The refined transfer, bundle structures and algebraic K-theory

THE REFINED TRANSFER, BUNDLE STR UCTURES AND ALGEBRAI C K -THEOR Y JOHN R. KLEIN A ND BRUCE WILLIAMS Abstract. W e give new ho motopy theo retic criteria for deciding when a fibration with homotopy finite fiber s a dmits a reduction to a fib er bundle with compact top ological manifold fiber s. The criteria lead to an unexpected result ab out homeo morphism gro ups of manifolds. A too l used in the pro of is a s urjective splitting of the a ssembly map for W aldhausen’s functor A ( X ). W e also give co ncrete exa mples of fibrations ha ving a reduction to a fibe r bundle with compact top olog ical manifold fibers but which fail to a dmit a compact fib er smo othing. The examples ar e detected by algebraic K -theory inv ariant s. W e co nsider a r efinement of the B eck er-Gottlieb tra nsfer. W e show that a v ersion of the axioms described b y Beck er a nd Sch ultz uniquely determines the refined transfer for the c lass of fibra tions admitting a reduction to a fiber bundle with compact top olog ical manifold fiber s. In an a ppendix, we sketch a theory of characteristic cla sses for fibrations. The classes are primary o bstructions to finding a com- pact fib er smoo thing. Contents 1. In tro duction 2 2. Con v en tions 9 3. Construction of a refined transfer 11 4. Characterization when B is a p oint 12 5. In terpretation 17 6. Pro of of Theorem A 18 7. Pro of of Theorems B and J 20 8. Pro of of Theorems F and G 23 9. Pro of of Theorem I 26 10. Pro of of Theorem H 27 11. Pro of of Theorem E 28 Date : August 16, 2021. 1991 Mathematics Su bje ct Classific ation. Primary : 57S05, 55R10, 19D1 0, 55R12. Secondar y: 5 5R15, 57 N65. 1 2 JOHN R. KLEIN AND B RUCE WILLIAMS 12. Appendix: c haracteristic classes for fibratio ns 31 References 32 1. Intro duction Let p : E → B b e a fibration whose base space B and whose fib ers hav e the homo- top y t yp e of a finite complex. The transfer construction of Bec k er a nd Gottlieb [BG1 ] asso ciates to p a “wrong wa y” stable homotopy class χ ( p ) : B + → E + suc h that the assignmen t p 7→ χ ( p ) is homotopy in v arian t a nd natural with resp ect to base c hange (here B + denotes B with the addition of a disjoin t basep oint). The transfer has sho wn itself to b e an imp ortant to ol in algebraic top ology . F or example, one of its early applications w as a simple pr o of of the Adams Conjecture [BG2]. A refinemen t of the transfer, a lso considered b y Bec k er and G ottlieb [BG1, top of p. 115], has recen tly surfaced in the Dwy er, W eiss and Williams approach to fib erwise smo othing problems and the theory of higher Reidemeister torsion (see [D WW],[BD], [I1]). Let E + denote the disjoin t union E ∐ B . Then E + is a retra ctiv e space o v er B . The category of suc h spaces is the sub ject of fib erwise homotop y theory (cf. [CJ]). The asso ciated stable homotop y category is thus the study of fib erwise stable phenomena (cf. [MS]). The r efine d tr ansfer of p is a certain fib erwise stable homoto p y class t ( p ) : B + → E + . The Bec k er-Gottlieb tra nsfer χ ( p ) is obtained from t ( p ) by collapsing the preferred sections B → E + and B → B + to a p oint. Bec k er and Sc h ultz [BS] gav e an axiomatic c haracterization of the Bec k er-Gottlieb transfer under the assumption that the fibration p is fib er ho motop y equiv alen t to a fib er bundle with compact top ological manifold fib ers. Their axioms inv olv e naturality , normalization, com- patibilit y with pro ducts and additivit y o f the transfer. Definition 1.1. If p : E → B admits a fib er homotopy equiv alence to a fib er bundle with compact top ological manifold fib ers, then p is said to hav e a c omp act TOP r e duction . THE REFINED TRANSFER 3 If p is fib er ho motop y equiv alen t to a fib er bundle with compact smo oth manifold fib ers, then p is said to ha v e a c omp a c t DIFF r e d uction or a c omp act fib er smo othing . R emark 1.2 . The fib ers of these bundles are p ermitted t o hav e non- empt y b oundary . O ur terminology in t he smo oth case differs f rom that of Casson and Gottlieb [CG], who instead use the term clo se d fib er smo othing. Our preference is t o use ‘compact’ instead of ‘closed’ so as to av oid p otential confusion. The following, comm unicated t o us b y G o o dwillie, g iv es fibrations with homotopy finite fib ers which fa il t o a dmit a compact TOP reduc- tion. Example 1.3 . Let F b e a connected based finite c omplex equipp ed with a based self homotopy equiv a lence θ : F → F . Assume θ induces the iden tit y map on fundamen tal g roups and has no n-trivial Whitehead torsion. Then the mapping torus F × θ S 1 → S 1 , con v erted into a fibration, do es not admit a compact TOP reduction. The e xample is v erified b y con tradiction. A c ompact TOP reduction w ould yield a homotop y equiv alence k : F → M , with M a compact top ological manifold, a nd a homotop y in v erse k − 1 : M → F suc h that the comp osite k ◦ θ ◦ k − 1 : M → M is homotopic t o a homeomorphism. But t his would show t hat the torsion of k ◦ θ ◦ k − 1 is trivial [Ch, th. 1]. Since θ induces the iden tit y on π 1 , the compo sition formula [Co, 22.4 ], sho ws tha t the torsion of θ is also tr ivial. This giv es the contradiction. F or sp ecific homotopy equiv a lences θ satisfying 1 .3, see [Co , 24.4 ]. The main result of this pap er is to giv e explicit homotopy theoretic criteria for deciding when a fibration admits a compact TOP reduction. Our a pproac h is to use the recen t w ork of Dwy er, W eiss and Williams, sp ecifically , the “Conv erse Riemann-Ro c h Theorem” whic h giv es an abstract c haracterization when suc h a reduction exists [DWW], a nd en tails an understanding of ho w the refined transfer relates to W ald- hausen’s algebraic K -theory of spaces. Along the w a y , we will extend the Bec k er-Sc h ulz axioms to the fib erwise setting and show ho w the axioms c haracterize the refined transfer for those fibrations admitting a compact TOP reduction. The pro of of this characterization follows along the lines of Bec ke r-Sch ulz, and w e do not claim a n y originality in this direction. As to whether t he axioms characterize the refined transfer for all fibrations with homotop y finite fib ers is an interesting op en question. 4 JOHN R . KLEIN AND BRUCE WILLIAMS The a xiomatic characterization of the refined transfer is indep en- den t of the r est o f the pap er and is included b ecause of Igusa’s re- cen t prog ress on axiomatizing higher R eidemeister torsion inv aria n ts [I2]. There is a close relationship b etw een hig her torsion and the re- fined transfer: when the fibration is fib er ho motop y equiv alent to a smo oth fib er bundle with compact fib ers, then the refined transfer ad- mits a lif t in to a gro up closely asso ciated with a lgebraic K -t heory , and this lift coincides with the higher torsion inv ariant of D wy er, W eiss and Williams. A curren tly unsolv ed problem is to determine whether the Dwy er-W eiss-Williams torsion coincides with Igusa’s torsion. The problem would b e solve d if one could v erify that Igusa’s axioms hold for the Dwy er-W eiss-Williams to rsion. Some evidenc e in fav or of this is that the a xioms we will shortly giv e f or the refined transfer seem to b e close in spirit to Igusa’s axioms, altho ugh further effort will b e needed to pin do wn the exact relatio nship. Here are the axioms. Definition 1.4. A r efi n e d tr ansfe r is a rule, whic h assigns to fibratio ns p : E → B with homotop y finite base and fib ers, a fib erwise stable homotop y class t ( p ) : B + → E + that is sub ject to the f ollo wing axioms: • A1 (Naturalit y). F o r comm utative homot op y pullbac k dia- grams E ′ ˜ f / / p ′   E p   B ′ f / / B in whic h p ′ and p are