼ 0 . By Lemma 3.20 we can assume th at c ( l ) + c ′ ( l ) ∼ ∂ b for some bordism datum b an d l ∈ N . W e extend b to a geometri c bordis m datum ˜ b by choosi ng a conn ection such that ∂ ˜ b ∼ = ˜ c ( l ) + ˜ c ′ ( l ) for some geomet ric exten sion ˜ c ′ of c ′ . Then we ha ve [ ˜ c ′ , T ( ˜ b ) − α ] + [ ˜ c , α ] = 0 . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1775 4.2.5 W e now deﬁne the structure maps a , R , I (see 2.2 ) of the smooth extensio n ˆ MU . Deﬁnition 4.9 (1) W e deﬁne R : ˆ MU ( A ) → Ω d = 0 ( A , R ) by R ([ ˜ c , α ]) : = T ( ˜ c ) − d α . (2) W e deﬁne a : Ω ( A , R ) → ˆ MU ( A ) by a ( α ) : = [ ∅ , − α ] . (3) W e deﬁne I : ˆ MU ( A ) → MU ( A ) by I ([ ˜ c , α ]) : = [ c ] (using the geometric m odel 3.19 ) Lemma 4.10 These maps are well deﬁned . W e ha ve R ◦ a = d . Pro of The only non-obv ious part is the fact that R is well deﬁned. T o this end consider a geometri c bordis m datum ˜ b . Then we ha ve R [ ∂ ˜ b , T ( ˜ b )] = T ( ∂ ˜ b ) − dT ( ˜ b ) = 0 by Equatio n ( 4–4 ). 4.2.6 W e no w extend A 7→ ˆ MU ( A ) to a contra-v ariant functo r on the category of smooth manifolds. Let f : B → A be a smooth map. Then we must cons truct a functo rial pull-back f ∗ : ˆ MU ( A ) → ˆ MU ( B ) su ch that the tr ansformatio ns R , I , a abov e become natur al. Let ( ˜ c , α ) be a smooth cycle with ˜ c = ( p , ν ) , p : W → A . W e can assume that p is transv erse to f . Otherwise w e replace p by a bor dant (homotopi c) map and correc t α corresp ondingl y so that the new pair represen ts the same class in ˆ MU ( A ) as ( ˜ c , α ) . Then we ha ve the Cartesia n diagram B × A W P   F / / W p   B f / / A . The map P is the underlyi ng map of a geometric cycle f ∗ ˜ c = ( P , f ∗ ν ) , w here f ∗ ν is th e pull-bac k of th e geometric normal G -structure as d eﬁned in 3.9 . W e wan t to deﬁne f ∗ [ ˜ c , α ] : = [ f ∗ ˜ c , f ∗ α ] . T he problem is that α is a distrib ution. In order to deﬁne the pul l-back f ∗ α of a distrib utional form we need the addi tional assumpti on that WF ( α ) ∩ N ( f ) = ∅ , where N ( f ) ⊆ T ∗ A \ 0 A is the normal set to f gi ven by N ( f ) : = clo { η ∈ T ∗ A \ 0 A | ∃ b ∈ B s . t . f ( b ) = π ( η ) and df ( b ) ∗ η = 0 } (where π : T ∗ A → A is the projec tion), and WF ( α ) denotes the wa ve front set of α . The wav e front set of a distrib utional form α on A is a conical subset of T ∗ A which measures t he locu s and the directions of the singulariti es of α . For a preci se deﬁnition A lgebraic & G eo metric T opology 9 (2009) 1776 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up and for the propertie s of distrib utions using the wav e front set needed belo w we refer to [ H ¨ or03 , S ection 8].. Note that we can change α by exact forms with distrib ution coef ﬁcients without altering the class of ( ˜ c , α ) . The idea is to show that one can choose α such that WF ( α ) ∩ N ( f ) = ∅ holds. B y [ H ¨ or03 , T heorem 8.2.4], in this case f ∗ α is deﬁned. It i s indep endent of the choice aga in up to exac t forms with distrib ution coef ﬁcients. The details will be e xplained in the follo wing paragraphs . 