PIECEWISE LINEAR STRUCTURES ON TOPOLOGICAL MANIFOLDS YULI B. RUDYAK Abstract. This is a survey paper where we expose the Kirby- Siebenmann results on classification of PL structures on topo- logical manifolds and, in particular, the homotopy equivalence T OP=P L = K(Z=2.3) and the Hauptvermutung for manifolds. Prologue In his paper [42 ] Novikov wrote: Sullivan's Hauptvermutung theorem was announced first in early 1967. After the careful analysis made by Bill Brow- der and myself in Princeton, the first version in May 1967 (before publication), his theorem was corrected: a necessary restriction on the 2-torsion of the group H3(M) was miss- ing. This gap was found and restriction was added. Full proof of this theory has never been written and published. Indeed, nobody knows whether it has been finished or not. Who knows whether it is complete or not? This question is not clarified properly in the literature. Many pieces of this theory were developed by other topologists later. In particu- lar, the final Kirby-Siebenmann classification of topological multidimensional manifolds therefore is not proved yet in the literature. I do not want to discuss here whether the situation is so dramatic as Novikov wrote. However, it is definitely true that there is no de- tailed enough and well-ordered exposition of Kirby-Siebenmann clas- sification, such that can be recommended to advanced students which are willing to learn the subject. The fundamental book of Kirby- Siebenmann [28 ] was written by pioneers and, in a sense, posthaste. It contains all the necessary results, but it is really "Essays", and one must do a lot of work in order to do it readable for general audience. ____________ Date: May 17, 2003. 1991 Mathematics Subject Classification. 57Q25. 1 2 YULI B. RUDYAK The job on hand is an attempt of (or, probably, an approximation to) such an expository paper. Contents Prologue 1 Introduction 3 Notation and conventions 9 Chapter 1. Architecture 10 1. A result from homotopy theory 10 2. Preliminaries on bundles and classifying spaces 13 3. Structures on manifolds and bundles 19 4. From manifolds to bundles 24 5. Homotopy PL structures on T kx Dn 27 6. The product structure theorem, or from bundles to manifolds 29 7. Non-contractibility of T OP=P L 31 8. Homotopy groups of T OP=P L 32 9. Do it 36 Chapter 2. Tools 37 10. Stable equivalences of spherical bundles 37 11. Proof of Theorem 4.6 39 12. Normal maps and F=P L 44 13. The Sullivan map s : [M, F=P L] ! PdimM 46 14. The homotopy type of F=P L[2] 49 15. Splitting theorems 54 16. Detecting families 58 17. A special case of the theorem on the normal invariant of a homeomorphism 59 Chapter 3. Applications 61 18. Topological invariance of rational Pontryagin classes 61 19. The space F=T OP 62 20. The map a : T OP=P L ! F=P L 64 21. The theorem on the normal invariant of a homeomorphism 66 22. A counterexample to the Hauptvermutung, and other examples 67 Epilogue 68 References 69 PL STRUCTURES 3 Introduction Throughout the paper we use abbreviation PL for "piecewise linear". Hauptvermutung (main conjecture) is an abbreviation for die Haupt- vermutung der kombinatorischen Topologie (the main conjecture of combinatorial topology). It seems that the conjecture was first for- mulated in the papers of Steinitz [54 ] and Tietze [59 ] in 1908. The conjecture claims that the topology of a simplicial complex determines completely its combinatorial structure. In other words, two simplicial complexes are simplicially isomorphic whenever they are homeomorphic. This conjecture was disproved by Milnor [35 ] in 1961. However, for manifolds one can state a refined version of the Haupt- vermutung. A PL manifold is defined to be a simplicial complex such that the star of every point (the union of all closed simplices containing the point) is simplicially isomorphic to the n-dimensional ball. (Equiv- alently, a PL manifold is a manifold with a fixed maximal PL atlas.) There are examples of simplicial complexes which are homeomorphic to topological manifolds but, nevertheless, are not PL manifolds (the double suspensions over Poincar'e spheres, see [6]). Moreover, there exists a topological manifold which is homeomorphic to a simplicial complex but do not admit a PL structure, see Example 22.5. Now, the Hauptvermutung for manifolds asks whether any two home- omorphic PL manifolds are PL isomorphic. Furthermore, the related question asks whether every topological manifold is homeomorphic to a PL manifold. Both these questions were solved (negatively) by Kirby and Siebenmann [27 , 28]. In fact, Kirby and Siebenmann classified PL structures on high-dimensional topological manifolds. It turned out that a topological manifold can have different PL structures, or not to have any. Below we give a brief description of these results. Let BT OP and BP L be the classifying spaces for stable topological and PL bundles, respectively. We regard the forgetful map p : BP L ! BT OP as a fibration and denote its fiber by T OP=P L. Let f : M ! BT OP classify the stable tangent bundle of a topolog- ical manifold M. It is clear that every PL structure on M gives us a p-lifting of f and that every two such liftings are fiberwise homotopic. (By the definition, a map bf: M ! BP L is a p-lifting of f if pfb= f.) It is remarkable that the inverse is also true provided that dim M 5. In greater detail, M admits a PL structure if f admits a p-lifting (the Existence Theorem 6.3), and P L structures on M are in a bijective correspondence with fiberwise homotopy classes of p-liftings of f (the 4 YULI B. RUDYAK Classification Theorem 6.2). Kirby and Siebenmann proved these the- orems and, moreover, they proved that T OP=P L is the Eilenberg-Mac Lane space K(Z=2, 3). Thus, there is only one obstruction -(M) 2 H4(M; Z=2) to a p-lifting of f, and the set of fiberwise homotopic p-liftings of f (if they exist) is in bijective correspondence with H3(M; Z=2). In other words, for every topological manifold M, dim M 5 there is a class -(M) 2 H4(M; Z=2) with the following property: M admits a PL structure if and only if -(M) = 0. Furthermore, given a homeomor- phism h : V ! M of two PL manifolds, there exists a class -(h) 2 H3(M; Z=2) with the following property: -(h) = 0 if and only if h is concordant to a PL isomorphism (or, equivalently, to the identity map 1M ). Finally, every class a 2 H3(M; Z=2) has the form a = -(h) for some homeo- morphism h. These results give us the complete classification of PL structures on a topological manifold of dimension 5. We must explain the following. It can happen that two different PL structures on M yield PL isomorphic PL manifolds (like that two p-liftings f : M ! BP L of f can be non-fiberwise homotopic). In- deed, roughly speaking, a PL structure on a topological manifold M is a concordance class of PL atlases on M (see Section 3 for accurate definitions). However, a PL automorphism of a PL manifold can turn the atlas into a non-concordant to the original one, see Example 22.3. So, in fact, the set of pairwise non-isomorphic PL manifolds which are homeomorphic to a given PL manifold is in a bijective correspondence with the set H3(M; Z=2)=R where R is the following equivalence re- lation: two PL structure are equivalent if the corresponding PL man- ifolds are PL isomorphic. The Hauptvermutung for manifolds claims that H3(M; Z=2)=R is one-element. But this is wrong. Namely, there exists a PL manifold M which is homeomorphic but not PL isomorphic to RP2n+1, see Example 22.1. So, here we have a counterexample to the Hauptvermutung. For completeness of the picture, we mention again that there are topological manifolds which do not admit any PL structure, see Exam- ple 22.4. Comparing the classes of smooth, PL and topological manifolds, we see that there is a big difference between first and second classes, and not so big difference between second and third ones. From the homotopy-theoretical point of view, one can say that the space P L=O PL STRUCTURES 5 (which classifies smooth structures on PL manifold, see Remark 6.7) has many non-trivial homotopy groups, while TOP/PL is an Eilenberg- Mac Lane space. Geometrically, one can mention that there are many smooth manifolds which are PL isomorphic to Sn but pairwise