fibrations, we ha v e ˜ f + ◦ f ∗ t ( p ′ ) = t ( p ) ◦ f + , where f + denotes f ∐ id B , ˜ f + denotes ˜ f ∐ id B and f ∗ t ( p ′ ) is the effect of making t ( p ′ ) in to a fib erwise stable homoto p y class B b y taking cobase change along f . • A2 (Normalization). Let 1 : B → B b e the iden tit y . Then t (1) : B + → B + is the iden tit y . • A3 (Pro ducts). F or a pro duct fibration p × p ′ : E × E ′ → B × B ′ , w e ha v e t ( p × p ′ ) = t ( p ) ∧ t ( p ′ ) . THE REFINED TRANSFER 5 where ∧ means external fib erwise smash pro duct. • A4 (Additivit y). If E ∅ j 2 / / i 1   E 2 i 2   E 1 j 1 / / E is a comm utative homoto p y pushout diagram of fibra tions o v er B , then t ( p ) = ( j 1 ) ∗ t ( p 1 ) + ( j 2 ) ∗ t ( p 2 ) − ( j ∅ ) ∗ t ( p ∅ ) , where for S ( { 1 , 2 } , p S : E S → B denotes the pro jection and ( j S ) ∗ : E + S → E + is the eviden t map. In Section 3, w e explain Bec k er a nd Gottlieb’s construction of a refined transfer. Their v ersion will b e called the r efined t ransfer, em- plo ying the definite ar ticle to distinguish it fro m o ther constructions satisfying the axioms. Theorem A. L et t and t ′ b e r e fine d tr ansfers define d on the class of fibr ations havin g homotopy fi n ite fib ers. The n t = t ′ for those fibr a tion s which adm i t a c omp act TOP r e duction. W e now giv e homotopy theoretic criteria for deciding when a fibra- tion a dmits a compact TOP r eduction. One should regard this as the main result of t he pap er. Theorem B. L et p : E → B b e a fibr ation with homotopy finite b ase and fib ers. Assume • p c omes e quipp e d with a se ction, • p is ( r + 1 ) -c onne cte d and • B has the homotopy typ e of a c el l c om plex of dimension ≤ 2 r . Then p admits a pr eferr e d c omp act TOP r e duction. Consequences. Com bining Theorems A and B, w e immediately ob- tain Corollary C. L et t and t ′ b e r e fi ne d tr ansfers. The t = t ′ for the fibr ations app e a ri n g i n The or em B. Here is a w a y to construct examples satisfying Theorem B. Star t with an y Hurewicz fibration p : E → B with homoto p y finite base and fib ers. The (unr e duc e d) fib erwis e susp e n sion of p is the fibrat ion 6 JOHN R . KLEIN AND BRUCE WILLIAMS S B p : S B E → B whose total space is the double mapping cylinder of the map p : S B E = ( B × 0) ∪ ( E × [0 , 1]) ∪ ( B × 1) (cf. [St]). The fib er of S B p at b ∈ B is g iv en b y the unreduced susp en- sion of the fib er of p at b . Consequen tly , the connectivit y of the map S B p is o ne more than that of p , so iteration of the fib erwise susp ension construction ev en tually yields a fibration whic h satisfies the conditions of Theorem B. Corollary D. Stably, any fibr ation p : E → B with homo topy finite b ase and fib ers admits a c om p act TOP r e duction. That is, ther e is an iter ate d fib e rw ise susp ensi o n S j B p : S j B E → B wh i c h admits a c omp a ct TOP r e duction. The metho d of pro of of Theorem B yields a new and unexpected result ab out a utomorphism groups of manifolds. F or a compact con- nected manifold M with basep oin t ∗ in its interior, let TOP( M , ∗ ) b e the simplicial group whose k -simplices a re the homeomorphisms of ∆ k × M whic h comm ute with pro jection t o ∆ k and whic h are the iden- tit y when restricted to ∆ k × ∗ . Let G ( M , ∗ ) b e defined similarly , using homotop y equiv alences in place of homeomorphisms. The forgetful ho- momorphism induces a map o f classifying spaces B TOP( M , ∗ ) → B G ( M , ∗ ) The surprise will b e that this map has a section up to homot op y along the 2 r - sk eleton of B G ( M , ∗ ) when M ⊂ R m is an r -connected compact co dimension zero manif old with a sufficien tly lo w dimensional spine (the exact dimensions will b e sp elled out in Section 11) . More precisely , define the stable home omorphism gr oup TOP st ( M , ∗ ) , to b e colimit of TOP( M × I k , ∗ ) via stabilization given b y taking carte- sian pro duct with the unit interv al. Similarly , one can define G st ( M , ∗ ), but in this case the asso ciated inclusion G ( M , ∗ ) → G st ( M , ∗ ) is a homotopy equiv alence. It follow s that one has a map of classifying spaces B TOP s t ( M , ∗ ) → B G ( M , ∗ ) . Theorem E. L et M ⊂ R m b e a c omp act c o dimension zer o smo oth submanifold. Assume M is r -c onne cte d. Then ther e is a sp ac e X M and a ma p X M → B TOP st ( M , ∗ ) such that the c omp osite X M → B TOP st ( M , ∗ ) → B G ( M , ∗ ) THE REFINED TRANSFER 7 is 2 r - c onne cte d. F urthermor e, ther e is a pr eferr e d d e c omp osition of homotopy gr oups π ∗ (TOP st ( M , ∗ )) ∼ = π ∗ ( G ( M , ∗ )) ⊕ π ∗ (map( M , TOP)) ⊕ π ∗ +1 ( Wh top ( M )) which is valid in de gr e es ∗ ≤ 2 r − 2 . In the ab ov e, Wh top ( M ) is the top olog ical Whitehead space ([W1, § 3], [H]), TOP is the gr oup of homeomorphisms of euclidean space stabilized with resp ect to dimens ion, and map( M , TOP) is the function space of ma ps M → TOP. Examples. W e no w giv e examples of fibrations whic h fail to ha v e a compact fib er smoo thing but whic h do admit a compact TOP r educ- tion. Theorem F. Ther e ex ist fibr ations p : E → B which admit a c omp act TOP r e duction but which d o n o t have a c omp act fib e r smo othing. The fib ers of thes e fibr ations have the homotopy typ e of a finite we dge of spher es ∨ k S n , for suitable ch o i c e of k and n . F urthermor e, these examples a r e dete cte d in the r ationalize d alge- br aic K -the ory of the inte gers. Theorem G. L et S 3 → E p → S 3 b e the spheric al fibr a tion c orr esp ond i n g to the gener ator of π 3 ( B F 3 ) ∼ = π 5 ( S 3 ) = Z 2 , wher e F 3 is the top olo gic al m onoid of b ase d self homotopy e quiva l e n c es of S 3 . Then p admi ts a c omp a c t TOP r e duction but do es not adm i t a c om- p act fib er s mo othing. The following result show s that the obstructions t o compact fib er smo othing are killed when taking the cartesian pro duct w ith fi nite com- plex hav ing tr ivial Euler c haracteristic. Theorem H. L et p : E → B b e a fibr a tion with homotopy finite fi b ers. L et X b e a finite c omple x with zer o Euler char acteristic. Then the fibr ation q : E × X → B given by q ( e, x ) = p ( e ) admits a c omp act fib e r smo othing. R emark 1.5 . A t the time the first draft of this paper w a s written, it w a s forgotten b y b oth authors that this result w as already stated in [WW2, Cor. 5.2.5] with a s k etc hed pro o f. This pap er con ta ins a differen t pro o f. 8 JOHN R . KLEIN AND BRUCE WILLIAMS The trace map. Giv en a fibration p : E → B , let p + : E + → B + b e the asso ciated map of r etractiv e spaces ov er B . Giv en a refined transfer t ( p ) : B + → E + , we take its comp osition with p + to obtain a fib erwise stable homotop y class p + ◦ t ( p ) : B + → B + . A straigh tforward unrav eling of definitions sho ws that p + ◦ t ( p ) is equiv- alen t to sp ecifying an ordinary stable cohomotop y class tr t ( p ) : B + → S 0 (b ecause B + coincides with B × S 0 ). The lat ter is called the tr ac e of the fibratio n p . (compare Brumfiel and Madsen [BM, p. 1 37]). The f ollo wing is a uniqueness result ab out the trace. Theorem I. L et t and t ′ b e r e fine d tr ansfers . Then tr t = tr t ′ on the class of fi b r ations wh ose b ase a n d fib e r have the homotopy typ e of a finite c o m plex. R emark 1.6 . F or further results, see D ouglas [D] and Dor abia la and Johnson. [DJ] Assem bly . The pro of of Theorem B uses the assem bly map for W ald- hausen’s algebraic of K -theory of spaces functor A ( X ). If f is a ho- motop y functor from spaces to sp ectra, the assem bly map is a na tural transformation f % ( X ) → f ( X ) whic h b est approximates f b y an excisiv e functor f % in the homotop y category of functors (recall that a functor is excisiv e if it preserv es homotop