4.2.7 Lemma 4.11 Let α ∈ Ω n −∞ ( A ) . Then there exists β ∈ Ω n − 1 −∞ ( A ) such that WF ( α − d β ) ⊆ WF ( d α ) . Pro of W e choose a Riemannian metric on A . Then w e can deﬁne the formal adjoi nt δ : = d ∗ of the de R ham diff erential and the Laplacia n ∆ : = δ d + d δ . Since ∆ is elliptic we c an choo se a proper pseud o-dif ferentia l parametrix P of ∆ . This is a pseud o-dif ferential operator of degr ee − 2 which is an in verse of ∆ up to pseudo- dif ferentia l operators of degree −∞ (smooth ing opera tors). A pseud o-dif ferential operat or on A is called prope r if the res triction of the two projec tions from t he supp ort (a subse t of A × A ) of its distrib ution kernel to the two fac tors A are proper maps. Then we form G : = δ P . This pseu do-dif ferenti al operator sati sﬁes dG + Gd = 1 + S , where S is a prop er smoothi ng operator . W e thus can set β : = G α and hav e α − d β = G d α − S α . Since S α is s mooth an d W F ( Gd α ) ⊆ WF ( d α ) (a ps eudo-dif ferential operator d oes n ot increa se wa ve front sets ) we see that WF ( α − d β ) ⊆ WF ( d α ) . If α ∈ Ω −∞ ( A , R ) , then we can write for some s ∈ N α = s X i = 1 α i ⊗ r i with α i ∈ Ω n −∞ , and with linea rly indepen dent r i ∈ R . In this case the wa ve front set of α is by deﬁnitio n WF ( α ) : = ∪ s i = 1 WF ( α i ) . It is now easy to see that L emma 4.11 ext ends to forms with coef ﬁcients in R . 4.2.8 Lemma 4.12 If ( ˜ c , α ) , ˜ c = ( p , ν ) , is a s mooth cycle, th en we can cho ose α such th at WF ( α ) ⊆ N ( p ) . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1777 Pro of It is a genera l fac t that the wa ve front set of the push- forward of a smooth distrib ution along a m ap is co ntained in the normal set of the map. In view of Deﬁnition 4.4 we ha ve WF ( T ( ˜ c )) ⊆ N ( p ) . Since T ( ˜ c ) − d α is smooth we ha ve WF ( d α ) = WF ( T ( ˜ c )) ⊆ N ( p ) , and by Lemma 4.11 we can change α by an e xact form such that WF ( α ) ⊆ N ( p ) . 4.2.9 A reformulation of the fact th at f and p are tran sverse is th at N ( f ) ∩ N ( p ) = ∅ . Using Lemma 4.12 we now tak e a representati ve of α such that WF ( α ) ⊆ N ( p ) . Then f ∗ α is a well deﬁned distrib ution. Deﬁnition 4.13 W e deﬁne f ∗ [ ˜ c , α ] = [ f ∗ ˜ c , f ∗ α ] , where we take represe ntati ves ˜ c = ( p , ν ) and α such that f and p are transv erse and WF ( α ) ⊆ N ( p ) . 4.2.10 Lemma 4.14 The pull-b ack is well deﬁned and functori al. Pro of First we s how that the pu ll-back is w ell deﬁned with respect to the cho ice of α . Let β ∈ Ω −∞ ( A , R ) and α ′ : = α + β be su ch that T ( ˜ c ) − d α ′ is smoo th. This implies that WF ( α ′ ) ⊆ N ( p ) , and hence WF ( d β ) ⊆ N ( p ) . By Lemma 4.11 we can modify β by a closed form such that WF ( β ) ⊆ N ( p ) . Then f ∗ α ′ = f ∗ α + df ∗ β . It is easy to see that the pull-back is additi ve and prese rves stabili zation. It remains to sho w th at it pr eserve s zero bordi sm. Let ˜ b = (( h , q ) , µ ) b e a geometric bordis m datum ov er A with ( h , q ) : W → R × A . W e deﬁne W 0 : = h − 1 ( { 0 } ) and assu me that q and q | W 0 are tr ansver se to f . W e th en ha ve t he geomet ric bordism datum ( id R × f ) ∗ ˜ b over B . Let us deﬁne the normal datum of b by N ( b ) : = clo { η ∈ T ∗ A \ 0 A | ∃ v ∈ W s . t . E ( v ) = π ( η ) and  dE ( v ) ∗ η = 0 or v ∈ W 0 and dE ( v ) ∗ η | T v W 0 = 0  } . Then we hav e WF ( T ( ˜ b )) ⊆ N ( b ) . Again, si nce q and q | W 0 are tr ansver se to f we hav e N ( b ) ∩ N ( f ) = ∅ so that f ∗ T ( ˜ b ) is well deﬁned. Using the fact that in a Cartesian diagram push-f orward of distrib utions commutes with pull-bac k we get f ∗ T ( ˜ b ) = T ( f ∗ ˜ b ) . It follo w s that ( f ∗ ∂ ˜ b , f ∗ T ( ˜ b )) = ( ∂ f ∗ ˜ b , T ( f ∗ ˜ b )) . This implies that the pull-back is well deﬁned on the le vel of equi valen ce classes. W e no w show functorialit y . L et g : C → B be a second smooth map. If ˆ x ∈ ˆ MU ( A ) , then we can cho ose the re presentin g smooth c ycle ( ˜ c , α ) with ˜ c = ( p , ν ) such that p is transv erse to f and f ◦ g . In this case one e asily sees that ( f ◦ g ) ∗ ( ˜ c , α ) an d g ∗ f ∗ ( ˜ c , α ) are isomorph ic cyc les. A lgebraic & G eo metric T opology 9 (2009) 1778 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 4.2.11 W e now ha ve deﬁned a functor A 7→ ˆ MU ( A ) from smooth manifolds to graded group s. Lemma 4.15 The transf ormations R , I and a are natural . Pro of Straigh tforward. 4.2.12 W e no w deﬁne the outer product × : ˆ MU ( A ) ⊗ ˆ MU ( B ) → ˆ MU ( A × B ) . Let ˆ x ∈ ˆ MU ( A ) be repre sented by ( ˜ c , α ) , and let ˆ y ∈ ˆ MU ( B ) b e represent ed by ( ˜ e , β ) . In 3.16 w e hav e alr eady deﬁned the pro duct of cycles c × e . Here we enhance this deﬁnitio n to the g eometric le vel. Write ˜ c = ( p , ν ) and ˜ d = ( q , µ ) . Then we de ﬁne ˜ c × ˜ d : = ( p × q , ν ⊕ µ ) , where the sum o f geometric normal G struct ures ν ⊕ µ is deﬁned similarl y as in the non-geo metric case. Note that we ha ve a grade d outer product × : Ω −∞ ( A , R ) ⊗ Ω −∞ ( B , R ) → Ω −∞ ( A × B , R ) . Deﬁnition 4.16 W e deﬁne the produc t of smooth cycle s ( ˜ c , α ) × ( ˜ e , β ) by ( ˜ c × ˜ e , ( − 1) | ˆ x | R ( ˆ x ) × β + α × T ( e )) , and we deﬁne t he product ˆ x × ˆ y ∈ ˆ MU ( A × B ) to be the correspondi ng equiv alence class. This cy cle lev el deﬁnition needs a few v eriﬁcations . Lemma 4.17 (1) The outer produc t is well deﬁned. (2) It is associa tiv e, i.e. ( ˆ x × ˆ y ) × ˆ z = ˆ x × ( ˆ y × ˆ z ) , where ˆ z ∈ ˆ MU ( C ) . (3) It is graded commutati ve i n t he sense that F ∗ ( ˆ x × ˆ y ) = ( − 1) | ˆ x || ˆ y | ˆ y × ˆ x , where F : B × A → A × B is the ﬂip F ( b , a ) : = ( a , b ) . (4) The product is natural, i.e. i f f : C → A is a smooth map, then we ha ve f ∗ ˆ x × ˆ y = ( f × id B ) ∗ ( ˆ x × ˆ y ) . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1779 Pro of W e ﬁ rst show that th e c ycle le vel d eﬁnition o f the outer p roduct pas ses thro ugh the equi v alence relatio n. It is obvi ous that the outer product is bilin ear and preserve s stabili zations in both ar guments. It remains to veri fy that it prese rves zero bordisms. Let ˜ b be a ge ometric bordism da tum. T hen we can form th e geometri c bordism datum ˜ b × ˜ e (see 3.16 ). W e hav e T ( ˜ b × ˜ e ) = T ( ˜ b ) × T ( ˜ e ) so that ( ∂ ˜ b , T ( ˜ b )) × ( ˜ e , β ) = ( ∂ ˜ b × ˜ e , T ( ˜ b ) × T ( ˜ e )) = ( ∂ ( ˜ b × ˜ e ) , T ( ˜ b × ˜ e )) ∼ 0 . In orde r to see that the produc t also prese rves zero bordism in the sec ond entry we re write (4–5) ( − 1) | ˆ x | R ( ˆ x ) × β + α × T ( ˜ e ) im ( d ) ≡ ( − 1) | ˆ x | T ( ˜ c ) × β + α × R ( ˆ y ) and apply the same ar gument as abo ve. Associati vity , graded commutati vity , and natura lity hold true on the lev el of smooth cycles . T o see this, for commutati vity w e use again ( 4–5 ), and the proof of associ ativ ity is base d on similar calc ulations. 