Obstructions to stably fibering manifolds

OBSTR UCTIONS TO ST ABL Y FIBER ING MANIF OLDS W OLFG ANG STEIMLE Abstract. Is a given map b et wee n compact topol ogical manifolds homotopic to the pro jection map of a fi b er bundle? In this paper obstructions to this question are int ro duced wi th v a lues in higher algebraic K -theory . Their v an - ishing implies that the given map fib ers stably . The methods also provide results for the corresp onding uniqueness question; moreo v er they apply to the fiberi ng of Hil bert cub e manifolds, generalizing results by Chapman-F err y . Contents 1. Int ro duction 1 2. Definition of the obstructions 3 3. Pro of o f Theorems 1 .1 a nd 1.2 6 4. Change of ba se a nd total space 9 5. Examples I: Elementary applications 12 6. Examples II: Stable vs. uns ta ble and blo ck fib ering and TO P vs. DIFF 13 7. Examples II I: Results o f Chapman-F erry 19 8. Compariso n with the obstructions b y F arrell-L ¨ uck-Steimle 21 Appendix A. Fiber ing Q -manifolds 22 References 25 1. Introduction Given a map f : M → B b etw een closed manifolds, is f homotopic to the pro jec- tion map of a fib er bundle of closed manifolds? Can the different wa ys of fib ering f b e cla ssified? These questions hav e a long tradition in geometric topo logy . In the research on high-dimensional ma nifolds, the in vestigation of these questions has accompanied the developmen t of the s ub ject since its b eginnings : The fib ering the- orem of Br owder-Levine [BL66] was an ea r ly application of sur gery tec hniques and the h -co bo rdism theorem. F urther results have been obtained by F a rrell [F ar72] and Siebe nma nn [Sie70] for B = S 1 , using the s - cob ordism theorem and co mputations of the Whitehead g roup of s e mi-direct pr o ducts G ⋊ α Z . Casson [Cas67] pioneered the study of fib ering questions for higher-dimensio nal base manifolds b y considering B = S n , applying techniques of sur gery theory . Quinn [Qui70] was the first to systematica lly de s crib e blo ck s tructure space s using the L -theoretic ass em bly map and to develop a general obstruc tio n theor y to “ blo ck fiber ing” a given map. Date : M a y 2011. 2010 Mathematics Subje ct Classific ation. 55R10, 19J10, 57N20. Key wor ds and phr ases. Fibering a m anif old, algebraic K -theory of spaces. 1 2 WOLF GANG STEIMLE In the Q -manifold world, Chapman- F err y [CF7 8] obtained the most general re- sults av ailable so far. Most r ecent ly , in the finite-dimensio nal case, joint work of the autho r with F arrell and L ¨ uc k [FLS0 9] shows how the o bs tructions defined by F a rrell and Sie b enmann ov er S 1 can be ge neralized to arbitrary base s paces (where, how ev er, they stop be ing a complete set of obstructions). In the lig h t of the development of parametrized h -cob ordism theor y since the 1970s , this work re - fo c us es o n the role of alg ebraic K -theor y in fib ering questions. As we will see, higher alg ebraic K -theor y of spaces provides o bstructions for b oth questions of existence and uniqueness . Moreover, the v anishing o f these o bstructions has a concrete geo metric meaning : The obstruc tio ns constr uc ted in this work for m a complete set o f obstructions to fib ering manifolds stably . Here stabilization refer s to cr ossing the total space with disks of sufficien tly high dimension, thus leaving the category of closed manifolds. In fact, the theo r y of stably fib ering manifolds is b est formulated and proved entirely in the world of compact manifolds with bo undary (whic h we call compact manifolds for short). More concre tely , let f : M → B be a map b etw een co mpact top ologica l mani- folds. Then, by definitio n, f stably fib ers if, for some n ∈ N , the comp osite f ◦ Pro j : M × D n → M → B is homotopic to the pro jection map o f a fib er bundle whose fib ers are compact top ological manifolds. The following questio ns will b e dealt with: • When do es f stably fib er? • How many different wa ys ar e there for f to stably fib er? Denote by C the set of a ll bundle maps g : M × D n → B for some n which ar e ho motopic to f ◦ Pro j. W e define tw o elements to b e e quivalent , a nd write g ∼ g ′ , if after further s tabilizing there is a bundle homeomorphism i : M × D N → M × D N from g to g ′ (i.e. i ◦ g = g ′ ), such that i is homotopic to the identit y map. The precise question is then: H ow can C / ∼ b e descr ibed? F a ctor f in to a homotopy equiv alence λ follo wed by a fibration p . Under a finiteness a ssumption on the fib e r F o f p , tw o o bstructions will b e defined: • W all( p ) ∈ H 0 ( B ; Wh( F )), which is a n obstruction to reducing p to a fib er bundle of compa ct manifolds. Here the term Wh is us e d to denote the (co n- nective top olo gical) Whitehead sp ectrum as defined by W aldhausen. It is defined in terms of algebraic K -theory of spa ces and is clo sely connected to the classification of parametr ized h -cob or dis ms. The term H 0 ( B ; Wh( F b )) denotes a sp ecific generalize d co homology gr oup of B with resp ect to the Whitehead spectrum of the fib ers, where the coefficients ar e t wisted ac - cording to the data of the fibration p . • If W all( p ) v anishes, then there is a second obstr uction o ( f ) lying in the cokernel of a sp ecific map π 0 ( β ) : H 0 ( B ; ΩWh( F )) → Wh( π 1 M ) . See s e c tion 2 for a pr ecise e x planation of ter ms. Theorem 1.1 (Existence) . The map f stably fib ers if and only if the fib ers of p ar e fin itely dominate d, and W all( p ) and o ( f ) b oth vanish. Theorem 1.2 (Classifica tion) . If f stably fib ers, then the set C / ∼ is in bije ct ion with the kernel of the map π 0 ( β ) . OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 3 In a s e nse, this paper is a companion pap er to [Ste11] since Theo rems 1.1 and 1.2 are r ather forma l co ns equences of the results from tha t pa per in combination with the “ Riemann-Ro ch theore m with conv erse” by Dwyer-W eiss-Williams [DWW03]. Apart from defining the obstructions (in section 2) and proving Theo rems 1.1 and 1.2 (in s ection 3), w e provide several examples, emphasizing the “change o f total space problem” where the mor e co mplicated W a ll obstruction do es not play a role. In sec tion 7 we show that the r esults of [CF78] on fibering co mpact Q -manifolds ov er compact ANRs can b e viewed a s sp ecia l cases of the re sults pr esented here. The conten t o f section 8 is to co mpa re the obstructions defined here with those of [FLS0 9]. An app endix collects the r e sults needed to relate the stable fiber ing problem presented here with the Q -manifold fib ering problem. Ac kno wledgemen t. This w ork is part of my PhD thesis, written at the Univ ersity of M ¨ unster. I thank my advisor W olfgang L ¨ uec k fo r his c onstant encourag emen t and supp ort and Bruce Williams for drawing my interest to the s table fib ering problem and sharing his ideas. Moreov er I am grateful to Arthur Bar tels, Diarmuid Crowley , Bruce Hughes, Matthias Kreck and Tibor Mack o for ma n y dis cussions and suggestions . 