non- diffeomorphic, while any PL manifold Mn , n 5 is PL isomorphic to Sn provided that it is homeomorphic to Sn. It is worthwhile to go one step deeper and explain the following. Let M4k be a closed connected almost parallelizable manifold (i.e. M becomes parallelizable after deletion of a point). Let oek denote the minimal natural number which can be realized as the signature of the manifold M4k. In fact, for every k we have three numbers oeSk, oePLk and oeTOPk while M4k is a smooth, PL or topological manifold, respectively. Milnor and Kervaire [37 ] proved that oeSk= ck(2k - 1)! where ck 2 N. On the other hand, oePL1= 16 and oePLk= 8 fork > 1. Finally, oeTOPk = 8 for allk. So, here we can see again the big difference between smooth and PL cases. On the other hand, oePLk = oeTOPk for k > 1. Moreover, we will see below that the number 2 = 16=8 = oePL1=oeTOP1 yields the group Z=2 = ß3(T OP=P L). It makes sense to say here about low dimensional manifolds, because of the following remarkable contrast. There is no difference between PL and smooth manifolds in dimension < 7: every PL manifold V n, n < 7 admits a unique smooth structure. However, there are infinitely many smooth manifolds which are homeomorphic to R4 but pairwise non- diffeomorphic, see [17 , 26]. The paper is organized as follows. The first chapter contains the architecture of the proof of the Main Theorem: T OP=P L ' K(Z=2, 3). In fact, we comment the following graph there: 6 YULI B. RUDYAK _____________________________________________________________________ | Theorem on the normal invariant of a homeomorphism for T kx Sn | |____________________________________________________________________| | | | _________________________________|||? ________________________________* *_|| | Classification of homotopy _|__________|- Classification of homotopy* * | | PL structures on T kx Dn | | k * * | |________________________________||| |___PL_structures_on_T__________* *_ | | | | | | | | ________________________ _____________________________* *|? | | Classification Theorem_|________________|_ProductoStructureeTheorem * * | | |_______________________ | |____________________________* *|| | | | | | | || | || | | _________________________________ | | _____________________|||? | Local contractibility of the | | | |_Reduction_Theorem___|____||oehomeomorphism group || | || | |________________________________ | || | | | | | | | | | _______________________________|?|? _____________________|? | T OP=P L = K(ß, 3), ß Z=2 | | Existence Theorem | |______________________________ | |____________________| H H H H H H HH___________________________j | | | | T OP=P L = K(Z=2, 3) || |_|_______________________||_| |6 | | _________________________________________________________________________| | Existence a high-dimensional topological manifold without PL structure | |________________________________________________________________________ | |6 |6 | | | | | | _____________________________| ______________________| | Rokhlin Signature Theorem | | Freedman's Example | |____________________________ | |_____________________ | PL STRUCTURES 7 Namely, we formulate without proofs the boxed claims (and provide the necessary definitions), while we prove all the implications (arrows), i.e., we explain how a claim can be deduced from another one. The second chapter contains a proof of the Sullivan Theorem on a normal invariant of a homeomorphism for T kx Sn, and also a proof of the Browder-Novikov Theorem 4.6 about homotopy properties of normal bundles. We need this theorem in order to define the concept of normal invariant. The third chapter contains several application of the Main theorem and, in particular, the counterexample to the Hauptvermutung. Let me tell something more about the graph. As we have already seen, the classification theory of PL structures on topological manifolds splits into two parts. The first part reduces the original geometric problem to a homotopy one (classification of p-liftings of a map M ! BT OP ), the second part solves this homotopy problem by proving that T OP=P L = K(Z=2, 3). The key result for the first