4.2.13 As usual, t he ou ter pro duct de termines a grade d commutati ve rin g str ucture by restr iction to the diagon al. Deﬁnition 4.18 W e de ﬁne the ring struc ture on ˆ MU ( A ) by ˆ x ∪ ˆ y : = ∆ ∗ ( ˆ x × ˆ y ) , wher e ∆ : A → A × A is the diago nal. The follo w ing assertion s are consequen ces of L emmas 4.14 and 4.17 . Cor ollary 4.19 A 7→ ˆ MU ( A ) is a contra -var iant functo r fr om the category of mani- folds to the cate gory of graded commutati ve rings. Lemma 4.20 The transfor mations R and I are multiplic ati ve, and we ha ve a ( α ) ∪ ˆ x = a ( α ∧ R ( ˆ x )) . Pro of Straigh tforward calcu lation. A lgebraic & G eo metric T opology 9 (2009) 1780 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 4.2.14 Recall that we ha ve ﬁxed in 3.22 , 3.26 , 3.4.7 a graded ring R over R and a formal po wer series φ ∈ R [[ z ]] 0 which determin es an R -va lued U -genus r φ . Theor em 4.21 The functor ˆ MU togeth er with the transfor mations R , I , a is a multi- plicati ve smooth extens ion of the pair ( MU , r φ ) . Pro of W e m ust verify the proper ties r equired in Deﬁnitions 2.2 and 2.3 . Most of them ha ve been sho w n abov e. W e are left with the commutati vity of (4–6) ˆ MU ( B ) R   I / / MU ( B ) r φ   Ω d = 0 ( B , R ) dR / / H ( B , R ) . and the exac tness of (4–7) MU ( B ) r φ → Ω ( B , R ) / im ( d ) a → ˆ MU ( B ) I → MU ( B ) → 0 . The commutati vity of the diagram ( 4–6 ) is a direct cons equence of ( 4–2 ). W e now discuss e xactne ss of ( 4–7 ). W e s tart with the su rjecti vity of I . Let x ∈ MU ( B ) be represente d by a cycle c . Then we can choose a geometric reﬁnement ˜ c . W e hav e dT ( ˜ c ) = 0 , a nd by L emma 4.11 there exists α ∈ Ω −∞ ( B , R ) such that T ( ˜ c ) − d α is smooth. Therefore ( ˜ c , α ) is a smoot h cyc le, and w e ha ve x = I [ ˜ c , α ] . W e now discuss exactn ess at ˆ MU ( B ) . It is clear that I ◦ a = 0 . Let ˆ x ∈ ˆ MU ( B ) , be such that I ( ˆ x ) = 0 . Then we can assume that ˆ x is of the form [ ∂ ˜ b , α ] for some geometric bordis m datum ˜ b . Hence ˆ x = a ( T ( ˜ b ) − α ) . W e now sho w exact ness at Ω ( B , R ) / im ( d ) . L et x ∈ MU ( B ) be represented by a cyc le c . Then we choose a geometric reﬁnement ˜ c , and by 4.11 a form α ∈ Ω −∞ ( B , R ) such that T ( ˜ c ) − d α is smooth. W e hav e r φ ( x ) = T ( ˜ c ) − d α . Let c = ( p , ν ) w ith p : V → B , and consider the constant map h : V → R with v alue 1 . The g eometric nor mal U - structu re of ( h , p ) : V → R × B can also be represen ted by ν . T hen ˜ b = (( h , p ) , ν ) is a geometri c bordis m datum with ∂ ˜ b = ∅ and T ( ˜ b ) = T ( ˜ c ) . It follo w s a ( d α − T ( ˜ c )) = [ ∂ ˜ b , T ( ˜ b ) − d α ] = [ ∂ ˜ b , T ( ˜ b )] = 0 . This pro ves that a ◦ r φ = 0 . Let no w α ∈ Ω ( A , R ) be such that a ( α ) = 0 . Then there exis t geometric bordism data ˜ b 0 , ˜ b 1 such that ∂ ˜ b 0 ∼ = ∂ ˜ b 1 and T ( ˜ b 0 ) − T ( ˜ b 1 ) − α ∈ im ( d ) . This alread y implies that α is closed. W e constr uct a geomet ric cy cle ˜ c such that T ( ˜ c ) = T ( ˜ b 0 ) − T ( ˜ b 1 ) by gluein g the bordism data along their common bounda ry . Then [ α ] = [ T ( ˜ c )] = r φ ([ c ]) in de Rham cohomol ogy . A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1781 4.3 Smooth MU -orientations 4.3.1 As before we ﬁx a graded ring R over R and a formal power series φ ∈ R [[ z ]] 0 . Let ˆ MU be the smo oth extens ion of ( MU , r φ ) as in Theor em 4.21 with str ucture maps R , a , I . If q : V → A is a proper MU -oriente d map, then we ha ve an integr ation q ! : MU ( V ) → MU ( A ) (see 3.3.10 ). Under the assumptio n that q is a submers ion we introduce the not ion of a smooth MU -orienta tion and deﬁne the in tegrati on map q ! : ˆ MU ( V ) → ˆ MU ( A ) . 