2. Definition of the obstructions Throughout this s e ction, let f : M → B b e a ma p betw een compact top olo gical manifolds, and let f = p ◦ λ b e a factorization int o a homotopy e q uiv alence follow ed by a fibr ation p : E → B . A functor fr om spa ces to spac e s is called homotopy inv ar iant if it sends ho- motopy equiv alences to homotopy equiv alences. Giv en such a functor Z a nd a fibration p : E → B , with fib er F , Dwyer-W eiss-Williams [D WW03] define a fibra - tion Z B ( E ) → B , with fib er Z ( F ), essentially by applying Z “ fiber -wise” to p . As W aldhausen’s functor A ( X ) is homotopy inv ariant, this co nstruction leads to a fibration A B ( E ) → B . Suppo se that the fib er F of p is finitely dominated (see b elow). In this situa- tion the “pa rametrized A -theory characteris tic” [DWW03] defines a sectio n of the fibration A B ( E ) → B , up to homo topy: χ ( p ) ∈ Γ  A B ( E ) ↓ B  The natural transformation from A ( X ) to the co nnec tiv e to po lo gical Whitehead sp ectrum Wh ( X ) induces a map Γ  A B ( E ) ↓ B  → Γ  Wh B ( E ) ↓ B  . Definition 2.1 . The p ar ametrize d W al l obstruction W a ll( p ) ∈ π 0 Γ  Wh B ( E ) ↓ B  =: H 0 ( B ; Wh( F )) of the fibration p is the image of the parametrized A -theor y characteris tic under this map. 4 WOLF GANG STEIMLE The par ametrized W a ll obstruction only depends of the fiber homotopy type of p in the following sense: If ϕ : p → p ′ is a fib er homoto p y equiv alence b etw een fibrations with fib ers F and F ′ resp ectively , then the induced isomor phism ϕ ∗ : H 0 ( B ; Wh( F )) → H 0 ( B ; Wh( F ′ )) sends W all( p ) to W a ll( p ′ ). In this sense, W a ll( p ) only dep ends on f rather than on the choice o f factorization f = p ◦ λ . It follows from the “Riemann-Ro ch theorem with conv erse” [DWW03]: Theorem 2.2 . The p ar ametrize d Wal l obstruction is zer o if and only if ther e is a factorization f = p ◦ λ wher e λ is a homotopy e quivalenc e and p is a fib er bund le, with fib ers c omp act m anifold s. Suppo se now that we a re given such a factorization. Supp o se for simplicity that B is connected and co ns ider the comp osite β : Γ  ΩWh B ( E ) ↓ B  → ΩWh( F ) χ ( B ) · i ∗ − − − − − → ΩWh( E ) where the first ma p is the r estriction map onto a chosen base p oint of b a nd the second map is induced by the inclusion F := p − 1 ( b ) → E followed by m ultiplication with the Euler characteris tic χ ( B ) ∈ Z . Definition 2.3. The fibering obstruction o ( f ) is the clas s of the Whitehead torsio n τ ( λ ) in the co kernel o f π 0 ( β ) : H 0 ( B ; ΩWh( F )) → π 0 ΩWh( E ) ∼ = Wh( π 1 E ) . Remark 2.4. (i) Rec a ll tha t a space X is called finitely dominated if there is a finite CW co mplex Y together w ith maps i : X → Y and r : Y → X s uc h that r ◦ i ≃ id X . (ii) If X is no t connected, then the group Wh( π 1 X ) should b e r ead as the direct sum of the Whitehead gro ups of π 1 ( C ) for all path comp onents C of X . (iii) If B is no t co nnected, the map β is defined a s the sum of the corr esp onding maps for the individual comp onents. (iv) See section 3 for a pro of tha t the fiber ing obstruction doe s not depe nd o n the choice of factoriza tion f = p ◦ λ . W e finish this section by a sp ectral sequence analysis of the par ametrized W all obstruction. Theorem 2.5 . (i) L et E → B b e a fibr ation over a CW c omplex, with fi b er F b over b . Ther e is a 4th quadr ant sp e ctra l se qu enc e E p,q 2 = H p ( B ; π − q Wh( F b )) = ⇒ H p + q ( B ; Wh( F b )) , wher e t he E 2 -term c onsists of or d inary c ohomolo gy with twiste d c o efficients in the system of ab eli an gr oups { b 7→ π − q Wh( F b ) } . (ii) If B is d -dimensional, d < ∞ , then the c orr esp onding filtr ation · · · ⊃ F p,q ⊃ F p +1 ,q − 1 ⊃ . . . OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 5 of H p + q ( B ; Wh( F b )) is finite, and the sp e ctr al se quenc e c onver ges in the str ongest p ossi ble sense, i.e. we have F 0 ,n = H n ( B ; Wh( F b )) for a ll n F d +1 ,n − d − 1 = 0 for a ll n F p,q / F p +1 ,q − 1 ∼ = E p,q ∞ for a ll p , q (iii) U nder t he e dge homomorphism e : H 0 ( B ; Wh( F b )) → H 0 ( B ; π 0 Wh( F b )) ⊂ Y [ b ] ∈ π 0 B ˜ K 0 ( Z [ π 1 F b ]) , the image of W all( p ) is the finiteness obstru ction of the fib er. (iv) Supp ose that al l the fib ers ar e homotopy e quival ent to fin ite CW c omplexes, so that e (W all( p )) = 0 . L et γ : S 1 → B b e a lo op. The natura l ly define d se c ondary homomorphi sm ker( e ) → H 1 ( B ; π 1 Wh( F b )) , fol lowe d by the r estriction map γ ∗ : H 1 ( B ; π 1 Wh( F b )) → H 1 ( S 1 ; π 1 Wh( F b )) ∼ = Wh( π 1 F b ) π 1 ( S 1 ) (c oi nvariants u nder the π 1 ( S 1 ) -action) sends W all( p ) to the element define d by the Whitehe ad t orsion of the fib er tr ansp ort t γ along γ . Remark 2.6. In the situatio n of (iv), the Whitehead tor sion τ ( t γ ) ∈ Wh( π 1 F b ) is not well-defined, since F b comes with no CW structure. How ever, after cho osing a homotopy equiv alence h : X → F b from some CW co mplex X , o ne may co ns ider the Whitehead torsion h ∗ τ ( h − 1 ◦ t γ ◦ h ) ∈ Wh( π 1 F b ); it is not har d to see that its class in Wh( π 1 F b ) π 1 ( S 1 ) is indep endent of the choice of X and h . Pr o of of The or em 2.5. F or par t (i) and (ii), as sume that B is the geometric re al- ization of a simplicial set B • . The rule which assig ns to a simplex σ of B • the pull-back E σ := | σ | ∗ E defines a functor on the simplex category s imp B • . Ther e is a w eak ho mo topy equiv a lence [DWW03] Γ  Wh B ( E ) ↓ B  ≃ holim σ ∈ sim p B • Wh( E σ ) . The sp ectra l sequence in question is the Bo usfield-Kan sp ectral sequence of the right-hand side. Part (iii) a nd (iv) fo llow from a clo se ex amination of the homo- morphisms in questio n and the identifi cation o f the higher Whitehead torsion with the cla ssical one in the unpar ametrized se tting [Ste11]. F or mor e details , consult [Ste10].  6 WOLF GANG STEIMLE 3. P r oof of Theorems 1.1 and 1.2 Given a fibration p : E → B , the structur e sp ac e S n ( p ) is de fined as the geometric realization of the s implicial set S n ( p ) • , where a k -simplex is given b y a comm utative diagram (1) E ′ q # # G G G G G G G G G λ ≃ / / E × ∆ k p × id ∆ k y y s s s s s s s s s s B × ∆ k in which q is a bundle of compact top ological manifolds and λ is a (fib er) homo topy equiv a lence. The simplicial op era tions are induced by pull-back. (Strictly sp eaking we alwa ys have to as s ume that E ′ is a subset of B × U for a chosen “ univ erse” U . See [Ste1 1 ] for mor e details .) If B is a p oint, then we wr ite S n ( E ) for S n ( p ). In the case wher e B is a compact top ological k -manifold, the ge o metric assembly map α : S n ( p ) → S n + k ( E ) is essentially given by “ forgetting B ”. More precis ely it sends a simplex ( q , λ ) as in the diagra m (1) to the simplex ( q ′ , λ ) where q ′ : E ′ → ∆ k is the comp osite of q with the pro jection, and λ is now consider ed a s a fib er homo top y equiv alence over ∆ k only . Here is the key observ ation that connects the fib ering question with the geo metric assembly map. Lemma 3.1. let f : M n + k → B k b e a map b etwe en c omp act top olo gic al manifolds, and let f = p ◦ λ b e a factorization of f into a homotopy e quiv alenc e, fol lowe d by a fibr atio n p : E → B . (i) The fibr ation p is fib er homotopy e quivalent to a bund le of c omp act t op o- lo gic al n -manifolds if and only if S n ( p ) is non-empty. (ii) f is homotopic t o a bun d le of n -manifolds if and only if the element define d by λ : M → E is in the image of the map π 0 ( α ) : π 0 S n ( p ) → π 0 S n + k ( E ) . (iii) If g , g ′ : M → B ar e two fib er bund le pr oje ctions homotopic to f , say t hat they ar e e quival ent if t her e is a c ommutative diagr am M i ∼ = / / g A A A A A A A A M g ′ ~ ~ } } } } } } } } B wher e i is a home omorph ism which is homotopic t o the identity. Then, the e quivalenc e classes of fib er bund le pr oj e ctio ns homotopic to f ar e in bije ction t o the pr eimage of [ λ ] under the m ap α : π 0 S n ( p ) → π 0 S n + k ( E ) . Pr o of. (i) is true b y definition, and (ii) follows fro m (iii). Statement (iii) is basically a close examination o f the definition. OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 7 Indeed, as S n ( p ) • is Kan, an element in the preima ge of [ λ ] under π 0 ( α ) is given by a co mm utative diagra m (2) N q @ @ @ @ @ @ @ @ ϕ ≃ / / E p   ~ ~ ~ ~ ~ ~ ~ B with q a bundle o f compact n -ma nifo lds, s uch that N is homeomor phic to M via a map under which ϕ co r resp onds to λ up to homotopy . It defines the same elemen t as the diagram N ′ q ′ A A A A A A A A ϕ ′ ≃ / / E p   ~ ~ ~ ~ ~ ~ ~ B if and only if b oth ele ments form the b oundar ies of a similar diagra m over B × I . This mea ns that b oth diag rams ex tend to a diagram N ′ q ′ ' ' N N N N N N N N N N N N N i ∼ = / / ϕ ′ % % N q   ϕ / / E p x x p p p p p p p p p p p p p B with i a homeomorphism o f bundles ov er B , such that the low er triangles commut e strictly and the upp er tr iangle commutes up to a homotopy ov er B . Suppo se that f is homotopic to a bundle g of n -manifolds. Then a choice of homotopy from f to g induces a fiber homotopy equiv alence ϕ : M → E from g to p together with a homotopy from ϕ to λ . Setting in the diag r am (2 ) N := M and q := g we obtain a co r resp onding element in the pre ima ge of [ λ ] under α . It is not hard to see that this r ule induces a bijection betw een equiv alence c la sses of fib er bundle pro jections homo topic to f and the preimage of [ λ ].  