THE STABLE PARAMETRIZED h-cobordism THEOREM. John Rognes. October 18th October 26th 1999

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1 THE STABLE PARAMETRIZED h-cobordism THEOREM Joh Roges October 18th October 26th 1999 These are the author s otes for lectures o the paper: Spaces of PL maifolds ad categories of simple maps (the maifold part) by F. Waldhause ad W. Vogell. Lecture I 1. Whitehead torsio ad simple homotopy equivalece. We first recall some of J.H.C. Whitehead s simple homotopy theory. A fiite polyhedro is the geometric realizatio of a fiite simplicial complex. Let K be a fixed fiite polyhedro ad cosider fiite polyhedral pairs (L,K). There is a elemetary collapse relk from (L 0,K) to (L 1,K) if L 0 is obtaied by attachig a simplex to L 1 alog a hor i its boudary, i.e., the complemet Λ i of the iterior of a face i of : L 0 = L 1 Λ We write L 0 ց L 1 relk. The equivalece relatio geerated by elemetary collapses rel K is called simple homotopy equivalece rel K. The space of deformatio retractios of L 0 oto L 1 equals the space of deformatio retractios of oto Λ, which is cotractible. I particular it is o-empty, so simple homotopy equivalece rel K implies homotopy equivalece rel K. A chai of elemetary collapses begiig with (L 0,K) ad edig with (L 1,K) thus determies a homotopy equivalece L 0 L 1 relk up to cotractible choice. Defiitio. The Whitehead group Wh 1 (K) is the set of simple homotopy classes relk of fiite polyhedral pairs (L,K) such that the iclusio K L is a homotopy equivalece. The sum operatio is give by uio alog K. The equivalece class τ(l,k) Wh 1 (K) is called its Whitehead torsio. The coditio that K L is a homotopy equivalece is equivalet to assertig that L cotais K as a deformatio retract. Suppose K is coected. Whitehead s mai theorem (1952?) is that Wh 1 (K) oly depeds o the fudametal group π 1 (K), i that there is a algebraically defied fuctor Wh 1 (π) = K 1 (Z[π])/{±π} ad a atural equivalece Wh 1 (K) = Wh 1 (π 1 (K)). There is a algebraic costructio of the Whitehead torsio class τ(l,k) Wh 1 (π 1 (K)) i terms of a model for (L,K) as a pair of simplicial complexes. 1 Typeset by AMS-TEX

2 2 JOHN ROGNES So for π 1 (K) = 1 or π 1 (K) = Z, whe Wh 1 (π) = 0, the otios of homotopy equivalece ad simple homotopy equivalece rel K agree. Geerally they do ot, as the calculatio Wh 1 (C 5 ) = Z idicates. (If π = C 5 = T T 5, a geerator is the image of 1 T +T 2 uder GL 1 (Z[π]) K 1 (Z[π]) Wh 1 (π).) 2. Parametrized h-cobordism theory. A h-cobordism (W,M) is a compact PL maifold W +1 with boudary W = M ( M I) M, such that the iclusios M W ad M W are homotopy equivaleces. The trivial h-cobordism is (M I,M), where we idetify M = M {0}. The s-cobordism theorem of Barde, Mazur ad Stalligs (1963) says that for a compact coected PL maifold M, 5, the isomorphism classes of h- cobordisms(w,m)areioe-to-oecorrespodecewiththeelemetsofwh 1 (M), via their Whitehead torsios. The trivial h-cobordism correspods to zero i Wh 1 (M). I particular there exists a isomorphism (W,M) = (M I,M) if ad oly if τ(w,m) = 0 Wh 1 (M). Such h-cobordisms are called s-cobordisms, ad such a isomorphism is called a trivializatio of (W,M). It provides (W,M) with a product structure. Such product structures have applicatios i the classificatio of maifolds, such as i the proof of the geeralized Poicaré cojecture i dimesios 5 by Smale i the smooth case (1960), ad Stalligs i the PL case. Likewise for the classificatio of acyclic maifolds such as les spaces ito PL homeomorphism vs. homotopy equivalece classes. Parametrized h-cobordism theory is cocered with the existece of product structures i parametrized families of h-cobordisms, i.e., the compariso of families of product structures o (M I,M). Defiitio. There is a space H(M) of h-cobordisms o M, or more precisely a simplicial set [q] H(M) q. Whe M is closed, a zero-simplex is a PL submaifold W M I such that (W,M) is a h-cobordism. A q-simplex is a q-parameter familyofsuchh-cobordisms, meaigaplfiberbudleover q that(i)iscotaied as a sub-budle i the trivial budle M I q q ad (ii) is a fiberwise h- cobordism. Whe M has o-empty boudary we also assume that W (U I) = U [0,t] for some eighborhood U of M M ad 0 < t < 1. The boudary eighborhood U ad height t may vary i a PL fashio withi a parametrized family. The the path compoets of H(M) are i bijective correspodece with the isomorphism classes of h-cobordisms o M, ad to elemets of Wh 1 (M) by meas of their Whitehead torsio. Let P be a fiite polyhedro. A PL fiber budle over P of h-cobordisms {(W p,m) H(M) p P} is classified by a PL map P H(M), ad vice versa. A fiber (W p,m) admits a product structure (W p,m) = (M I,M) if ad oly if p maps to the compoet of H(M) cotaiig the trivial h-cobordism (M I,M), which weviewasthebase poitih(m). Thefiber budleadmitsaglobalproduct structure, i.e., a isomorphism of PL fiber budles over P of {(W p,m)} p with the costat family {(M I,M)} p, if ad oly if the map P H(M) is homotopic to the costat map at the base poit.