4.3.2 Let q : V → A be a proper submersion. Deﬁnition 4.22 A representat iv e of a smooth MU -orien tation of q is a pair ˇ c : = ( ˜ c , σ ) , where ˜ c is a geomet ric cycle with und erlying map q : V → A and σ ∈ Ω − 1 ( V , R ) . A repres entati ve of a smooth MU -orientation of q induces in pa rticular an M U - orient ation of q . 4.3.3 W e no w introdu ce an equi valenc e relatio n ∼ called stable homo topy on t he set of repre sentati ves of smooth MU -orientati ons of q . Deﬁnition 4.23 W e deﬁne the l -fold stabi lization of ( ˜ c , σ ) by ( ˜ c , σ )( l ) : = ( ˜ c ( l ) , σ ) . Let h i : A → R × A d enote the inclusion s h i ( a ) : = ( i , a ) , i = 0 , 1 . Consider a geometric cyc le ˜ d = ( p , µ ) ove r R × A w ith u nderlyin g map p : = id R × q : R × V → R × A . It gi ves rise to a clos ed form φ ( ∇ µ ) ∈ Ω 0 ( R × V , R ) . Let ˜ c i : = h ∗ i ˜ d , ˜ c i = ( q , ν i ) . Deﬁnition 4.24 W e call ˜ d a homotop y between ˜ c 0 and ˜ c 1 . Deﬁnition 4.25 W e deﬁne the transg ression form ˜ φ ( ∇ ν 1 , ∇ ν 0 ) : = Z [0 , 1] × V / V φ ( ∇ µ ) ∈ Ω − 1 ( A , R ) / im ( d ) . Since the underlying cycle d of ˜ d is a product, and since the space of geometri c reﬁnements of d is contractible , the transgre ssion form is well deﬁned independ ent of the choice of the homotop y (this is a standard argumen t in the theory of characterist ic forms). By Stokes’ theore m the transgres sion satisﬁes (4–8) d ˜ φ ( ∇ ν 1 , ∇ ν 0 ) = φ ( ∇ ν 1 ) − φ ( ∇ ν 0 ) . Deﬁnition 4.26 W e call two representa tiv es of a smooth MU -orien tation ( ˜ c i , σ i ) ho- motopic if there ex ists a homotopy ˜ d from ˜ c 0 to ˜ c 1 , and σ 1 − σ 0 = ˜ φ ( ∇ ν 1 , ∇ ν 0 ) . A lgebraic & G eo metric T opology 9 (2009) 1782 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 4.3.4 W e no w deﬁne equi va lence of represe ntati ves of smooth MU -orient ations. Deﬁnition 4.27 Let ∼ be the minimal equi val ence relation on the set of representativ es of smooth MU -orien tations on q such that (1) ( ˜ c , σ ) ∼ ( ˜ c ( l ) , σ ) (2) ( ˜ c 0 , σ 0 ) ∼ ( ˜ c 1 , σ 1 ) , if ( ˜ c 0 , σ 0 ) and ( ˜ c 1 , σ 1 ) are homot opic. A smooth MU -orien tation of q is an equi valen ce class of repres entati ves of smooth MU -orien tations w hich we will usual ly write as o : = [ ˜ c , σ ] . 4.3.5 Let ˜ c : = ( q , ν ) and ˇ c : = ( ˜ c , σ ) be a representat iv e of a smooth MU -orientation . Deﬁnition 4.28 W e deﬁne A ( ˇ c ) : = φ ( ∇ ν ) − d σ ∈ Ω 0 ( V , R ) . Lemma 4.29 The form A ( ˇ c ) only depends on the smooth MU -orien tation [ ˇ c ] repre- sented by ˇ c . Pro of This immediately follo ws from ( 4–8 ) and the deﬁnition of homotop y . Belo w we will write A ( o ) : = A ( ˇ c ) , where o : = [ ˇ c ] . 