Denote by I the unit in terv a l a nd b y p × I the obvious fibration E × I → B . The sta bilization map σ : S n ( p ) → S n +1 ( p × I ) sends ( q , λ ) to ( q × I , λ × id I ); let S ∞ ( p ) := ho colim n S n ( p × I n ) . Clearly the geometric assembly map extends to a stable version α : S ∞ ( p ) → S ∞ ( E ) . Here is a stabilize d version of Lemma 3.2. I t follows from Lemma 3 .2 tog ether with the fact that colim n π 0 S n ( p ) ∼ = − → π 0 ho colim n S n ( p ) . Lemma 3.2 . (i) The fibr ation p is fib er homotopy e quiva lent to a bund le of c o mp act t op olo gic al manifolds if and only if S ∞ ( p ) is non empty. 8 WOLF GANG STEIMLE (ii) A map f : M → B stably fib ers if and only if t he element define d by λ : M → E is in the image of the map π 0 ( α ) : π 0 S ∞ ( p ) → π 0 S ∞ ( E ) . (iii) Re c al l the set C / ∼ fr om the intr o duction. Ther e is a bije ction fr om C / ∼ to the pr eimage of [ λ ] under the map α : π 0 S ∞ ( p ) → π 0 S ∞ ( E ) . The main result of [Ste1 1] was the construction o f a “ parametrized Whitehead torsion” τ : S ∞ ( p ) → Γ  ΩWh B ( E ) ↓ B  whenever p is a bundle of co mpact ma nifo lds , such that the following holds: Theorem 3.3 . The diagr am S ∞ ( p ) τ / / α   Γ  ΩWh B ( E ) ↓ B  β   S ∞ ( E ) τ / / ΩWh( E ) is a we ak homotopy pul l-b ack, with β as in se ction 2. Moreov er, if M is a c o mpact to p olo gical manifold, the map π 0 ( τ ) : S ∞ ( M ) → π 1 Wh( M ) ∼ = Wh( π 1 M ) agrees with the classica l Whitehead to rsion, s ending the c la ss of a homoto p y equiv- alence f : N → M to its Whitehead torsion τ ( f ). Pr o of of The or em 1.1. Assumption (i) is clearly necessa ry while assumption (ii) is necessary by Theorem 2.2. Now supp ose that assumptions (i) and (ii) hold, such that we can factor f = p ′ ◦ λ ′ where p ′ is a fib er bundle of compa c t top ologica l manifolds and λ ′ a homo topy eq uiv - alence. Denote b y F ′ b the fib er of p ′ ov er b a nd consider the following co mmutative diagram (3) π 0 S ∞ ( p ′ ) π 0 ( α ′ )   τ / / H 0 ( B ; Wh( F ′ b )) π 0 ( β ′ )   π 0 S ∞ ( E ′ ) τ / / Wh( π 1 E ′ ) which is π 0 of the pull-ba ck square from Theorem 3.3, applied to the bundle p ′ . By Lemma 3.2, f is homotopic to a bundle of compact manifolds if a nd only if the ele ment defined by λ ′ in the lower left-hand corner comes from an element in the upp er left-hand corner. Using the pull-back prop erty , this is equiv alent to saying that the co r resp onding element τ ([ λ ′ ]) in the lower right-hand cor ner comes from an element in the upp er rig h t-hand corner. T hus, if we define o ( f ) as the class of τ ([ λ ′ ]) in the cokernel of β ′ , f fiber s stably if and only if o ( f ) = 0. As the fibrations p and p ′ are fib er homotopy equiv alent, the c okernels o f π 0 ( β ) and π 0 ( β ′ ) are iso morphic. So we may think of o ( f ) as an element in the cokernel of π 0 ( β ). Finally w e ha ve to sho w that o ( f ) is well-defined. Indeed supp ose that w e c ho ose another facto rization f = ¯ p ′ ◦ ¯ λ ′ with ¯ p ′ : ¯ E ′ → B a fib er bundle o f compact OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 9 manifolds. Then by the comp osition rule the resulting torsion changes by the torsion of λ ′ ◦ ¯ λ ′− 1 : ¯ E ′ → E ′ , which is in the imag e of α ′ since it comes from a fib er homotopy equiv alence. Th us, w hen pas sing to the co kernel of π 0 ( β ), the element o ( f ) is not affected.  Pr o of of The or em 1.2. W e saw in Lemma 3.2 that the set C / ∼ is in bijection with π 0 ( α ′ ) − 1 ([ λ ]), which by square (3) is in bijection to π 0 ( β ′ ) − 1 [ τ ( λ )] and thus to the kernel of π 0 ( β ′ ) as β ′ is a n infinite lo op map. Now use that the kernels of π 0 ( β ) to π 0 ( β ′ ) are isomorphic.  4. Change of base and tot al sp ace The t w o pr oblems of “change of ba se” and “change of total space” are interesting sp ecial c ases wher e the para metrized W all obstruction do es no t pla y a ro le. W e first consider them in the light of the genera l theor y . After that we offer a second, more ge o metric p ers pective using families of h -co bo rdisms. This second pers pective makes it ea sier to find a n estima te for a sta ble ra nge. Theorem 4.1 (Cha ng e of total space) . L et p : M → B b e a fib er bund le of c omp act top olo gic al m anifolds over a c omp act top olo gic al manifold, and let N b e another c o mp act top olo gic al manifold, e quipp e d with a homotopy e qu ivalenc e f : N → M : N f ≃ / / pf ' ' N N N N N N N N N N N N N M p   B Then pf stably fib ers if and only if the Whitehe ad torsion τ ( f ) lies in the image of π 0 ( β ) : H 0 ( B ; ΩWh( F b )) → Wh( π 1 M ) for p . Theorem 4.2 (Chang e of base) . L et p : M → B b e a fib er bun d le of c omp act top olo gic al manifolds over a c omp a ct top olo gic al manifold, and let C b e another c o mp act top olo gic al manifold, e quipp e d with a homotopy e qu ivalenc e f : B → C : M p   f p ' ' N N N N N N N N N N N N N B f ≃ / / C Then f p st ably fi b ers if and only if the image of the W hitehe ad torsion f − 1 ∗ τ ( f ) ∈ Wh( π 1 B ) under the tr ansfer homomorphi sm p ∗ : Wh ( π 1 B ) → Wh( π 1 M ) lies in the image of π 0 ( β ) . In p a rticular, if t he fib er F of p is c onne cte d, π 1 ( B ) acts trivial ly on F , and f p stably fib ers, t hen χ e ( F ) · τ ( f ) = 0 ∈ Wh( π 1 C ) . Pr o of of The or em 4.1. Notice that pf is alrea dy a factorization in to a homo topy equiv a lence follow ed by a fib er bundle. So, conditions (i) and (ii) of Theor em 1.1 are satisfied, and the tors io n obstruction o ( f ◦ g ) is just the imag e of the Whitehead torsion of f in the cokernel.  10 WOLF GANG STEIMLE Pr o of of The or em 4.2. Denote by k : C → B a homo topy inv erse of f , and cons ider the pull-back k ∗ M / / k ∗ p   M p   C k / / B Now f induces a map ¯ f : M → k ∗ M s uc h tha t f ◦ p = k ∗ p ◦ ¯ f is a factoriza tion o f f ◦ p in to a homo topy eq uiv alence followed b y a fib er bundle. Thus o ( f ◦ p ) is given by the class of ¯ f − 1 ∗ τ ( ¯ f ), which satisfies ¯ f − 1 ∗ τ ( ¯ f ) = p ∗ f − 1 ∗ τ ( f ) by the geometric definitio n of the transfer map [And74]. Now s upp os e that F is co nnec ted and π 1 ( B ) ac ts trivia lly . In this cas e the comp osite p ∗ ◦ p ∗ is just m ultiplication with the Euler characteristic of F [L ¨ uc87]. W e saw that if f ◦ p stably fib ers, then p ∗ f − 1 ∗ τ ( f ) comes fro m some element κ ∈ Wh( π 1 F ) under the map induced by the inclusio n i : F → M . As the comp osite p ◦ i is nullhomotopic, we hav e 0 = p ∗ i ∗ κ = p ∗ p ∗ f − 1 ∗ τ ( f ) = χ e ( F ) · f − 1 ∗ τ ( f ) ∈ Wh( π 1 B ) .  