y pushouts). The crucial result used in the pro of of Theorem B is a functorial stable range splitting for the assem bly map for A ( X ) considered as a functor on the category o f based spaces. Theorem J. F or b ase d sp ac es X , the ass embly map A % ( X ) → A ( X ) is stably split. Mor e pr e cis ely, ther e is a ho m otopy functor X 7→ B ( X ) fr om b ase d sp ac es to sp e ctr a, and a natur al tr ansform ation B ( X ) → A % ( X ) such that the c omp osite map B ( X ) → A % ( X ) → A ( X ) is 2 r - c onne cte d whenever X is r -c o nne cte d. THE REFINED TRANSFER 9 Giv en what is already kno wn ab out A ( X ), this result is not hard to prov e. Ho w ev er, it is w orth stating here since it is one of our main to ols. The ro le of the basepo in t here is crucial; the result is fa lse on the category of un based spaces. A cknow le dgements. Both authors wish to thank Bill Dwye r for the metho d whic h pro duces the examples app earing in Theorem F. W e also wish to thank Ralph Cohen for miscellaneous discussions. An earlier draft of this pap er asserted that the splitting in Theorem J w as v a lid on the catego ry of unbased spaces. W e are indebted to a referee for explaining to us wh y this is no t true. The first author w as partially supp ort ed by NSF grants DMS05 03658 and DMS0803 363. 2. Conventions Spaces. W e w o rk in the category of compactly g enerated spaces. A map of spaces f : X → Y is a we ak homotopy e quivalenc e when it induces an isomorphism on homotop y in each degree. A space X is r -c o nne cte d , if for ev ery integer k suc h that − 1 ≤ k ≤ r , ev ery map S k → X extends to map D k +1 → X . In pa rticular, ev ery non-empt y space is ( − 1)-connected. The empt y space is considered to b e ( − 2)- connected. A map of spaces is r -connected if its homotopy fib ers (with resp ect to all basep oin ts) are ( r − 1)-connected. A space is homotopy finite if it has the homotopy t yp e of a finite cell complex. Although Quillen mo del categories a re barely men tioned in this pa- p er, w e will b e implicitly w orking in the mo del structure for spaces whose w eak equiv alences are the we ak homotopy equiv alences, whose fibrations are Serre fibrations, and whose cofibrations a re the Serre cofibrations. There is one notable exception to this p olicy: it is no t known whether the fib erwise susp ension of a Serre fibration is again a fibra- tion, but the analogous statemen t is tr ue in the Hurewicz fibratio n case. So unless otherwise men tioned, w e usually work with Hurewicz fibrations. A commutativ e square of spaces (or sp ectra) A / /   C   B / / D is said to b e r -c artesian if the map fr om A to the homotopy pullbac k P := holim ( B → D ← C ) is r -connected. More generally , supp ose 10 JOHN R . KLEIN AND BRUCE WILLIAMS the square only comm utes up to homotopy . Giv en a c hoice of homotopy A × [0 , 1] → D , one g ets a preferred map A → P . In this instance, w e say tha t the square to gether with its comm uting homot op y is r - cartesian provide d that A → P is r -connected. Fib erwise spaces. W e will b e assuming familiarity w ith fib erwise ho- motop y theory in its unstable and stable contex ts. The b o ok of Crabb and James [CJ] g iv es the foundatio nal material on this sub ject. F or a cofibrant space B , let T ( B ) denote the category of sp ac es over B . An ob j ect of T ( B ) consists of a space X and a reference ma p X → B (where the lat ter is typic ally suppresse d from the notat ion). A morphism X → Y is a map of spaces wh ic h is compatible w ith re ference maps. A morphism is a fibratio n or we ak equiv alence if and o nly if it is one when considered as a map of top ological spaces. This comes fro m a mo del structure on T ( B ), where t he cofibratio ns ar e defined using the lifting prop erty [Qu]. The ‘p oin ted’ v ersion of T ( B ) is the catego ry R ( B ) of r etr active sp ac es over B . This category ha s ob jects consisting of a space Y together with maps B → Y and Y → B suc h that the comp osite B → Y → B is the iden tit y map. A morphism is a map of underlying spaces whic h is compatible with the structure maps. Again R ( B ) is a mo del category by a pp ealing to the fo rgetful functor to spaces. An ob ject of R ( B ) is said to b e finite if it is obta ined from the zero ob ject b y att ac hing a finite num b er of cells. The (r e duc e d ) fib erwise susp ension functor Σ B : R ( B ) → R ( B ) is giv en by mapping an ob ject Y to the ob ject Σ B Y = ( Y × [0 , 1]) ∪ X × [0 , 1] X , where X × [0 , 1] → Y × [0 , 1] arises from the structure map X → Y b y taking cart esian pro duct with the iden tity map and X × [0 , 1] → X is first factor pro jection. R ( B ) also has smash pro ducts. If Y and Z are ob j ects, then their fib erwise smash pro duct is giv en b y Y ∧ B Z := ( Y × B Z ) ∪ ( Y ∪ X Z ) X where Y × B Z is the fib er pro duct and Y ∪ B Z → Y × B Z is the eviden t map. Note that the sp ecial case of Z = S 1 × B giv es Σ B Y . S. Sc h w ede [S] has sho wn that the catego ry of fib ered spectra ov er B , i.e., sp ectra formed using ob jects of R ( B ), a gain forms a mo del category , where the weak equiv alences in this case are the ‘stable w eak homotop y equiv a lences.’ THE REFINED TRANSFER 11 The recen t b o ok of Ma y and Sigurdsson equips the category of fib ered sp ectra o v er B with a w ell-b ehav ed internal smash pro duct [MS, § 11.2]. 3. Constr uction of a refined transfer Bec k er and G ottlieb define a refined transfer in their pap er [BG1, § 5]. The purp ose of this section is to sk etc h the idea b ehind their construction. First consider the case when B is a p oin t. Let F b e a ho motop y finite space, whic h f or conv enience we tak e to b e cofibrant. W e let F + denote the effect of adding a disjoint basep oint to F . The S -dual of F + is the sp ectrum D ( F + ) whic h is the mapping sp ectrum map( F + , S 0 ), where S 0 is the sphere sp ectrum. Explicitly , it is the sp ectrum whose k -th space is the space of maps F + → QS k , where Q = Ω ∞ Σ ∞ is the stable homotopy functor. More generally , w e use the notation D ( E ) for the function sp ectrum o f maps E → S 0 whenev er E is a homotop y finite sp ectrum. There is a map of sp ectra d : F + ∧ D ( F + ) → S 0 whic h defined as the adjoint to the identit y map of D ( F + ). The cano n- ical stable map F + → D ( D ( F + )) is a w eak equiv alence. F urthermore, w e ha v e a preferred weak equiv a- lence F + ∧ D ( F + ) ≃ D ( F + ∧ D ( F + )) whic h sho ws that F + ∧ D ( F + ) is self dual . Hence t he dualization of the map d ab ov e yields a map d ∗ : S 0 = D ( S 0 ) → D ( F + ∧ D ( F + )) ≃ F + ∧ D ( F + ) . The map d ∗ is w ell-defined in the ho motop y cat egory of sp ectra. No w fo rm the homotopy class t ( F ) : S 0 d ∗ − − − → F + ∧ D ( F + ) ∆ F + ∧ id − − − − → F + ∧ F + ∧ D ( F + ) id ∧ d − − − → F + ∧ S 0 . Then t ( F ) is ide ntifie d with a stable homotop y class S 0 → Q ( F + ). This giv es a refined transfer in the case when B is a p o in t. Pro ceeding to the case of a general fibration E → B , one app eals to a fib erwise v ersion of the ab ov e to get a refined transfer. As in the introduction, let E + denote E ∐ B , and define D B ( E + ) to b e the 12 JOHN R . KLEIN AND BRUCE WILLIAMS fib erwise mapping sp ectrum of maps E + → B × S 0 . Then analogo us to the ab ov e, one has a fib erwise stable map d : E + ∧ B D B ( E + ) → B + whic h is adjo in t to the iden tity . One then con t in ues in the same w ay as ab ov e, and the outcome is a fib erwise stable homotop y class B + → E + . This gives our roug h description of the refined transfer in the general case. V erification of the axioms. The only axiom w hic h is not s traightfor- w ard to v erify is the additivit y axiom A4. Bec k er and Sc h ultz remark that this axiom follows fro m fo rmal considerations inv olving S-duality . In our con text, the crucial p oin ts a re that the map of fib ered sp ectra E + → D B D B ( E + ) is a natur al transformatio n and the double dual D B D B preserv es ho- motop y pushouts. 