part is the Product Structure Theorem 6.1. Roughly speaking, this theorem establishes a bijection between PL structures on M and M x R. The Classification Theorem 6.2 and the Existence Theorem 6.3 are the consequences of the Product Structure Theorem. Passing to the second part, the description of the homotopy type of T OP=P L, we have the following. Because of the Classification Theo- rem, for n 5 there is a bijection between the set ßn(T OP=P L) and the set of PL structures on Sn. By the Smale Theorem, every PL man- ifold Mn , n 5, is PL isomorphic to Sn whenever it is homeomorphic to Sn. So, ßn(T OP=P L) = 0 for n 5. What about n < 5? Again, because of the Classification Theorem, the group ßn(T OP=P L) is in a bijective correspondence with the set of PL structures on Rk x Sn provided that k + n 5. However, this set of PL structures is uncontrollable. In order to make the situation more manageable, one can consider the PL structures on the compact manifold T kx Sn and then pass to the universal covering. We can't do it directly, but there is a trick (the Reduction Theorem 8.7) which allows us to estimate PL structure on Rk x Sn in terms of so-called ho- motopy PL structures on T kx Sn (more precisely, we should consider the homotopy PL structures on T kx Dn modulo the boundary), see Section 3 for the definitions. Now, using results of Hsiang and Shane- son [23 ] and Wall [62 , 63] about homotopy PL structures on T kx Dn, one can prove that ßi(T OP=P L) = 0 for i 6= 3 and that ß3(T OP=P L) 8 YULI B. RUDYAK has at most 2 elements. Finally, there exists a high-dimensional topo- logical manifold which does not admit any PL structure. Thus, by the Existence Theorem, the space T OP=P L is not contractible. Therefore T OP=P L = K(Z=2, 3) It is worthwhile to mention that the proof of the Product Structure Theorem uses the classification of homotopy PL structures on T k. Now I say some words about the top box of the above graph. Let Fn be the monoid of pointed homotopy equivalences Sn ! Sn, let BFn be the classifying space for Fn, and let BF = limn!1 BFn. There is an obvious forgetful map BP L ! BF , and we denote by F=P L the homotopy fiber of this map. For every homotopy equivalence of closed PL manifolds h : V ! M Sullivan [56 , 57] defined the normal invariant of h to be a certain homotopy class jF (h) 2 [M, F=P L], see Section 4. Sullivan proved that, for every homeomorphism h : V ! M, jF (h) = 0 whenever H3(M) is 2-torsion free. Moreover, this theorem implies that if, in addition, M is simply-connected then h is homotopic to a PL isomorphism. Thus the Hauptvermutung holds for simply-connected manifolds with H3(M) 2-torsion free. Definitely, the above formulated Sullivan Theorem on the Normal Invariant of a Homeomorphism is interesting by itself. However, in the paper on hand this theorem plays also an additional important role. Namely, the Sullivan Theorem for T kx Sn is a lemma in classifying of homotopy structures on T kx Dn. For this reason, we first prove the Sullivan Theorem for T kx Sn, then use it in the proof of the Main Theorem, and then (in Chapter 3) prove the Sullivan Theorem in full generality. You can also see that the proof of the Main Theorem uses the diffi- cult Freedman's example of a 4-dimensional almost parallelizable topo- logical manifold of signature 8. This example provides the equality oeTOP1 = 8. Actually, the original proof of the Main Theorem appeared before Freedman's Theorem and therefore did not use the last one. However, as we noticed above, the Freedman results clarify the rela- tions between PL and topological manifolds, and thus they should be incorporated in the exposition of the global picture. Acknowledgments. I express my best thanks to Andrew Ranicki who read the whole manuscript and did many useful remarks and com- ments. I am also grateful to Hans-Joachim Baues for useful discussions. PL STRUCTURES 9 Notation and conventions We work mainly with CW -spaces and topological manifolds. How- ever, when we quit these clases by taking products or functional spaces, we equip the last ones with the compactly generated topology, (follow- ing Steenrod [53 ] and McCord [33 ], see e.g.