Using the rela tion b et ween the par ametrized torsion and hig her h -cob or dism theory , we now g ive a s econd per sp ective on the change o f to tal space pro blem. Under smo othability conditions, this approach a llows to estimate a stable ra nge using the stability results of Igusa [Igu88]. In the change of total space problem as in Theore m 4.1, supp ose for simplicity that B is connected. Denote by k the smallest dimension o f a CW co mplex ho- motopy equiv alent to B , a nd b y n the the smallest dimension of a CW complex homotopy equiv a len t to the fiber s. Theorem 4.3. In t he situation of The o r em 4.1, supp ose that M , N , and the fib ers of f ar e smo othable. If τ ( f ) is in the image of π 0 ( β ) , then the c omp osite ¯ N := N × I l Pro j − − − → N f − → M p − → B fib ers as so on as dim ¯ N ≥ max { 2( n + k ) + 1 , dim M + n + k , dim M + k + 2 , dim B + 2 k + 6 , dim B + 3 k + 2 , dim N + 3 } . Pr o of. The first step is to replace f : N → M by ¯ f : N → ¯ M which is a stably tangential ho motopy equiv alence, i.e. ¯ f ∗ T ¯ M ∼ = T N stably . Therefore recall that M a nd N ar e supp osed to b e smoo tha ble, so we may choose a v ector bundle reduction ( f − 1 ) ∗ T N − T M : M → B O of the top o logical tangent bundle. I t a ctually has a further reduction to a O ( n + k )- bundle, the inclus ion B O ( N ) → B O b eing N -co nnec ted. Let therefore q : ¯ M → M be a disk bundle of this ( n + k )-dimensional vector bundle. W e obta in T ¯ M | M ∼ = T M ⊕ T fib q | M ∼ = T M ⊕ q ∼ = ( f − 1 ) ∗ T N stably , s o if we let ¯ f : N → ¯ M b e f followed by the zer o -section, then ¯ f is a stable tangential ho motopy equiv alence. OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 11 Notice that the map ¯ p = p ◦ q still is a fib er bundle o f compac t smo otha ble manifolds (with a fibe r we denote b y ¯ F b ), and tha t dim ¯ M = dim M + ( n + k ) . Now supp ose tha t τ ( ¯ f ) = τ ( f ) disassembles, i.e. that ther e is an element τ ∈ H 0 ( B ; ΩWh( ¯ F b )) such that that π 0 ( α )( τ ) = τ ( f ). By [Ste11, Coro llary 9.3], the element τ is the par ametrized torsion obtaine d fro m glueing a fiberwis e h -cob ordism E alo ng the vertical b oundar y bundle ∂ ¯ p , provided that b oth K + 1 ≥ k , l − 1 ≥ k where K is the co ncordance stable ra nge of ∂ ¯ F b and l is the connectivity of the pair ( ¯ F b , ∂ ¯ F b ). The following lemma (whose pro of is an exercise using the Blakers- Massey theo rem) shows that l ≥ n + k − 1. Lemma 4.4. If ¯ F → F is an L -disk bund le over a c omp a ct manifold, and the p ai r ( F, ∂ F ) is N -c onne cte d, t hen ( ¯ F , ∂ ¯ F ) is ( N + L ) -c onne cte d. By our a ssumptions the ma nifo ld ∂ ¯ F b is smo othable, so Igusa ’s s tabilit y res ult [Igu88] (see [WW01, Theor em 1.3.4 ] for the top ologica l rang e) says that k − 1 ≤ K whenever dim ∂ ¯ F b ≥ max { (2( k − 1) + 7 , 3( k − 1) + 4 } . Thu s, stabilizing further if necessa ry , w e o btain a parametrized h -cob ordism E over ∂ ¯ p such that the torsio n of the pro jection ¯ ¯ M := ¯ M ∪ ∂ ¯ p E → ¯ M is pr ecisely τ . Notice that dim ¯ ¯ M = max { dim M + n + k , dim M + k + 2 , dim B + 2 k + 6 , dim B + 3 k + 2 } . By the comp osition rule, the comp osite ¯ ¯ f : N → ¯ M → ¯ ¯ M has torsion zero (and is s till stably tang e n tial). Now sta biliz e N to obtain ¯ N := N × I l such that l ≥ 3 (so that ¯ N is π - π ) a nd dim ¯ N ≥ 2( n + k ) + 1, and stabilize either ¯ N o r ¯ ¯ M fur ther so that the dimensio ns agr ee. Letting K be a finite ( n + k )- dimensional CW c o mplex s imple homotopy equiv alen t to ¯ N , it follows that b oth ¯ N and ¯ ¯ M define thick enings of K in the sense of W all [W al66]. It is known tha t stably , thick enings a re c la ssified b y their tang e n t bundle. Now the dimension of the thick enings we consider exceeds 2( n + k ), so we ar e in the s table r ange. But ¯ ¯ g is stably tang ent ial, hence the thick enings agree. Thus ¯ ¯ g is homotopic to a homeomorphism. Summarizing all the necessary stabilizations, we see that pf fib ers as so on as dim ¯ N ≥ max { 2( n + k ) + 1 , dim N + 3 , dim M + n + k , dim M + k + 2 , dim B + 2 k + 6 , dim B + 3 k + 2 } .  12 WOLF GANG STEIMLE 5. E xamples I: Element ar y applica tion s In this section we give some immediate applications of our results on the sta ble fiber ing problem. The first one characterizes simple homotop y equiv alences b etw een compact ma nifolds as the homotopy equiv alences that stably fib er. After that we give s o me implicatio ns for closed manifolds. Prop ositio n 5.1. A homotopy e quival enc e f : M → N b etwe en c omp act manifolds stably fib ers if and only if τ ( f ) = 0 . If M and N ar e close d s m o othable of dimension k and τ ( f ) = 0 , then f fib ers after at most max { 2 k + 6 , 3 k + 2 } stabilizations. Pr o of of Pr op osition 5.1. This is a simple applicatio n of our res ults in the following change o f total s pace problem: M f ≃ / / f ' ' N N N N N N N N N N N N N N id   N As the fib ers o f the ident ity are contractible, their Whitehea d group v anishes. So π 0 ( β ) is the zero map and its c o kernel is just Wh( π 1 M ). Hence o ( f ) = f − 1 ∗ τ ( f ) ∈ Wh( π 1 M ).  Now w e turn to c lo sed manifolds and co nsider the change of total s pace pro blem M f ≃ / / g ' ' N N N N N N N N N N N N N N p   B If g stably fib ers, i.e. for some n ≫ 0 , M × D n +1 → B fib ers, then we may restrict to the b oundar y to see that ¯ g : M × S n → M g − → B fiber s for large enough n . So o ur theory gives sufficient co nditio ns fo r M × S n to fiber over B . On the other hand, if ¯ g fib ers, then it certainly stably fib ers. It follows: Prop ositio n 5. 2. (i) A ne c essa ry c ondition for M × S 2 N → B to fib er for lar ge N is that 2 o ( g ) = 0 . (ii) A su fficient c ondition for M × S 2 N → B to fib er for lar ge N is that o ( g ) = 0 . (iii) The su fficient c ondition is not ne c essary in gener al. Pr o of. (i) W e hav e τ ( f × id S 2 N ) = χ ( S 2 N ) · τ ( f ) = 2 τ ( f ), and its class in the cokernel of π 0 ( α ) defines the obstruction o ( ¯ g ) for ¯ g to stably fib er. (ii) Apply the res ults o n the change of tota l space problem and re strict to the bo undary . (iii) Cho ose a homotopy equiv alence f : M → K × S 1 betw een closed manifolds such that τ ( f ) 6 = 0 and 2 τ ( f ) = 0, and let p : K × S 1 → S 1 denote the pro jection. Hence o ( pf ) 6 = 0; in contrast, if q : M × S 2 N → M is the pro jection, then o ( pf q ) = 0. OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 13 W e will show in Theorem 8.1 that for B = S 1 , the s table fib ering o bstruction o ( pf q ) and the o bstructions τ fib ( pf q ) defined in [FLS0 9 ] agr ee. Now F arre ll’s fib er- ing theorem together with the compariso n of the differ en t obstructions [FLS09, Theorem 8 .1] s hows that pf q fiber s.  Prop ositio n 5. 3. If f : M → B is any map b etwe en close d manifolds whose ho- motopy fib er is finitely dominate d, then the c omp osite M × S 1 × S 1 × S N P r oj − − − → M f − → B fib ers for lar ge enough N . Pr o of. The para metrized W all obs tr uction b ecomes zero after taking pro duct with S 1 (see e.g. [WW01, Corollary 5 .2 .5]). Hence for the map M × S 1 → B , the fib ering obstruction is defined. As it is g iven by a Whitehead torsio n, it beco mes zero a fter taking pr o duct with another S 1 . Therefor e M × S 1 × S 1 × D N +1 → B fiber s. Now restrict to the bo undary .  6. E xamples I I: St able vs. un st able and bl ock fibering and TOP vs. DIFF In this sectio n we give examples o f maps f : M → B that fibe r stably but no t unstably . O f course, if the dimension o f M and B agr ee then fib ering f unstably just mea ns deforming the map to a homeo mo rphism, where a s f stably fib ers if and only if it is a s imple homotopy equiv alence (Pr op osition 5.1): This g ives obvious examples. More in terestingly , we will consider t wo types of situations o f a rbitrarily high co dimension. The first one considers the tangential data; supp osing that a recent conjecture of Reis-W eiss on topolo g ical rational P ontry agin clas s es ho lds , w e obtain a lower bo und on the num ber of stabilizations needed. The s e cond o ne applies surgery theo r y to pro duces an e x ample which actually do es not even blo ck fibe r . W e also co ns ider certa in maps to spher es with spherical fib ers where unsta ble fiber ing a nd blo ck fib ering are equiv a lent and we expand on an ex ample of Klein- Williams to pro duce examples that fiber sta bly in TO P but no t in DIFF. Bundle theory. Le t Z b e an exotic complex pro jective s pace equipp e d with a homotopy equiv a lence h : Z → C P 2 n +1 such that, for some k 6 = 0, the L -genus satisfies (4) ( h ∗ ) − 1 L ( Z ) = L ( C P 2 n +1 ) · (1 + 8 k e 2 n ) ∈ H ∗ ( C P 2 n +1 ) ∼ = Q [ e ] / ( e 2 n +2 ) . (T a ke a ll the cohomolo gy rings with ratio nal co efficients.) W e will s how b elow that such ob jects exis t. Conjecture 6.1 ([R W11]) . If ξ is a TOP( n ) -bund le over B , then t he i -th r ational Pontryagin class p i ( ξ ) ∈ H 4 i ( B ; Q ) vanishes pr ovi de d i > n/ 2 . Prop ositio n 6. 