4. Chara cteriza tion when B is a point W e show how the axioms c haracterize the refined transfer for the constan t fibration F → ∗ , where F is a homotopy finite cell complex This case actually follow s from the work of Bec k er and Sch ultz. Ho w- ev er, it will b e useful for what comes later to recast their pro of in a more co ordinate-free lang uage. The case B = ∗ captures the main features of the pro of in the general case. W e first digress with an observ atio n ab out the axioms in the case of a trivial fibration. F or a trivial fibration F → F × B p B → B , a refined transfer can b e regarded a s a fib erwise stable homotop y class t ( p B ) : B + → ( F + ) × B , and the asso ciated o rdinary transfer can b e regarded a s the asso ciated stable homotopy class t F ( B ) : B + → ( F × B ) + whic h is o btained fro m t ( p B ) b y collapsing t he preferred cop y of B to a p oint (note: ( F × B ) + = ( F + ) × B ) /B ). More generally , let ( B , A ) b e a cofibration pair. Cho o se an actual stable map ˆ t ( p B ) : B + → ( F + ) × B r epresen ting t he refined transfer THE REFINED TRANSFER 13 t ( p B ). The natur alit y axiom implies that the fib erwise homotopy class of the comp osite stable map A + − − − → B + ˆ t ( p B ) − − − → ( F + ) × B coincides with inclusion ( F + ) × A → ( F + ) × B comp osed with the r efined transfer t ( p A ) for the trivial fibration p A : F × A → A . F urthermore, since the diagram F × A / /   F × B   A / / B is a pullbac k, the space of c hoices consisting o f a c hoice of represen tativ e ˆ t ( p A ) of t ( p A ) to gether with a c hoice of homotop y making the diag ram of axiom A1 commute is a contractible space. This shows that once the represen tativ e ˆ t ( p B ) is chosen , then for a n y cofibratio n A ⊂ B , one obtains a preferred con tractible c hoice o f represen tativ es for all t ( p A ) equipped with a commuting homoto p y . In particular, the represen tative ˆ t ( p B ) determines a preferred stable homotop y class o f pairs ( B + , A + ) → (( F × B ) + , ( F × A ) + ) . whose comp onen ts are the transfers t F ( B ) and t F ( A ). The lat ter in turn induces a stable homotopy class o n quotients t F ( B , A ) : B / A → ( F + ) ∧ ( B / A ) . Note the sp ecial case when A is t he empt y space gives t F ( B , ∅ ) = t F ( B ). Axioms A2 and A3 straightforw ar dly imply t F ( B , A ) = t F ∧ id B / A , where t F : S 0 → F + coincides with t F ( ∗ , ∅ ). Also note that t F ( B , A ) = t F ( B / A, ∗ ). With this observ atio n, w e can no w return to the problem of c har- acterizing the r efined transfer when the base space is a p oin t. In this instance, a refined transfer is represen ted b y a homoto p y class of stable map t F : S 0 → F + , where t F = t F ( ∗ , ∅ ). Because F is a homotop y finite space, there is a co dimension zero compact smo oth manifold M ⊂ R d and a homotop y equiv alence F ≃ M . By homotopy in v ariance (i.e., axiom A1 when f is the iden tit y map of a p oint), it will suffice to 14 JOHN R . KLEIN AND BRUCE WILLIAMS c haracterize the homoto p y class t M := t M ( ∗ , ∅ ) : S 0 → M + . Consider the comm utative pullback diagram o f pairs (1) ( S d × M , M ) α × 1 / / π 1   (( M /∂ M ) × M , M ) π 1   ( S d , ∗ ) α / / ( M /∂ M , ∗ ) The v ertical maps of these diagr ams are fibrations. Applying the rela- tiv e transfer construction and using naturality , we see that the a sso ci- ated diagram of stable maps (2) S d ∧ M + α ∧ 1 / / M /∂ M ∧ M + S d α / / t M ( S d , ∗ ) O O M /∂ M t M ( M /∂ M , ∗ ) O O homotop y comm utes. The pro duct axiom for S d and F implies that t ∗ ( S d , ∗ ) ∧ t M ( ∗ , ∅ ) = t M ( S d , ∗ ) . The nor malization a xiom implies t ∗ ( S d , ∗ ) : S d → S d is the id en tity , and t M = t M ( ∗ , ∅ ) b y definition. Consequen tly , we see that the diagram of stable maps (3) S d ∧ M + α ∧ 1 / / M /∂ M ∧ M + S d α / / 1 S d ∧ t M O O M /∂ M t M ( M /∂ M , ∗ ) ≃ 1 M /∂ M ∧ t M O O is homotopy comm utativ e. Consider the diagonal em b edding ∆ : ( M , ∂ M ) → ( M × M , ( ∂ M ) × M ) . The asso ciated map of quotien ts M /∂ M → ( M /∂ M ) ∧ ( M + ) will also b e denoted ∆. The diagonal embedding has a compact tubular neigh- b orho o d isomorphic to the tota l space of the unit tangent disk bundle of M , whic h is a trivial bundle since M is a co dimension zero subman- ifold of euclidean space. Let D denote the unit disk bundle a nd let C denote the complemen t of the in terior of the tubular neigh b o rho o d. THE REFINED TRANSFER 15 Then w e hav e a pushout square S / /   C   D / / M × M . The inclusion ( ∂ M ) × M → M × M admits a factorization up to ho- motop y through C . The factor ization is giv en b y c ho osing an internal collar of ∂ M and letting M 0 denote the result of remo ving the o p en collar. Then M 0 ⊂ M is a homotopy equiv alence and the inclusion ∂ M × M 0 → M × M has image in C provide d that the tubular neigh- b orho o d has b een chose n sufficien tly small. The inclusion ( M × M 0 , ∂ M × M 0 ) → ( M × M , C ) then give s rise to a map of quotien ts M /∂ M ∧ M + ∼ = M /∂ M 0 ∧ ( M 0 ) + → ( M × M ) /C ∼ = D /S ∼ = S d ∧ M + . Denote it b y c : M /∂ M ∧ M + → S d ∧ M + . Lemma 4.1 (Compare [BS, lem. 2.5]) . The c omp o site S d ∧ M + α ∧ 1 − − − → ( M /∂ M ) ∧ M + c − − − → S d ∧ M + is homotopic to the identity. Pr o of. W e can assume that M is em b edded in the unit disk D d in such a w a y that M meets ∂ D d = S d − 1 transv ersely in ∂ M . Then we ha v e an embedding ( M , ∂ M ) ⊂ ( D d , S d − 1 ). Let ( W, ∂ 0 W ) b e the closure of the complemen t of ( M , ∂ M ) in ( D d , S d − 1 ). Consider the asso ciated embedding ( M × M , ∂ M × M ) ⊂ ( D d × M , S d − 1 × M ) giv en by taking the cartesian pro duct with M . Then the effect o f collapsing W × M in D d × M giv es rise to the ma p α ∧ 1. Consider the comp osite embedding ( M , ∂ M ) ∆ − − − → ( M × M , ∂ M × M ) ⊂ ( D d × M , S d − 1 × M ) . The comp osite c ◦ ( α ∧ 1) is t he effect of collapsing the complemen t of a tubular neigh b orho o d M in D d × M to a p oin t. But the complemen t is contractible, so this collapse map is homoto pic to the identit y .  Prop osition 4.2 (Compare [BS, 2 .9]) . The map c homotopic a l ly c o- e qualizes t M ( M /∂ M , ∗ ) and ∆ , i.e., c ◦ t M ( M /∂ M , ∗ ) ≃ c ◦ ∆ . 16 JOHN R . KLEIN AND BRUCE WILLIAMS Pr o of. The argumen t will use the comm utativ e pushout diagram of pairs ( S, S 0 ) / /   ( C , C 0 )   ( D , D 0 ) / / ( M × M , ( ∂ M ) × M ) where ( D , D 0 ) denotes unit tangen t disk bundle of ( M , ∂ M ), ( S, S 0 ) is the unit sphere bundle and ( C , C 0 ) is the complemen t. L et π 1 : ( M × M , ( ∂ M ) × M ) → ( M , ∂ M ) b e the first fa ctor pro jection. Let p D : ( D , D 0 ) → ( M , ∂ M ) b e its restriction to ( D , D 0 ); this is fibratio n pair with fib er D d . Similarly , let p C : ( C , C 0 ) → ( M , ∂ M ) and p S : ( S, S 0 ) → ( M , ∂ M ) b e the restric tions to ( C, C 0 ) and ( S, S 0 ) resp ectiv ely . Eac h of these is a lso a fibration pair. The fib er of p S is S d − 1 and the fib er of p C is M 0 , where M 0 is the effect of remo ving an op en ball from the in terior of M . W e therefore ha v e a pushout square of fib ers S d − 1 / /   M 0   D d / / M . Then the additivit y axiom implies t M ( M , ∂ M ) = j 1 t D d ( M , ∂ M ) + j 2 t M 0 ( M , ∂ M ) − j 12 t S d − 1 ( M , ∂ M ) , where j S for S ⊂ { 1 , 2 } is induced by the eviden t inclusion map into M . Then by the homoto p y in v ariance and normalization axioms t D d ( M , ∂ M ) = i ◦ t ∗ ( M , ∂ M ) = i , where i : M /∂ M → D /D 0 arises from the zero section. By definition, j 1 i is the reduced diagonal map ∆ : M /∂ M → M /∂ M ∧ M + . Conse- quen tly , j 1 t D d ( M , ∂ M ) = ∆ . In particular, c ◦ j 1 t D d ( M , ∂ M ) = c ◦ ∆ . T o complete t he pro of of the prop osition, it will suffice to show that c applied to eac h of the terms j 2 t M 0 ( M , ∂ M ) and j 12 t S d − 1 ( M , ∂ M ) is trivial, f or this will yield c ◦ t M ( M , ∂ M ) = c ∆. THE REFINED TRANSFER 17 T o see wh y c ◦ j 2 t M 0 ( M , ∂ M ) is trivial, recall that c is defined b y collapsing C ⊂ M × M to a p oin t, whereas j 2 t M 0 ( M , ∂ M ) is giv en by a comp osite of t he form M /∂ M t M 0 ( M ,∂ M ) − − − − − − − → C /C 0 − − − → ( M × M ) / ( ∂ M × M ) . The triviality of cj 2 t M 0 ( M , ∂ M ) therefore follow s from the fact that it factors t hrough C /C 0 . A similar argumen t shows that cj 12 t S d − 1 ( M , ∂ M ) is trivial.  