[48 ] for the exposition). All maps are supposed to be continuous. All neighbourhoods are supposed to be open. Given two topological spaces X, Y , we denote by [X, Y ] the set of homotopy classes of maps X ! Y . We also use the notation [X, Y ]o for the set of pointed homotopy classes of pointed maps X ! Y of pointed spaces. It is quite standard to denote by [f] the homotopy class of a map f. However, here we usually do not distinguish a map and its homotopy class and use the same symbol, say f for a map as well as for the homotopy class. In this paper this does not lead to any confusion. We use the term inessential map for null-homotopic maps; otherwise we say that a map is called essential. We use the sign ' for homotopy of maps or homotopy equivalence of spaces. We reserve the term bundle for locally trivial bundles and the term fibration for Hurewicz fibrations. Given a space F , an F -bundle is a bundle whose fiber is F , and an F -fibration is a fibration whose fibers are homotopy equivalent to F . Given a bundle or fibration , = {p : E ! B}, we say that B is the base of , and that E is the total space of ,. Furthermore, given a space X, we set , x X = {p x 1 : E x X ! B x X}. Given two bundles , = {p : E ! B} and j = {q : Y ! X}, a bundle morphism ' : , ! j is a commutative diagram g E --- ! Y ? ? p?y ?yq f B --- ! X. We say that f is the base of the morphism ' or that ' is a morphism over f. We also say that g is a map over f. If X = B and f = 1B we say that g is a map over B (and ' is a morphism over B). 10 YULI B. RUDYAK Given a map f : Z ! B and a bundle (or fibration) , over B, we use the notation f*, for the induced bundle over Z. Recall that there is a canonical bundle morphism If,,: , ! j over f, see [48 ] (or [16 ] where it is denoted by ad(f)). Following [16 ], we call If,,the adjoint morphism of f, or just the f-adjoint morphism. Furthermore, given a bundle morphism ' : , ! j with the base f, there exists a unique bundle morphism c(') : , ! f*j over the base of , such that the composition c(') * If,j , --- ! f j -- - ! j coincides with '. Following [16 ], we call c(') the correcting morphism. Given a subspace A of a space X and a bundle , over X, we denote by ,|A the bundle i*, where i : A X is the inclusion. Chapter 1. Architecture 1. A result from homotopy theory Recall that an H-space is a space F with a base point f0 and a multiplication map ~ : F x F ! F such that f0 is a homotopy unit, i.e. the maps f 7! ~(f, f0) and f 7! ~(f0, f) are homotopic to the identity rel {f0}. For details, see [4]. 1.1. Definition. (a) Let (F, f0) be an H-space with the multiplication ~ : F x F ! F . A principal F -fibration is an F -fibration p : E ! B equipped with a map m : E x F ! E such that the following holds: (i) the diagrams E x F x F -mx1--!E x F E x F -- m-! E ? ? ? ? 1x~?y ?ym p1?y ?yp p E x F --m-! E E -- - ! B commute; (ii) the map E -! E, e 7! m(e, f0) is a homotopy equivalence; (iii) for every e0 2 E, the map F - ! p-1(p(e0)), f 7! m(e0, f) is a homotopy equivalence. (b) A trivial principal F -fibration is the fibration p2 : X x F ! F with the action m : E x F ! E of the form m : X x F x F ! X x F, m(x, f1, f2) = (x, ~(f1, f2)). PL STRUCTURES 11 It is easy to see that if the fibration j is induced from a principal fibration , then j turns into a principal fibration in a canonical way. 1.2. Definition. Let ß1 : E1 ! B and ß2 : E2 ! B be two principal F -fibrations over the same base B. We say that a map h : E1 ! E2 is an F -equivariant map over B if h is a map over B and the diagram E1 x F -hx1--!E2 x F ? ? m1?y ?ym2 E1 --h-! E2 commutes up to homotopy over B. Notice that, for every b 2 B, the map hb : ß-11(b) ! ß-12(b), hb(x) = h(x) is a homotopy equivalence. Now, let p : E ! B be a principal F -fibration, and let f : X ! B be an arbitrary map. Given a p-lifting g : X ! E of f and a map u : X ! F , consider the map gxu m gu : X --- ! X x X --- ! E x F -- - ! E. It is easy to see that the correspondence (g, u) 7! gu yields a well- defined map (action) (1.1) [Liftpf] x [X, F ] ! [Liftpf]. In particular, for every p-lifting g of f the correspondence u 7! gu induces a map Tg : [X, F ] ! [Liftpf]. 