2. (i) The c omp osite Z × S N π − → Z h − → C P 2 n +1 of h with the pr oje ction fib ers stably. If Z is smo othable then it fib ers even unstably whenever N ≥ 12 n + 7 . 14 WOLF GANG STEIMLE (ii) If Conje ctur e 6.1 holds, then for N ≤ 2 n − 1 , the m ap fr om (i) do es not fib er (unst ably). Pr o of. (i) This is an a pplication of the change of total space problem. (ii) Let p : Z × S N → C P 2 n +1 be a fib er bundle homotopic to the ma p of (i). Then T ( Z × S N ) ∼ = p ∗ T ( C P 2 n +1 ) ⊕ η for an N -dimensio nal bundle η . Hence L ( Z × S N ) = p ∗ L ( C P 2 n +1 ) · L ( η ) . But L ( Z × S N ) = π ∗ L ( Z ) = p ∗ ( h ∗ ) − 1 L ( Z ) as the spher e is stably paralle liz able. It follows tha t L ( η ) = p ∗  L ( C P 2 n +1 ) − 1 · ( h ∗ ) − 1 L ( Z )  = 1 + 8 k p ∗ ( e ) 2 n using (4) for the last equality . Hence, since p ∗ is injective, the L -genus of η is non-zero in deg ree 4 n . Inductively o ne co ncludes p i ( η ) = 0 ( i < n ) , p n ( η ) 6 = 0 using the fact that the co e fficie n t of p i in L i is non-zero for all i [Hir78, I.1.(11 )]. So η must be at least 2 n -dimensional: N ≥ 2 n .  W e now indicate why a homotopy equiv alence h : Z → C P 2 n +1 with the prop erty (4) exists. This construction is due to Madsen-Milgra m [MM79]. Let f : X → C P 2 n be a top olo gical degree one no rmal map cor resp onding to the comp osite C P 2 n → C P 2 n / C P 2 n − 1 ∼ = S 4 n → G/ TOP where the la st map r e presents k times a generato r o f π 4 n ( G/ TOP) ∼ = Z . Let E → C P 2 n be the disk bundle o f the tautolog ical vector bundle. W e may pull back the normal map f to E . By the π - π -theorem, this pulled- back nor mal map is co bo rdant to a map g : Y → E which is a homotopy eq uiv alence and restricts to a homotopy equiv a lence over the b oundary ∂ E = S 4 n +1 . The Poincar´ e conjecture implies that ∂ Y ∼ = S 4 n +1 homeomorphica lly . Th us we may co ne off g at the b oundar ies to obtain a to p olo gical manifo ld Z and a homotopy equiv alence h : Z → C P 2 n +1 . W e have to show that (4) ho lds. T o do that, we will use the characteristic classes K 4 n ∈ H 4 n ( G/ TOP; Q ) given uniquely b y the prop erty that if γ : M → G/ TO P is a nor mal inv ar iant o n a closed 4 k -ma nifold, then its simply-connected surger y obstruction is given by the formula [MM7 9, Theo rem 4.9] s ( M , γ ) =  L ( M ) ·  X i ≥ 1 γ ∗ ( K 4 i )  , [ M ]  . Now the surgery o bstructions of h | C P i for i < 2 n are zero while the surge r y obstruction o f h | C P 2 n is k , hence inductiv ely one c oncludes that γ ∗ ( K 4 i ) = 0 (4 i < 2 n ) , γ ∗ ( K 4 n ) = k e 2 n ∈ H 4 n ( C P 2 n +1 ) . OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 15 Denote by L ∈ H ∗ ( B TOP ; Q ) the L -class of the universal bundle. By [MM7 9, Corollar y 4 .22], its r estriction along G/ TOP → B TOP is given by 1 + 8 K , where K = K 4 + K 8 + . . . . Hence ( h ∗ ) − 1 L ( Z ) = L ( C P 2 n +1 ) · γ ∗ (1 + 8 K ) = L ( C P 2 n +1 ) · (1 + 8 k e 2 n ) , as claimed. Remark 6. 3. T o the knowledge of the a uthor, it is unk nown in gener al which o f these fake C P 2 n +1 are smo othable. The following argument shows that there ar e infinitely many smo othable examples for even n . F o r each n ther e exists a num ber A n such that the normal inv ariant f is s mo o th- able if and only if k is a multiple of A n . (In fac t A n is the o rder of the g enerator of π 4 n ( G/ TOP) in the tor sion gr oup π 4 n − 1 (TOP /O ).) Hence the subgroup of all smo oth no rmal in v a riants o f C P 2 n satisfying (4) is infinite. Using the π - π -theor em in the smo oth setting, we o bta in a map [ C P 2 n , G/O ] ∼ = [ E , G/O ] ∼ = S DIFF ( E ) → S DIFF ( ∂ E ) = Γ 4 n +1 from the smo o th nor mal in v a riants of C P 2 n to the gr oup of homotopy spheres. By Brumfiel [Bru71, Cor ollary 6.6 ], this map is a gro up homomor phism if n is even. Hence in this ca se it ha s a n infinite kernel. If f : X → C P 4 n is a smo oth no rmal map of degre e one which represents a n element in the kernel, this means the fo llowing: The pull-back o f f to E is cob orda nt to a normal ma p g : Y → E which r e stricts to a diffeomor phism on the bounda ry . In this case the co ning pro cedure yields a homoto p y equiv alence h : Z → C P 4 n +1 where Z is smo oth. Surgery theory. Now we co me to the s urgery-theo retic example. Let γ : X → G/ TOP b e a normal inv ar iant on a closed manifold X , let M b e a clos ed manifo ld and le t h : M → X × S k be a s imple homotopy equiv alence which, co nsidered as a normal inv ariant, restricts to γ ov er X × {∗} . (Suc h a simple homotopy e quiv ale nce can b e obtained as follows: Pull-back of γ defines a normal inv ariant on X × D k +1 which, by the π - π theorem, can b e repre sent ed by a simple homotopy equiv a lence of pa irs ( k , h ) : ( N , M ) → X × ( D k +1 , S k ).) Prop ositio n 6. 4. If the sur gery obstruction of γ is non-zer o, then the c o mp osite f : M h − → X × S k Pro j − − − → S k do es not fib er. It always fib ers stably. Pr o of. The comp osite fib ers stably b y P r op osition 4.1. Supp ose there exis ts a fiber bundle F → M p − → S k homotopic to f . Then we c an lift the homotopy to obtain a homotopy H : M × I → X × S k × I ov er S k which res tricts to h at M × 0 and which is a fib er homo topy equiv alence F → X × S k ov er 1. T ak ing a transverse preimage ov er the base p oint of S k yields a degree o ne normal co bo rdism ( W ; N , F ) → ( X × S k × I ; X × S k × 0 , X × S k × 1) whose r e striction over 1 is a ho motopy eq uiv alence . The restric tion ov er 0 corre - sp onds to the element γ , which ther e fore ha s surg ery obstructio n 0 , co nt radicting the assumption.  16 WOLF GANG STEIMLE Spherical fibrations ov er s pheres. Prop ositio n 6. 5. L et f : M → S 2 k b e a m ap b etwe en close d t op olo gic al manifolds whose homotopy fib er is a 2 n -spher e, wher e 2 n ≥ 4 k . Supp ose t hat dim M ≥ 6 . Then, (i) f always fib ers stably. (ii) The fol lowing ar e e quivalent: (a) f fib ers unst ably, (b) f blo ck-fib ers, (c) for a r e gular pr eimage F of a p oint in S 2 k , the de gr e e one n ormal map F → S 2 n given by the inclusion of the fib er into the homotopy fib er has surge ry obstruction zer o . Remark 6.6. Casso n [Cas 67] sho wed tha t (b) and (c) are equiv ale n t in the smo oth case; in the top olo gical case this e q uiv alenc e ca n b e deduced from Quinn’s thesis [Qui70]. Addendum. If, in the situation of Pr op osition 6.5, the nu m b er n = 2 l is even, the fol lowing c ondi tions ar e e quival ent and e quiva lent to the ones fr om p art (ii): (d) F o r a r e gular pr eima ge F of a p oint in S 2 k , the signatur e σ ( F ) is zer o, (e) t he l -t h Pontryagin class p l ( M ) ∈ H 4 l ( M ; Q ) is zer o . The pro of o f Prop ositio n 6.5 will make use of the following lemma , which is due to Sieb enmann [KS77, E s say V, § 5] for n 6 = 4 and to F reedman-Quinn [FQ90, Theorem 8 .7A] for n = 4: Lemma 6.7 . The stabilization map B TOP( n ) → B TOP is n -c onne ct e d. Pr o of of Pr op osition 6.5. (i) Let f = p ◦ λ b e a facto r ization into a homo topy equiv a lence follow ed by a fibra tion. Since π 2 k − 1 G/ TOP = 0 a nd we are in the stable ra nge, we may as s ume that p is the sphere bundle of a TO P(2 n + 1 )-bundle η . Hence we are in a change of total space situation, a nd the result follows since E is s imply-connected. (ii) The implication from (a) to (b) is o b vious. The argument for the implica tion from (b) to (c) is very simila r to the pr o o f of Prop ositio n 6.4: a homoto p y from f to the pr o jection map of a blo