4.3.6 In the followin g two parag raphs we deﬁne the operatio ns of pull-back and composi tion of smooth MU -orientation s. W e start with the pull-back. Let f : B → A be a smooth map which is trans verse to q . T hen we ha ve the Cartes ian diagram W Q   F / / V q   B f / / A . Deﬁnition 4.30 W e deﬁne the pull-back of a representati ve of a smooth MU -orien tation of q by f ∗ ( ˜ c , σ ) : = ( f ∗ ˜ c , F ∗ σ ) (see 4.2.6 ) w hich is a represe ntati ve of a smooth MU - orient ation of Q . Lemma 4.31 The pull-back is compatib le with the equi v alence relation . It induce s a functo rial pull-bac k of smooth MU -orien tations. W e ha ve A ( f ∗ o ) = F ∗ A ( o ) . Pro of It is cl ear that the p ull-back is compa tible with stabiliz ation. Let ˜ d be a homo- top y from ˜ c 0 to ˜ c 1 . Then ( id R × f ) ∗ ˜ d is a homoto py from f ∗ ˜ c 0 to f ∗ ˜ c 1 . Furthermore, one checks that ˜ φ ( ∇ f ∗ ν 1 , ∇ f ∗ ν 0 ) = f ∗ ˜ φ ( ∇ ν 1 , ∇ ν 0 ) . These fo rmulas impl y th at th e pu ll- back preserves homotopic represent ativ es of smooth MU -orienta tions. W e conclud e that the pull-back is wel l deﬁned on the l ev el equiv alence class es. Functoriali ty and the fact that A ( f ∗ o ) = F ∗ A ( o ) are easy to see. A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1783 4.3.7 W e now d eﬁne the composition of smooth MU -orientat ions. Let p : A → B be a second proper submersion, and let ( ˜ d , θ ) , ˜ d = ( p , µ ) , be a representa tiv e of a smooth MU -orienta tion of p . Let o q = [ ˜ c , σ ] and o p : = [ ˜ d , θ ] . By ˜ d ◦ ˜ c we denote the composit ion of geometri c cycles which is bas ed on Deﬁnition 3.11 . Deﬁnition 4.32 W e deﬁne o p ◦ o q : = [ ˜ d ◦ ˜ c , A ( o q ) ∧ q ∗ θ + σ ∧ q ∗ φ ( ∇ µ )] . The deﬁnitio n require s some veriﬁca tions. Lemma 4.33 The composi tion of smo oth MU -orien tations is well deﬁned, compat ible with pull- back, and functori al. Pro of W e ﬁrst show t hat the compo sition is well deﬁne d. It is clear th at the com- positi on is compatib le with stabiliz ation. Let ˜ b be a homotop y f rom ˜ c 0 to ˜ c 1 . Then pr ∗ 2 ˜ d ◦ ˜ b is a homotopy fr om ˜ b ◦ ˜ c 0 to ˜ b ◦ ˜ c 1 , w here pr 2 : R × B → B is th e p rojection . W e furthe r calculate (using the properti es stated in Lemm a 4.2 ) ( σ 1 − σ 0 ) ∧ q ∗ φ ( ∇ µ ) = ˜ φ ( ∇ ν 1 , ∇ ν 0 ) ∧ q ∗ φ ( ∇ µ ) = ˜ φ ( ∇ µ ◦ ν 1 , ∇ µ ◦ ν 0 ) . This calcul ation implies that the compositio n ( ˜ d , θ ) ◦ . . . pres erves homotopic repre- sentat iv es. Let u s no w con sider a homot opy ˜ e from ˜ d 0 to ˜ d 1 W e get a homotop y ˜ e ◦ ˜ c from ˜ d 0 ◦ ˜ c to ˜ d 1 ◦ ˜ c . Furthermo re we re write (note that we work modulo im ( d ) ) A ( o q ) ∧ q ∗ θ + σ ∧ q ∗ φ ( ∇ µ ) = φ ( ∇ ν ) ∧ q ∗ θ + σ ∧ q ∗ A ( o p ) . W e ha ve φ ( ∇ ν ) ∧ q ∗ ( θ 1 − θ 0 ) = φ ( ∇ ν ) ∧ q ∗ ˜ φ ( ∇ µ 1 , ∇ µ 0 ) = ˜ φ ( ∇ µ 1 ◦ ν , ∇ µ 0 ◦ ν ) . Hence · · · ◦ ( ˜ c , ν ) preserve s homotopic re presentat iv es. This ﬁnishes th e pro of that the composi tion is well deﬁned. A lgebraic & G eo metric T opology 9 (2009) 1784 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up 4.3.8 The composition of smooth MU -orientati ons is associati ve and compatible with pull-b ack. For completen ess let us state the second f act in greater detail. Let r : Q → B be a map w hich is transv erse to q and p ◦ q . Then we hav e the composi tion o f pull-ba ck diagra ms Q × B V / /   V q   Q × B A   s / / A p   Q r / / B . In this situa tion we ha ve s ∗ o p ◦ r ∗ o q = r ∗ ( o p ◦ o q ) . W e lea ve the details of the straight forward proof to the reader . 