1.3. Theorem. Let , = {p : E ! B} be a principal F -fibration, and let f : X ! B be a map where X is assumed to be paracompact and locally contractible. If F is an H-space with homotopy inversion, then the above action (1.1) is free and transitive provided [Liftpf] 6= ;. In particular, for every p-lifting g : X ! E of f the map Tg is a bijection. Proof. We start with the following lemma. 1.4. Lemma. The theorem holds if X = B, f = 1X and , is the trivial principal F -fibration. Proof. In this case every p-lifting g : X ! X x F of f = 1X determines and is completely determined by the map __g: X :--g-! X x F --p2-! F. 12 YULI B. RUDYAK In other words, we have the bijection [Liftpf] ~=[X, F ], and under this bijection the action (1.1)turns into the multiplication [X, F ] x [X, F ] ! [X, F ]. Now the result follows since [X, F ] is a group. We finish the proof of the theorem. Consider the induced fibration f*, = {q : Y ! X} and notice that there is an [X, F ]-equivariant bijection (1.2) [Liftpf] ~=[Liftq1X ]. Now, suppose that [Liftpf] 6= ; and take a p-lifting g of f. Regarding Y as the subset of X x E, define the F -equivariant map h : X x F ! Y, h(x, a) = (x, g(x)a), x 2 X, a 2 F. It is easy to see that the diagram X x F --h-! Y ? ? p1?y ?yq X _______X commutes, i.e. h is a map over X. Since X is a locally contractible paracompact space, and by a theorem of Dold [9], there exists a map k : Y ! X x F over X which is homotopy inverse over X to h. It is easy to see that k is an equivariant map over X. Indeed, if m1 : X x F x F ! X x F and m2 : Y x F ! Y are the corresponding actions then m1(k x 1) ' khm1(k x 1) ' km2(h x 1)(k x 1) =' km2(hk x 1) ' km2, where ' denotes the homotopy over X. In particular, there is an [X, F ]-equivariant bijection [Liftq1X ] ~=[Liftp11X ] where p1 : X xF ! X is the projection. Now we compose this bijection with (1.2)and get [X, F ]-equivariant bijections [Liftpf]] ~=[Liftq1X ] ~=[Liftp11X ], __ and the result follows from Lemma 1.4. |__| 1.5. Example. If p : E ! B is an F -fibration then p : E ! B is a principal F -fibration. Here denotes the loop functor. PL STRUCTURES 13 2. Preliminaries on bundles and classifying spaces Here we give a brief recollection on Rn bundles, spherical fibrations and their classifying spaces. For details, see [48 ]. We define a topological Rn-bundle over a space B to be an Rn-bundle p : E ! B equipped with a fixed section s : B ! E. Given two topological Rn-bundles , = {p : E ! B} and j = {q : Y ! X}, we define a topological morphism ' : , ! j to be a commutative diagram g E --- ! Y ? ? (2.1) p?y ?yq f B --- ! X where g preserves the sections and yields homeomorphism of fibers. The last one means that, for every b 2 B, the map gb : Rn = p-1(b) ! q-1 (f(b) = Rn, gb(a) = g(a) for everya 2 p-1(b) is a homeomorphism. As usual, we say that f is the base of the mor- phism '. Topological Rn-bundles can also be regarded as (T OPn, Rn)-bundles, i.e. Rn-bundles with the structure group T OPn. Here T OPn is the topological group of self-homeomorphism f : Rn ! Rn, f(0) = 0. The classifying space BT OPn of the group T OPn turns out to be a clas- sifying space for topological Rn-bundles over CW -spaces. This means there exists a universal topological Rn-bundle flnTOP over BT OPn with the following 2.1. Universal Property. For every topological Rn-bundle , over a CW -space B, every CW -subspace A of B and every morphism _ : ,|A ! flnTOP of topological Rn-bundles, there exists a morphism ' : , ! flnTOP which is an extension of _. In particular, for every topological Rn-bundle , over B there exists a morphism ' : , ! flnTOP of topological Rn-bundles. We call such ' a classifying morphism for ,. The base f : B ! BT OPn of ' is called a classifying map for ,. It is clear that , is isomorphic over B to f*flnTOP. 2.2. Proposition. Let '0, '1 : , ! flnTOP be two classifying mor- phisms for ,. Then there exists a classifying morphism : , x I ! flnTOP such that |, x {i} = 'i, i = 0, 1. In particular, a classifying map f for , is determined by , uniquely up to homotopy. 