ck bundle g induces a nor mal b ordis m b etw een the degree o ne nor mal ma p F = f − 1 ( ∗ ) → S 2 n and the identit y map on S 2 n . Suppo se now that (c) holds. Aga in factor f = p ◦ λ as in (i). Again we can and will a ssume that p is the spher e bundle o f a TO P(2 n + 1)-bundle η . By the dimension a s sumptions, p ha s a se c tio n s : S 2 k → E , up to homotopy . Denote by i : S 2 n → E the inclusion o f the ho mo topy fib er. Collaps ing the low er skeleta de fines a map π : E → S 2 n +2 k ; homotopy-theoretic calculations show that the sequence [ S 2 n +2 k , G/ TO P] π ∗ − → [ E , G/ TOP] i ∗ ⊕ s ∗ − − − − → [ S 2 n , G/ TO P] ⊕ [ S 2 k , G/ TO P] is ex act. Lemma 6.8 . The map π ∗ is split by the sur gery obstruction [ E , G/ TOP] → L 2 n +2 k ( Z ) ∼ = [ S 2 n +2 k , G/ TO P] . OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 17 Let x := j ( λ ) where j : S ( E ) → [ E , G/ TO P] is the ca nonical map from the surgery structure set to the set of normal inv a r iants. As the surgery obstructio n map [ S 2 n , G/ TO P] → L 2 n ( Z ) is an isomor phism, condi- tion (c) says that i ∗ ( x ) = 0. The element s ∗ ( x ) is represented by some TOP(2 k + 1)- bundle ξ over S 2 k and a fibe r ho mo topy trivialization t : S ( ξ ) → S ( ε ) of the co rre- sp onding s phere bundle. Since 2 n ≥ 4 k , we can split off the bundle η a (2 k + 1 )- dimensional trivial bundle ε , s o that η ∼ = η ′ ⊕ ε and we can consider the homo topy equiv alence µ : S ( η ′ ⊕ ξ ) → S ( η ′ ⊕ ε ) ∼ = S ( η ) = E induced by the ident ity on η ′ and the trivializa tion t . Lemma 6.9. The homotopy e quivalenc e µ , c onsider e d as a normal invariant on E , agr e es with p ∗ s ∗ ( x ) . Now, p ∗ s ∗ ( x ) and x agre e after applying s ∗ but also after applying i ∗ (when both are zer o). It follows from the exact sequence ab ov e that the difference p ∗ s ∗ ( x ) − x lifts ov er the map π ∗ , which is split by the surg ery o bs truction. But the surger y obstruction of b oth x and p ∗ s ∗ ( x ) is zer o since they are repres e nted by ho motopy equiv a lences. Hence p ∗ s ∗ ( x ) = x . By the s urgery exact sequence, the map j is injectiv e, so the manifold structures λ and µ ag ree as w ell. But p ◦ µ is a fiber bundle pro jection.  Pr o of of L emma 6.8. Let α : S 2 n +2 k → G/ TOP repre sent a generator , corr esp ond- ing to a degree one nor mal map g : N → S 2 n +2 k . W e claim that the nor mal inv aria nt π ∗ g is represented by the deg ree one norma l map id ♯g : E ♯N → E ♯S 2 n +2 k = E . In fact if α is given b y the TOP-bundle ξ and a prop er fiber ho motopy trivia lization t : ξ → ε , then g is obtained by taking a re g ular preimag e of S 2 n +2 k ⊂ ε under t (see [MM7 9, The o rem 2.23]). W e may assume that t is a bundle isomorphism over the low er hemisphere which contains the p oint ∞ and that a round ∞ , g is the homeomo rphism from the zero section in ξ to the zer o section in ε . Since π maps E − D 2 n +2 k to the p oint ∞ ∈ S 2 n +2 k , the map q ∗ t : q ∗ ξ → q ∗ ε is then alre a dy transverse to E − D 2 n +2 k and E ♯N is a regular preimage. Since the simply- connected surger y obstruction is additive under c o nnected sum and the surgery obstruction o f f is a gener ator in L 2 n +2 k ( Z ), it follows tha t the surgery o bstruction of π ∗ α is also a genera tor.  Pr o of of L emma 6.9. The ho mo topy equiv a lence µ extends to a homotopy equiv a- lence ¯ µ : D ( η ′ ⊕ ξ ) → D ( η ′ ⊕ ε ) betw een disk bundles, which is the pull-back of the homotopy equiv alence ¯ t : D ( ξ ) → D ( ε ) under the bundle pr o jection D ( η ′ ⊕ ε ) → D ( ε ). The res triction o f ¯ t over a r egular pr e ima ge of S 2 k ∈ D ( ε ) is classified by s ∗ ( x ) : E → G/ TOP. Since the inclusion S 2 k ⊂ D ( ε ) is a homo topy equiv alence, the normal inv aria n t ¯ t is classified by the comp osite o f s ∗ ( x ) with the pro jection D ( ε ) → S 2 k . F rom the pull-back prop erty it follows that ¯ µ is cla ssified by the comp osite D ( η ) = D ( η ′ ⊕ ε ) → S 2 k s ∗ ( x ) − − − → G/ TOP . 18 WOLF GANG STEIMLE Now restrict to the bo unda ry .  Pr o of of A ddendum. Supp ose that f is a fib er bundle with fib er S 4 l . By the Poincar´ e conjecture , the fib er is actually a sphere. W e ca n extend f to a disk bundle ¯ f : ¯ M → S 2 k . Now T M ∼ = T ¯ M | M stably , so p l ( M ) is the res tr iction of p l ( ¯ M ) ∈ H 4 l ( ¯ M ) ∼ = H 4 l ( S 2 k ) = 0. Hence p l ( M ) = 0 . Now suppo se that p l ( M ) = 0 . Denote by F a r egular preimag e of a p oint ∗ ∈ S 2 k under f . Then F is framed in M , hence, for all i , p i ( F ) = p i ( M ) | F and in par ticular p l ( F ) = 0. Moreover the inclusion F → M factors, up to ho- motopy , over S 4 l , so a ll the Pont ryagin classes of F v a nish except for p 0 . Hence L ( F ) = 1 . The Hirzebruch signa ture formula yields σ ( F ) = 0 . Finally σ ( F ) − σ ( S 4 l ) is a multiple of the surgery o bstruction (in L 4 l ( Z ) ∼ = Z ) of the degree one normal map F → S 4 l . As σ ( S 4 l ) = 0, the v anishing of σ ( F ) is equiv a lent to the v anishing of the s urgery obstruction.  DIFF vs. TOP. Prop ositio n 6.10. L et x ∈ π s 4 i +1 such t hat η 2 x is in the image of the J -homomor- phism and let p : E → S 4 i +2 b e a b ase d n -spheric al fibr ation ( n > 4 i + 3 ) c orr esp ond- ing to x ∈ π 4 i +1+ n ∼ = π 4 i +2 B F n , wher e F n is the m onoid of p ointe d self-homotopy e quiva lenc es of S n . If M is a c omp act smo oth manifold and λ : M → E is a homo- topy e quivalenc e, then: (i) p ◦ λ fib ers stably in TOP. (ii) p ◦ λ do es not fib er stably in DIFF. Pr o of of Pr op osition 6.10. (i) Since n > 4 i + 3, the stabilization map π 4 i +1 G n / TOP n → π 4 i +1 G/ TOP is an is omorphism. But π 4 i +1 G/ TOP = 0. Hence the fibr ation p is the sphere bundle of a TOP n -bundle a nd we are in a change o f total space s itua tion. Th us, p ◦ λ fiber s s tably a s E is simply-connected. (ii) The following argument follows the lines of Klein-Williams [KW09]. Supp ose that p ◦ λ fibers stably in DIFF. Then by [D WW03], the parametrized A -theor y characteristic χ ( p ) ∈ H 0 ( S 4 i +2 ; A ( S n )) bec omes zero in H 0 ( S 4 i +2 ; Wh DIFF ( S n )). Calling y the image of χ ( p ) under the pro jection H 0 ( S 4 i +2 ; Wh DIFF ( S n )) → H 0 ( S 4 i +2 ; Wh DIFF ( ∗ )) ∼ = π 4 i +2 Wh DIFF ( ∗ ) we conclude that y = 0. As explained in [KW09, § 8], y is the image of x under the comp osite π 4 i +2 B G F − → π 4 i +2 A ( ∗ ) → π 4 i +2 Wh DIFF ( ∗ ) where F is the map defined by W aldhause n [W a l8 2]. B ut it was s hown in [BW8 7] that the imag e o f x is non-ze ro.  OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 19 It would be interesting to hav e an example of a map that fib ers in TOP w he r e the smo oth W all obstruction is zero, but the smo oth fib ering obstructio n is not. This would probably req uire a deep er analy sis of the higher homotopy type of Wh P L ( F ) and Wh DIFF ( F ) for a suitable F who se fundamental gr oup ha s non- v a nishing Whitehead g roup. 7. Examples I I I: Resul ts of Chapman-Ferr y The o bstruction theory develop ed in this chapter allows us to re- in terpret ob- structions obtained by Chapma n- F er ry [CF78 ] on fib ering c o mpact Q -manifolds ov er compact ANRs. E xplicitly , Cha pman-F erry deal with the cases wher e B is a wedge of copies of S 1 and where B is n -dimensio nal and the fib ers ar e n -connected. W e will see that in b oth cases the sp ectral sequence can b e used to ana ly ze our obstructions further. By definition, a Q -manifold is a separa ble metric spa ce which is lo cally homeo- morphic to op en subsets of the Hilb ert cube Q = Π ∞ i =1 I . The fib ering problem for Q -manifolds asks whether a given a map f : M → B from a compact Q -manifo ld to a compact ANR is homotopic to a fibe r bundle pro jection whose fib ers a re compact Q -manifolds aga in. (Since the pro jectio n Q × I → Q is homotopic to a homeomor- phism, there is no difference b etw een the stable and the unstable fiber ing problem.) It will be shown in App endix A that the obstruction theory for fiber fib ering compact Q -manifo lds ov er a compact ANR agrees with the obs tr uction theory for compact top olo g ical manifolds develop ed so far. Hence a ll the fib ering results may be applied in either context. Lemma 7.1. L et p : E → B b e a fi br ation. If the b ase sp ac e is homotopy e quivalent to a CW c o mplex of dimension at most n , and al l the fib ers F b ar e n -c onne cte d, then H 0 ( B ; Wh( F b )) = 0 . Henc e W all( p ) = 0 when define d. In the c ase n = 1 , it is en ou gh to su pp ose Wh ( π 1 F b ) = 0 . In the c ase n = 2 , it is enou gh to supp ose that F b is 1-c onne cte d. Pr o of. If a map X → Y is n - c onnected, then the induced map A ( X ) → A ( Y ) is n -connected by [W al78] and so is Wh ( X ) → Wh( Y ) b y a 5 -lemma type argument. As Wh( ∗ ) is contractible, it follows that Wh( X ) is n -c onnected whenever X is n - connected. If X is 1-co nnec ted, then Wh( X ) is 2-co nnected (see [Ha t78, section 3 ], with the co rrection in [Igu84]). Therefore the sp ectral seque nc e for H 0 ( B ; Wh( F b )) has v anis hing E 2 page.  