3 THE STABLE PARAMETRIZED h-cobordism THEOREM 3 Hece parametrized h-cobordism theory is cocered with the homotopy classes of maps P H(M), ad i the uiversal case, with the homotopy type of the represetig space of h-cobordisms H(M). We kow π 0 H(M) = Wh 1 (M), but desire a combiatorial, computable model for the full homotopy type of H(M). 3. Categories of simple maps. The composite of two elemetary collapses is typically ot a elemetary collapse, so the arrows L 0 ց L 1 relk do ot provide the morphisms i a category. Istead there is a weaker otio, called a simple map, that geerates the same equivalece relatio as simple homotopy equivalece. Defiitio. A PL map of polyhedra is simple if it has cotractible poit iverses. I particular the poit iverses are o-empty, so the map is surjective. A simplicial map of simplicial sets is simple if its geometric realizatio has cotractible poit iverses. Agai such a map is surjective. A elemetary collapse L 0 ց L 1 relk provides a simple map f: L 0 L 1, with poit iverses sigle poits or closed itervals. Coversely a simple map is a simple homotopy equivalece. Hece the existece of a chai of simple maps relk is equivalet to the existece of a chai of elemetary collapses relk, so simple maps geerate the same equivalece relatio as simple homotopy equivalece. Lemma [W4]. The composite of two simple maps of polyhedra is a simple map. ((Look at this proof later.)) Defiitio. Let K be a fiite polyhedro. Let se h 0(K) be the category of fiite polyhedra L that cotai K as a deformatio retract, ad simple maps relk. More geerally let se h q(k) be the category with objects the q-parameter families of such fiite polyhedra, meaig PL fiber budles over q with fiite polyhedral fibers, cotaiig the trivial budle K q q as a fiberwise deformatio retract. The morphisms are maps of PL fiber budles which are simple maps relk i each fiber. The fuctor [q] se h q(k) defies a simplicial category, briefly deoted se h (K), with geometric realizatio se h (K). Let X be a compact PL maifold. There is a forgetful fuctor f : H(X) se h (X) takig a h-cobordism (W,X) o X to the uderlyig polyhedral pair (W, X), ad likewise for parametrized families. Here we view the simplicial set H(X) as a discrete simplicial category, with oly idetity morphisms. It iduces a map f: H(X) se h (X) which i tur iduces a isomorphism o π 0 : π 0 H(X) = π 0 se h (X) = Wh 1 (X) Accordig to Hatcher (1975) the map f: H(X) se h (X) is k-coected wheever = dimx is large with respect to k, ad 3k + 5 will do. This is the parametrized h-cobordism theorem of Hatcher. If this is correct, the space se h (X) is a combiatorial model for the geometric space H(X) up to dimesio k, providig the desired model for parametrized h-cobordism theory i up to k parameters.