14 YULI B. RUDYAK Proof. This follows from the universal property 2.1 applied to , x_I,_if we put A = X x {0, 1} where X denotes the base of ,. |__|. A piecewise linear (in future PL) Rn-bundle is a topological Rn- bundle , = {p : E ! B} such that E and B are polyhedra and p : E ! B and s : B ! E are PL maps. Furthermore, we re- quire that, for every simplex B, there is a PL homeomorphism h : p-1( ) ~= x Rn with h(s( )) = x {0}. (For definitions of PL maps, see [24 , 47].) A morphism of PL Rn-bundles is a morphism of topological Rn- bundles where the maps g and f in (2.1)are PL maps. There exists a universal PL Rn-bundle flnPL over a certain space BP Ln. This means that the universal property 2.1 remains valid if we replace flnTOP by flnPLand öt pological Rn bundle" by "PL Rn-bundle" there. So, BP Ln is a classifying space for PL Rn-bundles. Notice that BP Ln can also be regarded as the classifying space of a certain group P Ln (which is constructed as the geometric realization of a certain simplicial group), [29 , 30]. A sectioned Sn-fibration is defined to be an Sn-fibration p : E ! B equipped with a section s : B ! E. Morphisms of sectioned Sn- fibrations are defined to be diagrams like (2.1) where each map gb is assumed to be a pointed homotopy equivalence. There exists a universal sectioned Sn-fibration flnFover a certain space BFn. This means that the universal property 2.1 remains valid if we replace flnTOP by flnF and öt pological Rn bundle" by "sectioned Sn- fibration" there. So, BFn is a classifying space sectioned Sn-fibrations. The space BFn can also be regarded as the classifying space for the monoid Fn of pointed homotopy equivalences (Sn, *) ! (Sn, *). Be- cause of this, we shall use the brief term "(Sn, *)-fibration" for sectioned Sn-fibrations. Furthermore, we will also use the term "Fn-morphism" for morphism of sectioned Sn-fibrations. Finally, an Fn-morphism over a space is called an Fn-equivalence. We need also to recall the space BOn which classifies n-dimensional vector bundles. The universal vector bundle over BOn is denoted by flnO. This space is well-known and described in many sources, e.g. [38 ]. Since flnPLcan be regarded as the (underlying) topological Rn-bundle, there is a classifying morphism (2.2) ! = !PLTOP(n) : flnPL! flnTOP. PL STRUCTURES 15 We denote by ff = ffPLTOP(n) : BP Ln ! BT OPn the base of this mor- phism. Similarly, given a topological Rn-bundle , = {p : E ! B}, let ,o denotes the Sn-bundle ,o = {po : Eo ! B} where Eo is the fiber- wise one-point compactification of E. Notice that the added points ("infinities") give us a certain section of ,o. In other words, the T OPn-action on Rn extends uniquely to a T OPn-action on the one-point compactification Sn of Rn, and ,o is the (T OPn, Sn)-bundle assoc* *iated with ,. Furthermore, the fixed point 1 of the T OPn-action on Sn yields a secti* *on of ,o. So, ,o can be regarded as an (Sn, *)-fibration over B. In particular, (flnTOP)o can be regarded as an (Sn, *)-fibration over BT OPn. So, there is a classifying morphism !TOPF(n) : flnTOP ! flnF. We denote by ffTOPF(n) : BT OPn ! BFn the base of !TOPF(n). Finally, we notice that an n-dimensional vector bundle over a poly- hedron X has a canonical structure of PL Rn-bundle over X. Similarly to above, this gives us a (forgetful) map ffOPL(n) : BOn ! BP Ln. So, we have a sequence of forgetful maps 0 ff00 ff000 (2.3) BOn -ff!BP Ln -! BT OPn -! BFn where ff0= ffOPL(n), etc. 2.3. Constructions. 1. Given an F -bundle , = {p : E ! B} and an F 0-bundle ,0 = {p0 : E0 ! B0}, we define the product , x ,0 to be the F x F 0-bundle p x p0: E x E0 ! B x B0. 2. Given an F -bundle , = {p : E ! B} with a section s : B ! E and an F 0-bundle ,0 = {p0 : E0 ! B0} with a section s0 : B0 ! E0, we define the smash product , ^ ,0 to be the F ^ F 0-bundle as follows. The map p x p0 : E x E0 ! B x B0 passes through the quotient map q : E x E0 ! E x E0=(E x s(B0) [ E0x s(B), and we set , ^ j = {ß : E x E0=(E x s(B0) [ E0x s(B) ! B x B0, where ß is the unique map with p x p0= ßq. Finally, the section s and s0 yield an obvious section of ß. 16 YULI B. RUDYAK 3. Given an Rm -bundle , and an Rn-bundle j over the same space X, the Whitney sum of , and j is the Rm+n -bundle , j = d*(, x j) where d : X ! X x X is the diagonal. Notice that if , and j are a PL Rm and PL Rn-bundle, respectively, then , x j is a PL Rm+n -bundle. 