If the fibr ation p has a s e ction, then this result can b e considerably str e ng h tened: Theorem 7.2 ([KW09]) . If p has a se ction, al l the fib ers ar e n -c onne cte d, and B is homotopy e quivalent t o a CW c omplex of dimension at most 2 n , then the map H 0 ( B ; A ( F b )) → H 0 ( B ; Wh( F b )) is zer o. Henc e, W all( p ) = 0 when define d. Example 7.3. A tw o-connected map f : M → S 2 stably fibers . In fact, such a map is split up to homotopy , so Theore m 7 .2 applies. The homo topy fib er F has the homotopy t ype of a CW co mplex b y Milnor [Mil59]. Mo reov er H ∗ ( F ) is finitely generated. In fact, the E 2 -term of the A tiyah-Hirzebruch sp ectral sequence E 2 pq = H p ( S 2 , H q ( F )) ⇒ H p + q ( M ) 20 WOLF GANG STEIMLE consists o f t wo columns only , so there is a n exact sequence 0 → E ∞ 2 ,n − 1 → E 2 2 ,n − 1 → E 2 0 ,n → E ∞ 0 ,n → 0 with E 2 2 ,n − 1 = H n − 1 ( F ) , E 2 0 ,n = H n ( F ) . The E ∞ -page is finitely generated (since H ∗ ( M ) is); o ne co ncludes b y induction that H n ( F ) is also finitely g enerated. As π 1 ( F ) = 0, it follows [W al65] that F is ho motopy finite. So W all( p ) is defined and z e r o. Mor eov er o ( f ) = 0 as M is simply-connected. Let alwa ys f : M → B b e a map either betw een compact top ologica l manifolds, or from a compac t Q -manifold to a compa ct ANR. In the Q -manifold s e tting, the following results are due to Chapman-F erry . Prop ositio n 7 .4. Supp ose B is homotopy e quiva lent to a finite n - c o mplex ( n ≥ 1 ). If the homotopy fib er F of f is homotopy finitely dominate d and n -c onne cte d, then the torsion obstruction o ( f ) ∈ Wh ( π 1 M ) is define d and vanishes if and only if f stably fib ers. If n = 1 , we c an r epla c e t he assu mption “ F 1 - c onn e ct e d” by “ Wh( π 1 F ) = 0 ”. If n = 2 , we c an r eplac e the assumption “ F 2 -c onne ct e d” by “ F 1 -c onne cte d”. Pr o of. By the a ssumptions and Lemma 7 .1, the parametr ized W all obs truction is defined and zero. Moreover, the ma p β : H 0 ( B ; Wh( F b )) → Wh( π 1 E ) is zero since it factors thr ough Wh( π 1 F b ) = 0.  Prop ositio n 7. 5. (i) Supp ose t hat B is homotopy e quivalent to a we gde of n c opies of S 1 ( n ≥ 1 ), and su pp ose that t he homotopy fib er F of f is c o nne cte d and homotopy e quiva lent to a fi nite c omplex. Then, W all( p ) is an element of L n Wh( π 1 F ) α n (c oi nvariants under the action of the fi b er tr ansp ort along the c orr esp onding c o py of S 1 ). (ii) The torsion obstruction (whenever define d) is an element in the quotient Wh( π 1 M ) / ( n − 1) · Wh( π 1 F ) π 1 B . In p a rticular, if n = 1 , the t orsion obstruction lives in Wh ( π 1 M ) . Pr o of. (i) By the spec tral sequence, there is an exact sequence 0 → H 1 ( B ; π 1 Wh( F b )) → H 0 ( B ; Wh( F b )) e − → H 0 ( B ; π 0 Wh( F b )) → 0 , and the image of W all( p ) under the edg e homomorphism e is given b y the finiteness obstruction o f the fiber , which is zero by assumption. Ther efore, W all( p ) lifts to H 1 ( B ; π 1 Wh( F b )) ∼ = M n H 1 ( S 1 ; π 1 Wh( F b )) ∼ = M n Wh( π 1 F b ) α n . (ii) The map β factors as H 0 ( B ; ΩWh( F b )) → H 0 ( B ; Wh( π 1 F b )) ∼ = Wh( π 1 F b ) π 1 B χ e ( B ) · i ∗ − − − − − → Wh( π 1 M ) , where the first map is a sur jection a nd the Euler characteristic of B is n − 1.  Prop ositio n 7. 6. L et f , g : M → B b e two homotopic pr oje ctions of bund les of c o mp act manifolds. S upp ose that B is homotopy e quivale nt t o a we dge of n c opies of S 1 . Denote by F the homotopy fib er of f , which is homotopy e quivalent to the OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 21 homotopy fib er of g . The obstruction gr oup A for f and g b eing e quivalent (in the sense of the intr o duction) fits into the fol lowing exact se quenc e: 0 → M n π 2 Wh( F ) α n → A → Wh( π 1 F ) π 1 B ( n − 1) · i ∗ − − − − − → Wh ( π 1 M ) Pr o of. By Theorem 1.2, A is given by the kernel of β : H 0 ( B ; ΩWh( F b )) γ − → H 0 ( B ; Wh( π 1 F )) ∼ = Wh( π 1 F ) π 1 B ( n − 1) · i ∗ − − − − − → Wh ( π 1 M ) . By the spec tr al s equence, we have 0 → H 1 ( B ; π 2 Wh( F b )) → H 0 ( B ; ΩWh( F b )) γ − → H 0 ( B ; π 1 Wh( F b )) → 0 . Thu s ker γ ∼ = H 1 ( B ; π 2 Wh( F b )) ∼ = M n π 2 Wh( F ) α n . Since γ is surjective, ther e is a sho rt ex a ct seq uence 0 → k er γ → A → ker(( n − 1) · i ∗ ) → 0 . The cla im follows.  8. Comp arison with the obstructions by F arrell-L ¨ uck-Steimle The conten t o f the author ’s Diploma thesis [Ste07] was to define Whitehead torsion obstructions to fib ering a manifold ov er another manifold. See [FLS09] for a published and extended version. The goal of this section is to compare these obstructions. Given a ma p f : M → B of topo logical manifo lds , factor as usual as f = p ◦ λ , a homotopy equiv alence follow ed by a fibra tio n. In [FLS09], tw o obstructions for f to b e homo topic to a fib er bundle ar e defined: (i) An element θ ( f ) ∈ H 1 ( B ; Wh( π 1 M )) which is defined whenever the homo- topy fib er F of f is ho mo topy finite (an obvious necessar y c o ndition). It is defined by the rule that whenever γ : S 1 → B is a lo op in B , then under the restriction map H 1 ( B ; Wh( π 1 M )) γ ∗ − → H 1 ( S 1 ; Wh( π 1 M )) ∼ = Wh( π 1 M ) θ ( f ) maps to i ∗ ( τ ), where τ is the Whitehead tor sion of the fib er transp or t on p along γ (c ho osing a n arbitrar y s imple structure on the fib er F ). (ii) If θ ( f ) = 0, ther e is defined a n element τ fib ( f ) ∈ coker (Wh( π 1 F ) χ e ( B ) · i ∗ − − − − − → Wh ( π 1 M )) where i : F → M is the inclusion of the homoto p y fib er, and χ e ( B ) denotes the Euler characteristic. It is defined as follows: Cho ose a simple structure on the homotopy fiber of f and per form a c e rtain cons truction (inductively ov er the cells o f B ) to o btain a simple s tructure o n E . Then τ fib ( f ) is the image of the Whitehead torsion o f λ : M → E , whic h is well-defined in the quotient. Theorem 8.1 . (i) The image of W all( p ) u nder t he r estriction H 0 ( B ; Wh( F b )) → H 0 ( { b } ; Wh( F b )) ∼ = ˜ K 0 ( Z [ π F b ]) is t he finit en ess obstruction of the fib er. 22 WOLF GANG STEIMLE (ii) S upp ose that F is homotopy finite. The image of the Wal l obstru ction W a ll( p ) u nder t he se c ondary homomorphism ker  H 0 ( B ; Wh( F b )) → H 0 ( B ; π 0 Wh( F b ))  → H 1 ( B ; Wh( π 1 F b )) i ∗ − → H 1 ( B ; Wh( π 1 M )) is θ ( f ) . (iii) S upp ose that W all( p ) = 0 . The definition of the map β as a c omp osite H 0 ( B ; ΩWh( F b )) → Wh( π 1 F b ) χ e ( B ) · i ∗ − − − − − → Wh ( π 1 E ) ∼ = Wh( π 1 M ) induc es a map coker ( π 0 ( β )) → co ker  Wh( π 1 F ) χ e ( B ) · i ∗ − − − − − → Wh( π 1 M )  under which o ( f ) maps to τ fib ( f ) . In p art icular, if χ e ( B ) = 0 or Wh( π 1 F ) = 0 , then o ( f ) = τ fib ( f ) ∈ Wh( π 1 M ) . Pr o of. (i) and (ii) were prov ed in Theorem 2.5. (iii) If W all( p ) = 0, then we may assume that p is a bundle of compac t topolog ical manifolds, and it follows fro m [FLS09, Lemma 3.19] that the simple structur e on E is just the cano nical simple structure o f the top ologic a l manifold E . Therefor e bo th o ( f ) and τ fib ( f ) ar e given by the r esp e ctiv e cla sses of the Whitehead torsio n of λ .  