Pr o of of T h e or e m A when B is a p oint. By 4.1 and 4.2 and diagram (3) w e ha v e t M = c ◦ ( α ∧ 1 ) ◦ t M = c ◦ t M ( M /∂ M , ∗ ) ◦ α = c ◦ ∆ ◦ α . This shows that t M is determined b y the maps c, ∆ and α whose defi- nition is indep enden t of t M .  5. Interpret a tion Motiv ated b y P eter May’s pap er on the Euler characteristic in the setting of deriv ed categories [M], w e giv e an alternativ e interpretation of what w e ha ve just s how n in terms of the algebra of the stable homotop y category . This section is indep enden t of the rest of the pa p er. Giv en F as ab o v e, recall that D ( F + ) denotes the S -dual of o f F + . Then D ( F + ) is a ring sp ectrum with unit u : S 0 → D ( F + ) whic h rep- resen ts a desusp ension of the map α app earing ab ov e. In what follo ws we consider F + as an ob ject of t he stable homotopy category . Then we ha v e an action µ : D ( F + ) ∧ F + → F + whic h is defined a s the formal a djoin t to the ma p D ( F + ) → hom( F + , F + ) giv en b y ma pping a stable map f : F + → S 0 to f ∧ 1 F + comp osed with the diago nal of F + . The map µ is a homotop y theoretic v ersion of the collapse map c describ ed ab o v e. Lemma 4.1 in this lang uage asserts that F + is a D ( F + )-mo dule. F urthermore, F + is a coalgebra in the stable category , and one has a co-action map κ : D ( F + ) → D ( F + ) ∧ F + whic h expresses D ( F + ) as an F + -como dule: it can b e defined as the linear dual of µ (= maps in to S 0 ). Then κ is a homotopy theoretic v ersion of the diagonal map ∆. Corollary 5.1. With r esp e ct to ab ove, we have: 18 JOHN R . KLEIN AND BRUCE WILLIAMS • the diag r am S 0 ∧ F + u ∧ 1 / / D ( F + ) ∧ F + S 0 u / / t F O O D ( F + ) ∧ S 0 1 D ( F + ) ∧ t F O O is ho motopy c om mutative (cf. diagr am (3 ) ); • The c omp osite S 0 ∧ F + u ∧ 1 − − − → D ( F + ) ∧ F + µ − − − → F + = S 0 ∧ F + is ho motopic to the identity; • The map µ hom otopic al ly c o e qualizes 1 D ( F + ) ∧ t F and κ (cf. 4.2). F rom 5.1 w e immediately infer t F = µ ◦ κ ◦ u . 6. Proof of The orem A Let p : E → B b e a fibration with homotop y finite fib ers. Assume p admits a compact TOP reduction q : W → B . When B is homotopy finite, one can replace q with its topolo gical s table normal bundle along the fib ers to obtain a new compact TOP reduction whic h is a co dimen- sion zero subbundle of the trivial bundle B × R j for j sufficien tly large (cf. [BS, p. 5 99], [RS]). Consequen tly , w e can assume without loss in generalit y that q comes equipped with a fib erwise co dimension zero top o logical em b edding W ⊂ B × R d . W e let ∂ v W → B b e the fib erwi s e b oundary of q . This is the bundle whose fib er at b ∈ B is give n b y ∂ W b . The idea of the pro of of Theorem A will b e to adapt the metho d of § 4 to t he fib erwise top ological setting. The pro of will hinge up on the follow ing structure, whic h is a ssumed to v ary contin uously in b ∈ B : • The fib ers W b come equipped with a degree one collapse map S d → W b /∂ W b and • The diag onal map ( W b , ∂ W b ) → ( W b × W b , ( ∂ W b ) × W b ) has a compact tubular neighborho o d. The first of these prop erties is giv en b y taking the Thom-P on try agin collapse of the embedding W b ⊂ { b } × R d . T he second prop erty is discusse d in [BS, p. 599]. THE REFINED TRANSFER 19 Pr o of of T h e or e m A. As in § 4, t he pro of b egins b y considering a com- m utativ e diagram (the fib ered analogue of diagram (1 )): ( S d × W, W ) ( α × B 1 , 1) / / (1 × q, q )   (( W / /∂ v W ) × B W , W ) ( q ∗ ,q )   ( S d × B , B ) ( α, 1) / / ( W / /∂ v W , B ) . Here W / /∂ v W denotes the pushout of t he diagra m B ← ∂ v W ⊂ W , and q ∗ denotes t he pullbac k of q : W → B along the map W / /∂ v W → B . Note that the fib ers of W / /∂ v W → B are giv en by W b /∂ W b . The map α is giv en b y the fib erwise Thom-P on tryagin collapse map o f the co dimension zero embedding W ⊂ R d . Giv en a refined transfer t , we apply it to t he ab o v e and app eal to the naturality and pro duct a xioms to obtain a homotop y commutativ e diagram Σ d B W + α ∧ B 1 / / ( W / /∂ v W ) ∧ B W + Σ d B B + α / / Σ d B t ( q ) O O W / /∂ v W , t ( q ∗ ,q ) O O where Σ d B W + denotes the d - fold fib erwise suspension of W + → B (note that Σ d B B + = B × S d ). The vertical maps are refined transfer maps asso ciated with fibration pairs. This is the fib ered analog of diagram (3). Let c : ( W / /∂ v W ) ∧ B W + → Σ d B W + b e the fib erwise collapse ma p which on eac h fib er, maps the comple- men t data of the fib erwise em b edding of the diagonal to a p oin t. T o complete the pro of, w e a pp eal to the follow ing tw o assertions, whic h are fib erwise v ersions o f 4.1 and 4.2, and are pr o v ed similarly (alternativ ely , these are prov ed in the smo oth case in [BS, lem. 2 .5, eq. 2.9] and their pro of a dapts in our case). Assertion 1. The comp osition c ◦ ( α ∧ B 1) : Σ d B W + → Σ d B W + is fib erwise homotopic t o the iden tit y . Assertion 2. The map c homotopically co equalizes the t ( q ∗ , q ) and the fib erwise reduced diagonal map ∆ : W / /∂ v W → ( W / /∂ v W ) ∧ B W + . That is, c ◦ t ( q ∗ , q ) ≃ c ◦ ∆. 20 JOHN R . KLEIN AND BRUCE WILLIAMS Giv en these assertions, we obtain the equation Σ d B t ( q ) ≃ c ◦ ∆ ◦ α , whic h uniquely determines t ( q ).  7. Pr oof of Theorems B and J W e first g iv e the definition of the a ssem bly map. Giv en a homotop y functor f : T op → Sp ectra from spaces to sp ectra, the assem bly map is an asso ciated natural tra ns- formation of homotopy functors f % ( X ) → f ( X ) giving the b est appro ximation to f on the left by a homology theory . The simplest construction of f % ( X ) is to take the homotopy colimit ho colim ∆ k → X f (∆ k ) where the indexing is give n b y the category whose ob jects are the sin- gular simplices ∆ k → X and whose morphisms are giv en b y restriction to faces. The natural transformation f % ( X ) → f ( X ) is induced from the eviden t maps f (∆ k ) → f ( X ) asso ciated with eac h singular simplex ∆ k → X . Using the we ak equiv alence f (∆ k ) → f ( ∗ ), w e o btain a w eak equiv alence of functors f % ( X ) ∼ → X + ∧ f ( ∗ ) . Consequen tly , w e may think o f the a ssem bly map as a natural trans- formation X + ∧ f ( ∗ ) → f ( X ) . F or more details, see [WW1]. Pr o of of T h e or e m J. W e will give tw o pro ofs. The first, suggested by a referee, sho ws that the assem bly map is (2 r − c )-split fo r some constan t c ≥ 0. Recalling the decomp osition A ( X ) ≃ Σ ∞ ( X + ) × Wh diff ( X ) ([W3]), it will suffice to split the assem bly map of each factor. The assem bly map f or Σ ∞ ( X + ) is the iden tity map, so it clearly splits. The assem bly map for Wh diff ( X ) ha s the form X + ∧ Wh diff ( ∗ ) → Wh diff ( X ) . THE REFINED TRANSFER 21 There is a nat ural map Wh diff ( ∗ ) → X + ∧ Wh diff ( ∗ ) whic h arises fro m the base p o in t of X . The comp osite with the assem bly map yields