4.4 The push-f orward 4.4.1 Let p : V → A be a proper submers ion with a smooth MU -orienta tion o p : = [ ˜ d , σ ] , ˜ d = ( p , ν ) . In the follo wing, ( ˜ c , α ) denot es a smooth cycle on V , and w e use the notatio n Z V / A : = p ! : Ω −∞ ( V , R ) → Ω −∞ ( A , R ) for the inte gration of forms. Deﬁnition 4.34 W e deﬁne the push-f orward on the le vel of c ycles by p ! ( ˜ c , α ) = ( ˜ d ◦ ˜ c , Z V / A ( φ ( ∇ ν ) ∧ α + σ ∧ R ( ˜ c , α ))) . Lemma 4.35 For ﬁxed ( ˜ d , σ ) the push-forw ard preserv es eq uiv alence of smooth cyc les. Furthermore, the induced map p ! : ˆ MU ( V ) → ˆ MU ( A ) only depends on the equi valenc e class [ ˜ d , σ ] of repre sentati ves of the smooth MU -orien tation. Pro of It is clear that the push-forw ard is addit iv e and compatib le with stabilizat ion. Let now ˜ b be a geometric bordism datum over V . Let p r : R × A → A be the A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1785 projec tion and form ( ˜ e , θ ) : = pr ∗ ( ˜ d , σ ) . Then ˜ e ◦ ˜ b is a bord ism datum, and we ha ve T ( ˜ e ◦ ˜ b ) = R V / A φ ( ∇ ν ) ∧ T ( ˜ b ) . W e calculate p ! ( ∂ ˜ b , T ( ˜ b )) = ( ˜ d ◦ ∂ ˜ b , Z V / A φ ( ∇ ν ) ∧ T ( ˜ b )) = ( ∂ ( ˜ e ◦ ˜ b ) , T ( ˜ e ◦ ˜ b )) . This equal ity implies that p ! preser ves zero bord isms. For a ﬁx ed r epresenta tiv e ( ˜ d , σ ) of the s mooth MU -orientat ion we no w ha ve a well deﬁned map p ! : ˆ MU ( V ) → ˆ MU ( A ) . Next we sho w that it only depend s on the smo oth orient ation represen ted by ( ˜ d , σ ) . Again it is clear that stabili zation o f the representativ e of the smooth orienta tion do es not change p ! . W e no w consider a homotopy ˜ b from ( ˜ d 0 , σ 0 ) to ( ˜ d 1 , σ 1 ) . The idea of the argu ment is to translate this homotopy into a bordis m datum. T o thi s end w e ﬁrst co nsider a model case. L et κ : R → R be deﬁned by κ ( x ) : = x − x 2 . Then κ − 1 ( { [0 , ∞ ) } ) = [0 , 1] . W e choose a representa tiv e o f the stable normal bundle of κ with a geometric U -structu re µ such that ˜ κ = ( κ, µ ) is a geometri c bordism datum. Let pr 1 : R × A → R denote the projection. The composi tion ˜ r : = p r ∗ 1 ˜ κ ◦ ˜ b is no w a bordism datum. Let ρ denote the represent ativ e of the geometr ic U -structur e on the normal b undle of r . W e consider ˜ r ◦ pr ∗ 2 ˜ c as a geometric bordism datum with ∂ ( ˜ r ◦ pr ∗ 2 ˜ c ) = ˜ d 0 ◦ ˜ c + ( ˜ d 1 ◦ ˜ c ) op , where ( · ) op indica tes a ﬂip of the orien tation. Fix ˜ c = ( q , ν ) with q : U → V and ˜ d i = ( p , λ i ) . T ( ˜ r ◦ pr ∗ 2 ˜ c ) = Z q − 1 r − 1 ([0 , ∞ ) × V ) / A φ ( ∇ ρ ) ∧ φ ( ∇ ν ) = Z V / A ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ Z U / V φ ( ∇ ν ) ! On the othe r hand Z V / A ( φ ( ∇ λ 1 ) − φ ( ∇ λ 0 )) ∧ α + ( σ 1 − σ 0 ) ∧ R ( ˜ c , α ) = Z V / A d ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ α + ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ R ( ˜ c , α ) = Z V / A ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ d α + ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ R ( ˜ c , α ) = Z V / A ˜ φ ( ∇ λ 1 , ∇ λ 0 ) ∧ Z U / V φ ( ∇ ν ) ! A lgebraic & G eo metric T opology 9 (2009) 1786 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up These two equ ations toge ther sho w that ( ˜ d 1 , σ 1 ) ◦ ( ˜ c , α ) ∼ ( d 0 , σ 0 ) ◦ ( ˜ c , α ) . Indee d ( ˜ d 0 ◦ ˜ c + ( ˜ d 1 ◦ ˜ c ) op , Z V / A ( φ ( ∇ λ 1 ) − φ ( ∇ λ 0 )) ∧ α + ( σ 1 − σ 0 ) ∧ R ( ˜ c , α )) = ( ∂ ( ˜ r ◦ pr ∗ 2 ˜ c ) , T ( ˜ r ◦ pr ∗ 2 ˜ c )) . 4.4.2 Lemma 4.36 The follo wing diagram commutes. Ω ( V , R ) / im ( d ) R V / A A ( o p ) ∧ ...   a / / ˆ MU ( V ) p !   I / / R * * MU ( V ) p !   Ω ( V , R ) R V / A A ( o p ) ∧ ...   