4 4 JOHN ROGNES Ufortuately the published proof is ot satisfactory. Hatcher s argumet ivolves formig mappig cyliders of geeral PL maps, which is ot a fuctorially meaigful costructio. Istead, Waldhause ad Vogell provide a proof of a stabilized versio of this result, passig to a it over maps takig X to X I. This is the stable parametrized h-cobordism theorem of Waldhause, or rather its maifold part. 4. Stable parametrized h-cobordism theory. There is a (lower) stabilizatio map σ: H(X) H(X I) essetially takig a h-cobordism(w, X) to(w I, X I), but makig adjustmets ear (X I). Let H(X) = H(X I ) deote the homotopy it of the resultig direct system. This is the stable h-cobordism space of X. Theorem (Waldhause, Vogell). Let X be a compact PL maifold. There is a homotopy equivalece H(X) se h (X). This effects the traslatio from compact maifolds to fiite polyhedra. The o-maifold part of the theory cocers the traslatio from fiite polyhedra to fiite simplicial sets. LetsCf h (X)bethecategoryoffiitesimplicialsetscotaiigX asadeformatio retract, adsimplemaps. ItremaistoproduceahomotopyequivalecesCf h(x) se h (X) takig a simplicial set to a associated polyhedro. This is the subject of [W4]. Uio alog X makes scf h (X) a category with sums, ad a Γ-category i the sese of Segal. It admits a deloopig Wh PL (X) = sn Cf h(x), where N refers to Segal s simplicial category of sum diagrams. The Wh 1 (X) = π 0 H(X) = π 0 ΩWh PL (X)adsoWh PL (X)isacoectedspacecarryigtheWhiteheadgroup Wh 1 (X) as its fudametal group. I [W2], Waldhause establishes a homotopy equivalece ad a fiber sequece sn C h f(x) ss R h f(x ) ΩsS R h f(x ) ΩsS R f (X ) α ΩhS R f (X ). The right had term is a model for A(X), the middle term is the homology theory i X with coefficiet spectrum A( ), ad the map α is the assembly map. The left had term is homotopy equivalet to ΩWh PL (X) sc h f (X). Give all this, the full stable parametrized h-cobordism theorem asserts that for compact PL maifolds X there is a homotopy equivalece H(X) ΩWh PL (X). Here the left had side is geometrically defied, while the right had side is defied i terms of algebraic K-theory, ad is homotopy equivalet to the fiber of the assembly map for the fuctor X A(X).