the map Wh diff ( ∗ ) → Wh diff ( X ) that is induced b y the inclusion o f the basep oin t. As noted b y W ald- hausen [W2, p. 1 53], for r - connected X , this map is approxim ately 2 r -connected. This completes t he first pro o f. Our second pro o f sho ws that the assem bly map for A ( X ) is 2 r -split. It uses the commu tative diag ram of sp ectra (4) A ( X ) / /   X + ∧ S 0   A ( ∗ ) / / S 0 where the vertical maps are induced by t he map X → ∗ , and the hori- zon tal ones are giv en using W aldhausen’s splitting of A ( X ) men tioned ab ov e. Supp ose that X is r -connected. Then G o o dwillie has sho wn that the square (4) is (2 r + 1)-cartesian ([G, cor. 3.3]). It follo ws that the square X + ∧ S 0 / /   A ( X )   S 0 / / A ( ∗ ) is 2 r -cartesian, where the horizontal maps are giv en b y the natura l transformation from stable homotopy to the algebraic K -theory of spaces induced from the inclusion functor fro m finite sets to finite spaces. Using the basep oint for X , it fo llo ws that X ∧ S 0 → A ( X ) → A ( ∗ ) is a homotop y fib er seq uence up through dime nsion 2 r in the s ense that the map from X ∧ S 0 to the homotopy fib er of the map A ( X ) → A ( ∗ ) is 2 r -connected. F urthermore, we ha v e a homotopy fib er sequence A ( ∗ ) ∧ X → A % ( X ) → A ( ∗ ) . 22 JOHN R . KLEIN AND BRUCE WILLIAMS Consider the diagram X ∧ A ( ∗ ) / /   A % ( X ) / /   A ( ∗ ) X ∧ S 0 / / A ( X ) / / A ( ∗ ) where the middle v ertical m ap is the assem bly map, and the left vertical map is giv en b y smashing the map A ( ∗ ) → S 0 with the iden tit y map of X . Since the lo w er row is a ho motop y fib er sequence up throug h dimension 2 r , and the map A ( ∗ ) ∧ X → S 0 ∧ X is homotopically split, it follows that the assem bly map A % ( X ) → A ( X ) is 2 r -split. The functor B ( X ) in this case is given b y the wedge ( X ∧ S 0 ) ∨ A ( ∗ ) , and the map B ( X ) → A % ( X ) is giv en using the eviden t map X ∧ S 0 → X ∧ A ( ∗ ) ≃ A % ( X ) together with the map A ( ∗ ) → A % ( X ) coming from the basep oint of X .  R emark 7 .1 . There is no such stable splitting of the assem bly map for A ( X ) on t he category of un based spaces. F or supp ose there w ere a homotop y functor B ( X ) defined on unbase d spaces equipp ed with a natural transformation B ( X ) → A % ( X ) suc h that the comp osite B ( X ) → A % ( X ) → A ( X ) is (2 r − c )-connected fo r r -connected X . T a king the first stage of the Go o dwillie tow er of these functors yields maps B ( X ) → P 1 B ( X ) → P 1 A % ( X ) → P 1 A ( X ) suc h that the comp osite B ( X ) → P 1 A ( X ) is a w eak equiv alence. Since A % ( X ) = P 1 A % ( X ), we infer that A % ( X ) → P 1 A ( X ) has a section up to ho motop y . But this is imp ossible when X = ∅ is the empt y space, since A % ( ∅ ) = ∗ whereas P 1 A ( ∅ ) is not contractible. W e are indebted to a referee f or explaining this argumen t to us. Pr o of of T h e or e m B. Consider the fibration p : E → B with section whose underlying map is ( r + 1)-connected. Then t he fib erwise v ersion of the assem bly map A % B ( E ) → A B ( E ) is defined a nd is a map of fib ered sp ectra ov er B (cf. [D WW, p. 51]). Applying the metho d of our second pro of of Theorem J in a fib erwise THE REFINED TRANSFER 23 manner, w e obtain a fib ered sp ectrum B B ( E ) and a map B B ( E ) → A % B ( E ) suc h that the comp osite B B ( E ) → A % B ( E ) → A B ( E ) is 2 r -connected (note: the fib er of B B ( E ) at a p oin t b ∈ B is iden tified with B ( E b )). By sligh t abuse of notation, consider the fib erwise assem bly map as a map of (asso ciated) fib erwise infinite lo op spaces. Then a ssuming that B ha s the homotop y t yp e of a cell complex of dimension ≤ 2 r , the previous paragr aph sho ws that induced map of section spaces sec( A % B ( E ) → B ) → sec( A B ( E ) → B ) is surjectiv e on path comp onen ts. By the “Con v erse Riemann-Ro ch Theorem” of Dwy er, W eiss and Williams ([DW W, cor . 10.18]) , it follow s that p : E → B admits a compact TOP reduction.  8. Proof of The orems F and G Pr o of of T h e or e m F. Recall tha t W aldha usen’s space A ( ∗ ) ha s the same rational homoto p y type as K ( Z ), the algebraic K -theory space of the in tegers. Borel sho w ed that the rational cohomology of K ( Z ) is a n ex- terior algebra on classes b 4 k + 1 in degree 4 k + 1 for k > 0 [B]. Therefore, H 4 k + 1 ( A ( ∗ ); Z ) has a nontrivial torsion free summand for k > 0. Let ∨ k S n b e a k -fold w edge of n -spheres and let H n k denote the to p o- logical monoid o f based homotopy equiv alences of ∨ k S n (cf. [W1, p. 385]. Then w e hav e a homomorphism H n k → H n +1 k giv en by suspension and a ho momorphism H n k → H n k +1 giv en b y we dging on a single cop y of the iden tit y map. One of the standard definitions of A ( ∗ ) is Z × lim k ,n B H n k + , where “ B ” denotes the classifying space functor, and “ + ” denotes Quillen’s plus construction. In particular, the natural ma p ι : Z × lim k ,n B H n k → A ( ∗ ) , (from the space to its plus construction) is a rational cohomology iso- morphism. Let x ∈ H 4 i +1 ( A ( ∗ ); Z ) b e an y non-t orsion elemen t. Then the restriction ι ∗ x ∈ H 4 i +1 ( B H n k ; Z ) is a lso non- torsion when k and n are c hosen sufficien t ly large. Fina lly , let B ⊂ B H n k 24 JOHN R . KLEIN AND BRUCE WILLIAMS b e a connected, finite sub complex whic h supp orts the class ι ∗ x . This inclusion can b e though t of as a classifying map f or a fibration p : E → B whose fib er at the basep oin t is ∨ k S n . Theorem F will now follow from: Claim. T h e fibr ation p : E → B do es not admit a c omp act fib er smo oth- ing. Howeve r, p admits a c omp a c t TOP r e duction pr ovide d n is suffi- ciently lar ge. The second part of the claim follo ws directly fro m Theorem B. T o pro v e the first part, it will b e sufficien t b y the work of D wy er, W eiss and Williams to prov e t hat the A -value d tr ac e ma p χ A ( p ) : B → A ( ∗ ) (cf. b elow) do es not admit a factorization up to homotopy as B → Q ( S 0 ) → A ( ∗ ) . This is sufficien t b ecause the theory (cf. [D WW, § 12]) sho ws tha t the existence of a compact fib er smo othing w ould imply the existence of suc h a factorization. Since Q ( S 0 ) has trivial rational cohomology in p ositive degrees, it suffices to sho w that the A -v alued trace is rationally no n-trivial in cohomology in degree 4 i + 1. W e first giv e a quic k sk etc h of the construction of the A -v alued trace map using the alternativ e definition of A ( ∗ ) as the algebraic K -theory of the category of homotop y finite based spaces (with cofibrations and w eak equiv alences). Deferring to W aldhausen’s notat ion, let w R hf ( ∗ ) b e the cat egory whose o b jects are based, homotopy finite cofibran t top ological spaces, and whose morphisms are we ak ho motop y equiv a- lences. In particular, a homotopy finite space F determines an ob ject of w R hf ( ∗ ), namely F + . W aldhausen a lso gives a ‘1- sk eleton’ inclusion map j : | w R hf ( ∗ ) | → A ( ∗ ) ([W1, p. 3 29]), so w e can regard F + as p oint of either the realization | w R hf ( ∗ ) | or of A ( ∗ ). Apply this construction to each fib er of the fibration p . This give s for eac h x ∈ B , a p o in t ( F x ) + ∈ A ( ∗ ) which can b e arranged so as t o v ary con tin uously in x (the reader is referred to [DWW, § 1.6] for the details). This yields the desired trace map χ A ( p ) : B → A ( ∗ ). THE REFINED TRANSFER 25 F rom the construction w e ha v e g iv en, it is immediate that ha v e a factorization of χ A ( p ) as B v − − − → | w R hf ( ∗ ) | j − − − → A ( ∗ ) , where v is a contin uous rectification o f the map x 7→ ( F x ) + . On t he other hand, since the fibration p : E → B comes equipp ed with a preferred section (arising from the w edge p oint), each fib er F x is auto matically a based space. W e therefore ha v e ano ther o b ject F x ∈ w R hf ( ∗ ), so w e hav e another map u : B → | w R hf ( ∗ ) | giv en b y x 7→ F x . W aldhausen has a lso sho wn ([W1 , prop. 2.2.5]) that the comp onen t of | w R hf ( ∗ ) | whic h con tains the fib ers F x is homotopy equiv alent to the classifying space B H n k , and with respect to this iden- tification, u can b e regarded as the classifying map o f the fibration p . In particular, the comp osition B ⊂ − − − → B H n k → | w R hf ( ∗ ) | j − − − → A ( ∗ ) coincides with j ◦ u up to homotopy and is therefore rationally non- trivial in degree 4 i + 1 . No w, using W aldhausen’s additivity theorem ( [W1, prop. 1.3.2]), j ◦ v coincides with the map x 7→ ( F x ) ∨ S 0 , whic h is we dge sum of the map j ◦ u and the constan t map with v alue S 0 ∈ A ( ∗ ), and recall that w edge sum gives the H -space structure on A ( ∗ ) ([W1, p. 330 ]). In particular, the maps j ◦ v and j ◦ u coincide on rational cohomology in p ositiv e degrees, so j ◦ v is rationa lly non-trivial in degree 4 i + 1. This completes b oth the pr o of of the claim and a lso the pro of of Theorem F.  