Ω ( A , R ) / im ( d ) a / / ˆ MU ( A ) I / / R 4 4 MU ( A ) Ω ( A , R ) Pro of Commutati vity of the left square follo ws from partial integra tion Z V / A ( φ ( ∇ ν ) ∧ α − σ ∧ d α ) = Z V / A ( φ ( ∇ ν ) − d σ ) ∧ α = Z V / A A ( o p ) ∧ α . For t he right square w e use T ( ˜ d ◦ ˜ c ) = Z V / A φ ( ∇ ν ) ∧ T ( ˜ c ) , which implies R ( p ! ( ˜ c , α )) = T ( ˜ d ◦ ˜ c ) − d Z V / A ( φ ( ∇ ν ) ∧ α + σ ∧ R ( ˜ c , α )) = Z V / A ( φ ( ∇ ν ) ∧ T ( ˜ c ) − φ ( ∇ ν ) ∧ d α − d σ ∧ R ( ˜ c , α )) = Z V / A ( φ ( ∇ ν ) − d σ ) ∧ R ( ˜ c , α ) = Z V / A A ( o p ) ∧ R ( ˜ c , α ) . Commutati vity of the middle square is a direct conseq uence of geometric descripti on of p ! : M U ( V ) → MU ( A ) (see 3.3.10 ). A lgebraic & G eo metric T opology 9 (2009) Landweber e xact formal g r ou p laws and smooth cohomology theories 1787 4.4.3 Let p : V → A be as before with the smooth MU -orientat ion o p : = [ ˜ d , σ ] . W e furthe rmore c onsider a proper submersion q : A → B with a smooth MU -orientatio n o q : = [ ˜ e , ρ ] , ˜ e = ( q , µ ) . Let r : = q ◦ p : V → B be equ ipped w ith the compo sed smooth MU -orientatio n o r : = o q ◦ o p (see Deﬁnition 4.32 ) Lemma 4.37 The push- forward is functo rial, i.e. we ha ve the equalit y r ! = q ! ◦ p ! : ˆ MU ( V ) → ˆ MU ( B ) . Pro of The equality holds on the smooth cycle lev el. The proof is a straightforw ard calcul ation of both sides by inserti ng the deﬁnitions and using the right square in Lemma 4.36 . 4.4.4 Let p : V → A be a proper smoothly MU -oriente d map as above , and let f : B → A be a seco nd smooth map so that we get a Cartesi an diagra m W P   F / / V p   B f / / A . The map P has an in duced smooth MU -orientat ion o P : = f ∗ o p (see Deﬁnition 4.30 ). Lemma 4.38 The push- forward commutes with pull- back, i.e. w e ha ve the equali ty P ! ◦ F ∗ = f ∗ ◦ p ! : ˆ MU ( V ) → ˆ MU ( B ) . Pro of The equality holds true on the lev el of smooth cyc les ( ˜ c , α ) whose underly ing map is transv erse to F . By deﬁnition we ha ve o P = [ f ∗ ˜ d , f ∗ σ ] . Further more, it follo ws immediatel y fro m the deﬁnitions that f ∗ ( ˜ d ◦ ˜ c ) = f ∗ ˜ d ◦ F ∗ ˜ c . The ﬁnal ingred ient of the veri ﬁcation is the identity f ∗ ◦ Z V / A . . . = Z W / B ◦ F ∗ . . . . 4.4.5 Let p : V → A be a smoothly MU -oriented prope r submersi on as abov e. Lemma 4.39 The projec tion formula holds true, i.e. for x ∈ ˆ MU ( A ) and y ∈ ˆ MU ( V ) we ha ve p ! ( p ∗ x ∪ y ) = x ∪ p ! y . 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NWF I - Mathematik, Universit ¨ at R egensburg, 9304 0 Regensburg, Germa ny Mathematisches Institut, Georg-August-Un iv ersit ¨ at G ¨ ottingen, Bunsenstr . 3, 37 073 G ¨ ottingen, German Mathematisches Institut, Georg-August-Un iv ersit ¨ at G ¨ ottingen, Bunsenstr . 3, 37 073 G ¨ ottingen, Germany A lgebraic & G eo metric T opology 9 (2009) 1790 Ulrich Bunke, Thomas Schick, I ngo Schr ¨ oder and Moritz W ietha up Mathematisches Institut, Georg-August-Un iv ersit ¨ at G ¨ ottingen, Bunsenstr . 3, 37 073 G ¨ ottingen, Germany ulrich .bunk e@mathematik.uni-regensburg.de , schi ck@un i-math .gwdg.de , ischro ed@un i-math.gwdg.de , wie thaup@ uni-m ath.gwdg.de http:/ /www. mathematik.uni-regensburg.de/Bunke/ , http:/ /www. uni-math.gwdg.de/schick Receiv ed: 24 September 2008 Revised: 15 July 2009 A lgebraic & G eo metric T opology 9 (2009)

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