5 THE STABLE PARAMETRIZED h-cobordism THEOREM 5 Lecture II 5. Categories of PL maifolds. Let M be a -dimesioal PL maifold, i.e., a polyhedro that is locally PL homeomorphic to the ball B = [ 1,1] R. Equivaletly, M is a combiatorial maifold, so it admits a triagulatio with all liks PL-homeomorphic to either B = S 1 or B 1 = D 1. Let τ M M be the taget PL microbudle of M, which is a germ of PL eighborhoods of the diagoal (M) M M. A stable framig of M is a equivalece class of isomorphisms φ: τ M ǫ k = ǫ +k of PL microbudles over M. Here ǫ k M is the trivial k-dimesioal microbudle over M. The trivializig isomorphism φ is equivalet to its stabilizatio τ M ǫ k ǫ l = τ M ǫ k+l φ id = ǫ +k ǫ l = ǫ +k+l. For stably framed maifolds M ad N, a framed map f: M N is oe such that the composite map of budles over M ǫ +k = τm ǫ k f id f τ N ǫ k = ǫ +k is the idetity. Cosider the category M 0 with objects all stably framed -dimesioal compact PL maifolds, ad morphisms all framed PL homeomorphisms of such maifolds. This category is i fact a groupoid, i.e., all morphisms are ivertible. Let hm 0 be the category with the same objects as M 0, i.e., stably framed -dimesioal compact PL maifolds, ad morphisms all framed homotopy equivaleces of such maifolds. There is a forgetful fuctor M 0 hm 0 regardig PL homeomorphisms as special cases of homotopy equivaleces. We shall eed parametrized versios of these categories. Let P be a polyhedro. A family over P of compact -dimesioal PL maifolds is a PL fiber budle π: E P with fibers M p = π 1 (p) which are compact - dimesioal PL maifolds. A stable framig of such a family of maifolds is a family of stable framigs φ p : τ Mp ǫ k = ǫ +k which vary i a PL maer with p P. This meas that there is a isomorphism Φ: τ v E ǫk = ǫ +k of PL microbudles over E, restrictig to φ p over M p E. Here τ v E = p P τ M p E. Let M q be the groupoid with objects all families over q of stably framed - dimesioal compact PL maifolds, ad morphisms all framed PL homeomorphisms ofsuch. LethM q bethecategorywiththesameobjects, admorphismstheframed PL budle maps which are fiberwise homotopy equivaleces. The ow obvious fuctor [q] M q is a simplicial groupoid, deoted M. Likewise there is a simplicial category hm, ad a forgetful fuctor M hm. We shall also eed to stabilize these categories with respect to the maifold dimesio. The stabilizatio map σ: M M +1 takes a PL maifold M to M I, ad likewise for parametrized families, etc. There is a iduced forgetful fuctor M hm.

6 6 JOHN ROGNES 6. Categories of simple maps. Let se 0 be the category of fiite polyhedra ad simple maps, ad let he 0 be the category of fiite polyhedra ad homotopy equivaleces. The forgetful fuctor se 0 he 0 regards a simple map as a homotopy equivalece. More geerally let se q be the category of PL fiber budles over q of fiite polyhedra, ad fiberwise simple PL budle maps. The [q] se q is a simplicial category, deoted se. Likewise let he q be the category of PL fiber budles over q of fiite polyhedra, ad PL budle maps that are fiberwise homotopy equivaleces. The [q] he q is a simplicial category, deoted he, ad there is a forgetful fuctor se he. There are stabilizatio maps σ: se se ad σ: he he takig a fiite polyhedro K to K I, ad similarly for parametrized families etc. 7. The mai theorem. There is a fuctor f: M se that takes a compact PL maifold M to its uderlyig fiite polyhedro, ad likewise for parametrized families. It is well defied, sice a PL homeomorphism is a simple map. Likewise there is a fuctor g: hm he, ad the two are compatible uder the forgetful fuctors. The fuctors commute with stabilizatio, ad iduce maps ad f: M se g: hm he. Hece we have the followig commutative diagram: M f se i 0 se hm g he i 0 he Lemma [WV, 1.1]. The caoical maps i 0 : se se ad i 0 : he he are homotopy equivaleces. Proof. The stabilizatio map σ: se se is homotopic to the idetity via the atural trasformatio σ id give o a polyhedro K by the projectio K I K away from I. Its poit iverses are itervals, which are cotractible, so the projectio is a simple map. The proof i the h-case is the same. The stable parametrized h-cobordism theorem will follow from the followig four claims: Theorem. Let X be a stably framed compact PL maifold. (a) The homotopy fiber of M hm at X is homotopy equivalet to the stable h-cobordism space H(X). (b) The homotopy fiber of se he at X is homotopy equivalet to the space se h (X). (c) The stabilized map f: M se