Pr o of of T h e or e m G. W aldhausen [W2, § 3] constructs a map B F → A ( ∗ ) where F is the top ological monoid of based stable self homotopy e quiv- alences of the sphere. Let F k denote the top olog ical monoid of based (unstable) self ho- motop y equiv alences of S k . Then, up to ho motop y , the comp o site B F k → B F → A ( ∗ ) can b e con v enien tly described as follows . Recall that w R hf ( ∗ ) 26 JOHN R . KLEIN AND BRUCE WILLIAMS is the category of we ak equiv alences of cofibrant based spaces. Let w R hf ( ∗ ) ( S k ) denote the comp onen t of w R hf ( ∗ ) that contains t he sphere S k . Then the realization | w R hf ( ∗ ) ( S k ) | has the ho motop y t yp e of B F k ([W1, pro p. 2 .2.5]), and with resp ect to this iden tification, the map B F k → A ( ∗ ) is the ‘1- sk eleton’ inclusion j : | w R hf ( ∗ ) ( S k ) | ⊂ A ( ∗ ) (cf. [W1, p. 329]). B¨ okstedt and W aldhausen [BW, p. 419] ha v e sho wn that the com- p osite map B F → A ( ∗ ) → Wh diff ( ∗ ) is non-trivial on homotopy gro ups in degree three. The second map in the comp o site is the splitting map for A ( ∗ ) ≃ Q ( S 0 ) × Wh diff ( ∗ ). By the F reuden thal susp ension theorem, the homomorphism π 3 ( B F 3 ) → π 3 ( B F ) = Z 2 is an isomorphism. On the lev el of spherical fibrat ions, this gro up is generated b y the clutc hing construction of a map S 2 × S 3 → S 3 whose asso ciated Hopf construction S 5 → S 3 represen ts η 2 , where η ∈ π st 1 ( S 0 ) is represen ted b y the Ho pf map S 3 → S 2 . The clutc hing construction pro duces the spherical fibration S 3 → E → S 3 stated in Theorem G. It follo ws from the computation of B¨ okstedt and W aldhausen that the image of this generator under the homomorphism π 3 ( B F 3 ) → π 3 ( A ( ∗ )) is not an elemen t of the subgroup Z 24 = π 3 ( QS 0 ) ⊂ π 3 ( A ( ∗ )) . F rom the theory o f Dwy er, W eiss and Williams [DWW, § 12], w e infer the fibrat ion fails to hav e a compact fib er smo othing. On the other hand, the fibration is admits a compact TOP reduction b y Theorem B. Since this fibration represen ts a torsion elemen t , it is not detected rationally .  9. Pr oof of Theorem I Let t ( p ) , t ′ ( p ) : B + → E + b e r efined transfers asso ciated t o a fibra- tion p : E → B . W e will show that the traces tr t ( p ) , tr t ′ ( p ) : B + → S 0 THE REFINED TRANSFER 27 coincide. Let S B p : S B E → B b e the fib erwise susp ension of p . Let C B p : C B E → B b e the mapping cone of p . Then the inclusion B → C B E is a fib er homotop y equiv alence. Apply t he additivity and normalization axioms to the pushout E / /   C B E   C B E / / S B E and then tak e the asso ciated tr aces t o get tr t ( S B p ) = 1 + 1 − tr t ( p ) , where 1 : B + → S 0 is the unit map. Applying fib erwise susp ension again, we obtain tr t ( S 2 B p ) = tr t ( p ) . Iterating this last equation j -times, w e get tr t ( S 2 j B p ) = tr t ( p ) . If j is sufficien tly large, the fibration S 2 j B p a dmits a compact TOP reduction by Theorem B. By Theorem A, t and t ′ agree o n S 2 j B p . T a king tra ces w e conclude tr t ( p ) = tr t ′ ( p ). 10. Pr oof of Theore m H Let m b e the dimension of the finite complex X . Let X k denote the k -sk eleton, and let X ( k ) denote the quotient X k /X k − 1 . Consider t he fibration p : E → B . A t each fib er E x , t here is a cofibration sequence of retractiv e spaces o v er E x of the form ( E x × X k ) ∐ E x → ( E x × X k +1 ) ∐ E x → E x × X ( k +1) , for k ≥ 0. Denote the sum operation in the category o f retractiv e spaces b y +; this is give n b y fib erwise w edge. By the additivit y theorem [W1, prop. 1.3.2], w e obtain a preferred homot op y class of path in A ( E x ) from the sum ( E x × X k ) ∐ E x + ( E x × X ( k +1) ) to ( E x × X k +1 ) ∐ E x . Summing o ver k , w e get a preferred homotop y clas s of path in A ( E x ) connecting the p oints (5) ( E x × X ) ∐ E x and m X k =0 E x × X ( k ) . 28 JOHN R . KLEIN AND BRUCE WILLIAMS Let T k denote a based set ha ving cardinalit y one more than the n umber of k -spheres in X ( k ) . Then we get an iden tification T k ∧ S k ∼ = X ( k ) . As fib erwise suspension induces the ho motop y in v erse to t he H - m ultiplication defined b y the sum (see [W1 , pro p. 1.6 .2]), and t he sum op eration is homotopy commutativ e, there is a preferred pa th b et w een the ab o v e and the sum X k E x × T 2 k + X k E x × Σ T 2 k + 1 . The latter can b e rewritten as E x × ( T 0 ∨ T 2 ∨ · · · ) + E x × Σ( T 1 ∨ T 3 ∨ · · · ) where the based sets ( T 0 ∨ T 2 ∨ · · · ) and ( T 1 ∨ T 3 ∨ · · · ) hav e the same cardinalit y under the assumption that the Euler c haracteristic o f X is zero. Consequen tly , if w e let T denote ( T 0 ∨ T 2 ∨ · · · ), the ab o v e is iden tified with ( E x × T ) ∨ Σ E x ( E x × T ) whic h, b y the additivit y theorem has a preferred homotop y class of path to the zero ob ject. No w the assignmen t x 7→ ( E x × X ) ∐ E x giv es rise to the g eneralized Euler c haracteristic of the fibration q : E × X → B , whic h is a section of the fibration A B ( E × X ) → B ([DWW , I.1]). The ab o v e arg umen t sho ws t hat this section is v ertically homotopic to the constant section giv en b y the basep oin t of each fib er A ( E x ). But the basep oin t section clearly factors through the map Q B E → A B E via the base p o in t section of the fibrat ion Q B E → B . W e now apply t he conv erse Riemann-Ro ch theorem in the smo oth case ( [D WW, § 12]) to conclude that q admits a compact fib er smo othing. This completes the pro of of Theorem H. R emark 1 0.1 . When X = ( S 1 ) × k is a torus of sufficien tly large di- mension, Theorem H b ecomes the ‘closed fib er smo ot hing theorem’ of Casson and Gottlieb [CG, p. 160]. 