Appendix A. Fibering Q -manifolds The g oal of this appe ndix is to show that the obstructio n theor y for b oth ex- istence a nd uniqueness develop ed in this pap er als o a pplies to fib ering compa c t Q -manifolds over compact ANRs. Here is a collection of results from the theory of Q -manifolds (see [Cha76]): (i) If X is a lo ca lly compact ANR (e.g. a top ologica l manifold), then X × Q is a Q -manifold. (ii) E very compact Q -manifold is of the fo rm X × Q , where X is a compac t po lyhedron. (iii) A map f : X → Y be t ween co mpact CW complexes is a simple homoto p y equiv a lence if a nd only if f × id Q : X × Q → Y × Q is homotopic to a homeomorphism. This prop erty may b e taken as a definition o f simple homotopy equiv alence b etw een c ompact ANRs. (iv) Any cell-like map [Lac69] be t ween Q - manifolds is arbitra rily clo se to a homeomorphism. This also shows that M × Q ∼ = Q . A compa ct Q -manifold bundle is a fib er bundle whose fiber s are compact Q - manifolds. Given a fibr ation p : E → B over a para compact s paces, the s tructure space S Q ( p ) of compact Q -ma nifold bundles is defined in analo gy to the structure space S n ( p ). F a cts (i) and (iv) show that the total space of a compact Q -manifold bundle ov er a compact ANR is a compact Q -manifold. As a consequence there is a geometr ic assembly map α : S Q ( p ) → S Q ( E ) OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 23 whenever B is a co mpact ANR. Of co urse, the re-interpretation of the fib er ing problem in terms of the g eometric a ssembly map (Lemma 3.1) r emains v alid in the Q -manifold setting. The relation b e tw een the fib ering problem for compac t Q -manifo lds and the fiber ing pr oblem o f compact topo logical manifolds is given by the map ( × Q ) : S n ( p ) → S Q ( p ) which sends a simplex ( q , λ ) to ( q × Q, λ ′ ) there λ ′ is the obvious comp osite of λ × id Q with the pro jection E × Q → E . Since Q × I ∼ = Q , it facto rs ca nonically through S ∞ ( p ). Theorem A.1. If B is a c omp act top olo gic a l manifold and p : E → B is a bu nd le of c omp act top ol o gi c al manifolds, then the fol lowing diagr am is a we ak homotopy pul l-b ack: S ∞ ( p ) ( × Q ) / / α   S Q ( p ) α   S ∞ ( E ) ( × Q ) / / S Q ( E ) Pr o of. The dia gram commutes up to homotopy . Moreov er the vertical tangent bundle defines a map T v : S ∞ ( p ) → map( E , B TOP) such that the following diagram co mm utes up to homotopy (see [Ste11, Prop osition 8.3] for the T v -comp onent): S ∞ ( p ) (( × Q ) ,T v ) / / α   S Q ( p ) × map( E , B TOP) α × (+ p ∗ T B )   S ∞ ( E ) (( × Q ) ,T v ) / / S Q ( E ) × map( E , B TOP) Since the map (+ p ∗ T B ) is an equiv a lence, we can pro of the theo rem by showing that the hor izontal lines in the diagram are weak homotopy equiv alences. T o do that, denote by S f r n ( p ) the space o f framed manifold struc tur es on p : It is the geometric realization of a simplicial set where a k -s implex is a commutativ e diagram E ′ q # # G G G G G G G G G ϕ ≃ / / E × ∆ k p y y s s s s s s s s s s B × ∆ k together with a bundle map T v q → ǫ from the vertical tangent bundle of q to the trivial bundle n -dimensiona l ov er E × ∆ k which covers ϕ . The usual sta bilization pro cedure pr o duces a spa ce S f r ∞ ( p ). The forgetful map from S f r ∞ ( p ) to S ∞ ( p ) fits int o a commutativ e diagra m (5) S f r ∞ ( p ) / / ( × Q ) ( ( P P P P P P P P P P P P P S ∞ ( p ) T v / / ( × Q )   map( E , B TOP) S Q ( p ) 24 WOLF GANG STEIMLE The pr o of is now a co nsequence of the following t wo cla ims: Claim (i ): The hor izontal line in (5) is a weak homoto p y fibra tion s equence, and Claim (i i): The diagona l ar row in (5) is a weak homotopy equiv alence. Claim (i) is Pro po sition 1 .2.1 fro m [Ho e0 9]. T o prov e claim (ii), we will use the notation and results from [Ste11, section 2]. The structure spa ce S n ( p ) is weakly homotopy equiv a len t to a s pa ce o f lifts in the diagram Bun n ( ∗ ; F )   B p / / 6 6 Fib( ∗ ; F ) where F is the fib er of p . Since B is compact, it follows that S ∞ ( p ) ≃ Lift Bun ∞ ( ∗ ; F ) ↓ B p − → Fib( ∗ ; F )) ! with Bun ∞ ( ∗ ; F ) := co lim n Bun n ( ∗ ; F ), the colimit ov er the stabiliza tion. F o r a spa c e X , let Fi b f r n ( X ; F ) b e the catego r y where an ob ject is a fibratio n p : E → X with fiber F together with a TOP( n )-bundle ov er E , and a morphis m is a fib er homotopy equiv a lence of fibrations which is cov ered by a bundle map. Let Bun Q ( X ; F ) be the categ o ry of bundles of compa ct Q - manifolds over X where the fiber s are homotopy equiv alent to F , with bundle homeomorphisms a s morphisms. The arg umen ts from [Ste11, sec tion 2] pr o duce classifying spaces Fib f r n ( ∗ ; F ) and Bun Q ( ∗ ; F ). Letting Fib f r ∞ ( ∗ ; F ) := co lim n Fib f r n ( ∗ ; F ) , the colimit ov er the stabilization of the euclidean bundle, we obtain ho motopy equiv a lences S f r ∞ ( p ) ≃ Lift   Bun ∞ ( ∗ ; F ) ↓ B ( p,ǫ ) − → Fib f r ∞ ( ∗ ; F )   , S Q ( p ) ≃ Lift Bun Q ( ∗ ; F ) ↓ B p − → Fib( ∗ ; F ) ! where F denotes the fib er of p . Hence, claim (ii) follows from Claim (i i’): The diagr am Bun ∞ ( ∗ ; F ) ( × Q ) / /   Bun Q ( ∗ ; F )   Fib f r ∞ ( ∗ ; F ) / / Fib( ∗ ; F ) is homotopy cartesia n. W e will show cla im (ii’) by considering the hor izontal homotopy fib ers. Again by the arguments of [Ste11, section 2 ], the lower horizo n tal fib er is g iven by the mapping space map( F, B TOP). On the other hand, the uppe r homotopy fib er over the p oint deter mined b y a Q -manifold N is the stable structure space o f compact top ologica l manifolds M OBSTR UCTIONS TO ST ABL Y FIBERING M ANIFOLDS 25 equipp e d with a homeomo rphism M × Q → N . The path comp onents of this space, in view of pro p er ty (iii) a b ove, a re the stable homeomor phism classes of compact top olog ical manifolds equipp ed with a simple homoto p y equiv alence to N . There a re in bijection to [ F , B TOP] by a n a rgument in volving the h -cob or dism theorem (see e.g. [W al66, pr op. 5.1]). T o deter mine the higher homo topy groups of this str ucture s pa ce, r ecall that Bun ∞ ( ∗ ; F ) and Bun Q ( ∗ ; F ) are disjoint unions of classifying spaces, and use the homotopy fibration sequence [WW01, p. 171] colim n TOP( M × D n ) → TOP( M × Q ) → map( M , B TO P) for a compact to po lo gical manifold M , which is due to Chapman-F erry and Burghe- lea [Bur83].  This theorem shows that for a given map f : M → B b et ween compact top olog i- cal manifolds, the fib ering problem for f and the one for M × Q → B are equiv ale n t. Since by r e sult (ii) above, every compact Q -manifold is of the form M × Q , this reduces the fib ering pro blem fo r Q -manifolds to the o ne for top olog ic al manifolds provided the base space is a top ological manifold. The case wher e B is a compact ANR which is not a top olo gical ma nifold follows by the following “ change o f base” r esult [CF78]: Prop ositio n A.2. L et f : M → B a map fr om a c omp act Q -manifold to a c o mp act ANR, and let h : B → B ′ b e a simple homotopy e quivale nc e b etwe en c omp act A NRs. Then, f fib ers if and only if h ◦ f fib ers. Mor e over, the e qu ivalenc e classes of Q - manifold fib er bund le pr oje ctions homotopic t o f ar e in bije ction t o those homotopic to h ◦ f . Pr o of. Prop erties (i) and (iv) ab ov e allow a reduction to the c ase where B is a compact Q -manifold. But in this case the claim is obvious since we may assume that h is a ho meomorphism.  References [And74] Douglas R. Anderson. 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