7 THE STABLE PARAMETRIZED h-cobordism THEOREM 7 is a homotopy equivalece. (d) The stabilized map is a homotopy equivalece. g: hm he The proofs of (a) ad (d) are left as exercises, to which we shall retur. We shall also retur to why we ca assume X is stably framed. The proof of (b) appears i the o-maifold part [W4]. We ow tur to the proof of (c), which is [WV, 1.2]. 8. Simple maifolds over polyhedra. We aim to prove that the map f: M se is a homotopy equivalece. As a first step, we shall costruct a simplicial category ME ad fuctors α: ME M ad β: ME se such that α is a homotopy equivalece ad fα is homotopic to β. Upo stabilizatio we get a map β: ME se such that f α i 0 β. Hece f is a homotopy equivalece if ad oly if β is oe. This replaces f by a map where the target category is ot stabilized. Let K be a fiite polyhedro. A simple -maifold over K is a -dimesioal compact PL maifold M together with a PL simple map M K. If M is also stably framed, this is a stably framed simple -maifold over K. LetME 0 bethecategorywithobjectsu: M K wherek isafiitepolyhedro ad M is a stably framed simple -maifold over K. The morphisms from u: M K to u : M K are commutative squares M M u K K where M M is a framed PL homeomorphism, ad K K is a simple map. Let ME q be the q-parameter versio of this category, with objects the simple maps M K q over q where M q is a PL fiber budle of stably framed -dimesioal compact PL maifolds, as usual, ad K is a fiite polyhedro. The morphisms are the obvious commutative squares of PL fiber budles over q. Let ME be the resultig simplicial category. Let the fuctors α: ME M ad β: ME se take the object u: M K to M ad K, respectively. Lemma. α is a homotopy equivalece. Proof. Let ι: M ME be the fuctor takig M to the idetity map M = M, viewig the target M as a fiite polyhedro. The αι = id. There is a atural u

8 8 JOHN ROGNES trasformatio ια id of edo-fuctors o ME, takig a simple map u: M K to the commutative square: M = M Hece α is a homotopy equivalece. = M u K Lemma. The maps fα ad β are homotopic. Proof. The structural map M K of a object of ME defies a atural trasformatio from fα to β of fuctors ME se. There is a stabilizatio map u σ: ME ME +1 takig the object u: M K to M I K, i.e., the product map u id: M I K I followed by projectio pr: K I K away from the iterval. This is compatible uder α with the dimesio-icreasig stabilizatio σ: M M +1, ad uder β with the trivial stabilizatio id: se se. Hece we obtai a (o-commutative) diagram: ME β se α M f se i 0 The lower left composite takes u: M K to M, ad the upper right composite takes u: M K to K. Restricted to a fixed ME, these maps are homotopic sice the simple map u: M K defies a homotopy from fα to i 0 β. However, this homotopy is ot compatible with stabilizatio, so we have to work a little harder. There is a correspodig diagram of mappig telescopes: Tel ME Tel β se Tel M Tel α i 0 Tel f Tel se The composite maps Tel ME Tel fα i 0 Tel β Tel se take (,u: M K) to (,M) ad (0,K), respectively. The edge i the source telescope from (,u: M K) to ( + 1,σ(u): M I K) maps to the edge from (,M) to (+1,M I) ad to the degeerate edge at (0,K) i the target telescope, respectively.

9 THE STABLE PARAMETRIZED h-cobordism THEOREM 9 There is a path i Tel se from (,M) to (0,K), alog the edges: (,M) u (,K) pr (,K I ) σ (0,K). The followig diagram i Tel se describes how to exted this path to a homotopy from Tel fα to i 0 Tel β. (,M) u (,K) pr (,K I ) σ (0,K) σ σ (+1,K I) σ u id pr pr (+1,M I) (+1,K) (+1,K I +1 ) σ(u) pr σ +1 = (0,K) The caoical maps ad κ: Tel ME ME κ: Tel se se are homotopy equivaleces sice the stabilizatio maps are cofibratios. Now suppose we ca show that β is a homotopy equivalece. The so is i 0 β, ad i 0 Tel β. This is homotopic to Tel fα, so also fα is a homotopy equivalece. Thus f is a homotopy equivalece, which is what we wat to prove. Lecture III 9. Spaces of thickeigs. LetS 0(K)bethesetofstablyframedsimple-maifoldsoverK. Moregeerally let S q (K) the the set of PL fiber budles over q of stably framed -dimesioal compact PL maifolds together with a PL simple map over q to K q. The simplicial set [q] S q (K) is deoted S (K).