11. Pr oof of Theore m E By replacing M b y M × D 2 if necessary , w e can assume that M ⊂ R m is a co dimension zero compact connected smo oth submanifold suc h that ∂ M ⊂ M is 2-connected. Let E ( M , ∗ ) denote the geometric realization of the simplicial mo noid whose k - simplices a re fa milies of top ological em b eddings e : ∆ k × M → ∆ k × M THE REFINED TRANSFER 29 suc h that e comm utes with pro jection to ∆ k , e is a homoto p y equiv a- lence and is the iden tity when restricted to ∆ k × ∗ . Similarlt y , let TOP ( M , ∗ ) b e t he geometric realization of the simpli- cial gr oup whose k -simplices are self homeomorphism of ∆ k × M that preserv e ∆ k × ∗ . Then one has a for getful ho momorphism TOP( M , ∗ ) → E ( M , ∗ ) of top ological monoids whic h induces a map of classifying spaces B TOP( M , ∗ ) → B E ( M , ∗ ) , whose homotopy fib er is identified with the Borel construction E TOP( M , ∗ ) × TOP( M , ∗ ) E ( M , ∗ ) . The latter may also b e iden tified the orbit space E ( M , ∗ ) / TOP( M , ∗ ), b ecause the action of TOP( M , ∗ ) on E ( M , ∗ ) is f ree. The orbit space ma y also b e iden tified with H ( ∂ M ), the space of top ological h -cob ordisms of ∂ M . This can b e seen as follo ws: le t E ′ ( M , ∗ ) b e defined j ust as E ( M , ∗ ) but where w e now require the em- b edding e to ha v e image in ∆ k × int( M ), where int( M ) is the in terior of M . Using a choice o f colla r neighborho o d of ∂ M , one sees that the inclusion E ′ ( M , ∗ ) ⊂ E ( M , ∗ ) is a deformat ion retract. Therefore, the orbit space is also iden tified with the Bor el construction of TOP( M , ∗ ) acting on E ′ ( M , ∗ ). Define a map E ′ ( M , ∗ ) → H ( ∂ M ) b y sending an em b edding e : ∆ k × M → ∆ k × in t( M ) to the k -parameter family of h -cob ordisms (∆ k × M ) − e (∆ k × in t( M )) . Then TOP( M , ∗ ) → E ′ ( M , ∗ ) → H ( ∂ M ) is a homotop y fiber se quence (compare [WW2, p. 170]), so the assertion follo ws. T a king classifying spaces, we extend to the rig h t to obtain a homo- top y fib er sequence H ( ∂ M ) → B TOP( M , ∗ ) → B E ( M , ∗ ) . Let B = B E ( M , ∗ ), and let E → B b e the asso ciated univ ersal fibration with fib er M , obta ined as follows: the tautological action of E ( M , ∗ ) on M giv es a Borel construction E E ( M , ∗ ) × E ( M , ∗ ) M → B E ( M , ∗ ) whic h is a quasifibration. Then E → B is the effect o f con v erting the quasifibration into a fibrat ion. Using this fibration, obtains a fib erwise generalized Euler character- istic B χ → A B ( E ) . 30 JOHN R . KLEIN AND BRUCE WILLIAMS The restriction of χ to B TOP( M , ∗ ) factors through t he fib erwise as- sem bly map A % B ( E ) → A B ( E ) via an “excisiv e c haracteristic” χ % ([D WW, 7.11 ]; here we are consid- ering the fib erwise assem bly map as a map of fib erwise infinite lo op spaces). The resulting diagra m (6) B TOP( M , ∗ ) χ % / /   A % B ( E )   B χ / / A B E . is preferred homotopy comm utativ e. T a king ve rtical ho motop y fib ers, w e o btain a map of spaces H ( ∂ M ) → Ω Wh top ( M ) . This map is a comp o site of the fo rm H ( ∂ M ) ( a ) → Ω Wh top ( ∂ M ) ( b ) → Ω Wh top ( M ) , where the map ( a ) is an equiv alence in the top o logical concordance stable range (whic h is appro ximately m/ 3 b y [I3 ] and [BL]). In partic- ular, b y taking the cartesian pro duct of M with a disk of sufficien tly large dimension, we can assume that the ma p ( a ) is highly connected. The map ( b ) is is induced by applying the functor Ω Wh top to the in- clusion ∂ M → M . If w e r eplace M b y M × D k , it to o b ecomes highly connected when k -grow s b ecause ∂ ( M × D k ) → M × D k is at least ( k − 1)-connected, a nd after applying the functor the result is also ap- pro ximately k - connected (since the same is true for the functor A % and also the functor A using, sa y , [G, cor. 3.3]). The upshot of this is that w e can, b y replacing M b y M × D k , assume the comp osite map ( b ) ◦ ( a ) a w eak equiv alence up t hrough any given dimension. W e will assume this to b e the case. No w let M b e r -connected. It follows with resp ect t o o ur assump- tions that the diagra m (6) is 2 r -cartesian. By the metho d of pro o f of Theorem B, we kno w that the fib erwise assem bly map A % B ( E ) → A B ( E ) is 2 r - split in a preferred w ay . This sho ws t hat the map B TOP( M , ∗ ) → B E ( M , ∗ ) is also 2 r -split. It follows that w e hav e a preferred decomp osition of homotop y groups π ∗ (TOP( M , ∗ )) ∼ = π ∗ ( E ( M , ∗ )) ⊕ π ∗ +1 ( Wh top ( M )) THE REFINED TRANSFER 31 for ∗ < 2 r − 1 . T o complete the pro o f w e need to iden tify E ( M , ∗ ). Let I ( M , ∗ ) b e the realization of the sim plicial monoid defined jus t as E ( M , ∗ ) b ut no w with immersions in place of embeddings. By top ological transv ersalit y [KS], the inclusion map E ( M , ∗ ) → I ( M , ∗ ) is a w eak equiv a lence in our range after replacing M with M × D k for k sufficien tly larg e. Fina lly , let τ M b e the top ological tangent microbun- dle of M , whic h is a trivial fib er bundle M × R m → M since M is a co dimension zero submanifold of euclidean space. Let G ( τ M , ∗ ) b e the (simplicial) monoid whose zero simplices are pairs ( f , φ ) suc h that f : M → M is a based self homotopy equiv alence and φ : τ M → τ M is a fib er bundle isomorphism cov ering f . The k -simplices of G ( τ M ) ar e families of suc h pairs parametrized b y the standar d k -simplex. Then w e hav e an iden tification G ( τ M , ∗ ) = G ( M , ∗ ) × maps( M , TOP m ) , and the eviden t map I ( M , ∗ ) → G ( τ M , ∗ ) , is kno wn to b e a we ak equiv alence b y immersion theory [L , p. 137 ]. Assem bling t his information completes the pro of of Theorem E . R emark 11.1 . A more careful statemen t of T heorem E is as follows. Let M had dimension m and spine dimension d . Let c b e the concordance stable range of M (t his is the connectivit y of the stabilization map C ( M ) → C ( M × I ), where C ( M ) is the smo o th concordance space of M ; b y [I3 ] one has c ≥ max(2 m + 7 , 3 m + 4)). Then the map B TOP( M , ∗ ) → B G ( M , ∗ ) has a section up to homotopy on the (2 r )-sk eleton provided that b oth m − d and c are greater than 2 r . Conseque n tly , if the homotop y ty p e of M is held fixed, o ne needs the dimension of M to approx imately exceed b oth 6 r and d + 2 r for there to b e a section. 12. Appe ndix: chara cte ristic clas ses for fibra tions This section, whic h migh t b e of indep enden t inte rest, sk etches a theory of c haracteristic classes f or fibratio ns with homotopy finite base and fib ers. These classes were implicitly used in section 8. Let B b e a connected finite complex. Then a s in section 8, a fibration p : E → B with homotopy finite fib ers g iv es a n A -v alued trace map χ A ( p ) : B → A ( ∗ ) . 32 JOHN R . KLEIN AND BRUCE WILLIAMS Pulling bac k the Borel classes y 4 k + 1 ∈ H 4 k + 1 ( A ( ∗ ); Q ), w e obtain ra- tional cohomolog y classes y 4 k + 1 ( p ) ∈ H 4 k + 1 ( B ; Q ) , k > 0 . These classes v anish whenev er p admits a compact fib er smo othing. F urthermore, they satisfy the follo wing axioms: • (Naturalit y). The classes y 4 k + 1 ( p ) are natura l with resp ect to base c hange. • (Pro ducts). F or a pro duct fibration p × p ′ : E × E ′ → B × B ′ with fib er F × F ′ , w e hav e y 4 k + 1 ( p × p ′ ) = y 4 k + 1 ( p ) ⊗ χ ( F ′ ) + χ ( F ) ⊗ y 4 k + 1 ( p ′ ) , where χ ( F ) ∈ H 0 ( B ) ∼ = Z is the Euler characteristic. • (Additivit y). If E ∅ / /   E 2   E 1 / / E is a homotopy pushout of fibratio ns ov er B ha ving homotopy finite fib ers, t hen y 4 k + 1 ( p ) = y 4 k + 1 ( p 1 ) + y 4 k + 1 ( p 2 ) − y 4 k + 1 ( p 12 ) , where p S : E S → B for S ( { 1 , 2 } . R emarks 12.1 . (1). The classes y 4 k + 1 ( p ) ∈ H 4 k + 1 ( B ; Q ) are primary obstructions to finding a compact fib er smo othing. When there is a compact fib er smo othing , one has the higher Reidemeister torsion classes τ 4 k ( p ) ∈ H 4 k ( B ; Q ) defined b y Igusa [I1]. 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[W3] W aldha usen, F.: Algebraic K -theory of s paces, concordanc e, a nd stable homotopy theory . Algebraic top olog y a nd a lgebraic K -theory (Princeton, N.J., 1 983), 39 2–417 , Ann. of Math. Stud., 11 3, 1987 [WW1] W eiss, M; Williams, B.: Assembly . Novik ov conjectures, index theo rems and rigidity , V ol. 2 (Ober wolfac h, 1993), 332–3 52, LMS Lecture Notes 22 7, Cambridge Univ. Pr ess, 1995 [WW2] W eiss, M; Williams, B.: Automorphisms of Manifolds. Surveys in Sur gery Theory , V ol. 2, Ann . of Math. Studies 149 Princeton Univ ersity Press, 2 001 W a yne St a te University, Detroit, MI 48202 E-mail addr ess : k lein@ math. wayne.edu University of Notre Dame, Notre D ame, IN 46556 E-mail addr ess : w illia ms.4@ nd.edu