4. Given a sectioned Sm -bundle , and sectioned Sn-bundle j over the same space X, we set , y j = d*(, ^ j). We denote by rn = rTOPn : BT OPn ! BT OPn+1 the map which classifies flnTOP `1BTOPn. The maps rPLn : BP Ln ! BP Ln+1 and rOn: BOn ! BOn+1 are defined in a similar way. We can also regard the above map rn : BT OPn ! BT OPn+1 as a map induced by the standard inclusion T OPn T OPn+1. Using this approach, we define rFn : BFn ! BFn+1 as the map induced by the standard inclusion Fn Fn+1, see [31 , p. 45]. 2.4. Remarks. 1. Regarding Rm as the bundle over the point, we see that (Rm )o = (Sm ) and, moreover, (Rm x Rn)o = Sm ^ Sn, i.e.(Rm Rn)o = Sm y Sn. Therefore (, j)o = ,o y jo for every Rm -bundle , and Rn-bundle j. 2. Generally, the smash product of (sectioned) fibrations is not a fibrations. But we apply it to bundles only and so do not have any troubles. On the other hand, there is an operation ^h, the homotopy smash product, such that , ^h j is the (F ^ G)-fibration over X x Y if , is an F -fibration over X and j is an G-fibration over Y , see [48 ]. In particular, one can use it in order to define an analog of Whitney sum for spherical fibrations and then use this one in order to construct the map BFn ! BFn+1. The spaces BOn, BP Ln, BT OPn and BFn are defined uniquely up to weak homotopy equivalence. However, it is useful for us to work with more or less concrete models of classifying spaces BOn, etc. In greater detail, we do the following. Choose classifying spaces B0Fn for (Sn, *)-fibrations (i.e., in the weak homotopy type BFn) and consider the maps rFn : B0Fn ! B0Fn+1 as above. We can assume that every B0Fn is a CW -complex and every rn is a cellular map. We define BF to be the telescope (homotopy direct limit) of the sequence . . .---! B0Fn --rn-! B0Fn+1 --- ! . .,. PL STRUCTURES 17 see e.g. [48 ]. Furthermore, we define BFn to be the telescope of the finite sequence rn-1 0 . . .---! B0Fn-1 --- ! B Fn. (Notice that BFn ' B0Fn.) So, we have the sequence (filtration) . . .BFn BFn+1 . ... S Here BF = BFn, BFn is closed in BF and BF has the direct limit topology with respect to the filtration {BFn}. Moreover, if f : K ! BF is a map of a compact space K then there exists n such that f(K) BFn. Now, for every n consider a CW -space B0T OPn in the weak ho- motopy type BT OPn and define B00T OP to be the telescope of the sequence . .-.-- ! B0T OPn --rn-! B0T OPn+1 -- - ! . . .. Furthermore, we define B00T OPn to be the telescope of the finite se- quence rn-1 0 . . .--- ! B0T OPn-1 --- ! B T OPn. So, we have the diagram . . . B00T OPn B00T OPn+1 . . .B00T OP ? ? ? ? ? ? y y y p . . . BFn BFn+1 . . . BF where the map p is induced by maps ffTOPF(n). Now we convert every vertical map to a fibration (using Serre construction). Namely, we set fi BT OP = {(x, !) fix 2 B00T OP, ! 2 (BF )I, p(x) = !(0)} and define ffTOPF : BT OP ! BF by setting ffTOPF(x, !) = !(1). Fi- nally, we set fi BT OPn = {(x, !) 2 BT OP fix 2 B00T OPn, ! 2 (BFn)I (B00T OP )I} So, we have the commutative diagram . . .BT OPn BT OPn+1 . . .BT OP ? ? ? ? ? ? y y y p . . . BFn BFn+1 . . . BF where all the vertical maps are fibrations. 18 YULI B. RUDYAK Now it is clear how to proceed and get the diagram . . . BOn BOn+1 . . . BO ? ? ? ? ? ? O y y y ffPL . . . BP Ln BP Ln+1 . . . BP L ? ? ? ? ? ? PL (2.4) y y y ffTOP . . .BT OPn BT OPn+1 . . .BT OP ? ? ? ffTOPF(n)?y ?y ?yffTOPF . . . BFn BFn+1 . . . BF where all the vertical maps are fibrations and all the filtrations have nice properties. Moreover, each of limit spaces has the direct limit topology with respect to the corresponding filtration, and every com- pact subspace of, say, BO is contained in some BOn. Furthermore, the fiber of ffOPLis denoted by P L=O, the fiber of ffPLTOP is denoted by T OP=P L, etc. Similarly, the fiber of the composition, say, ffPLF:= ffTOPFOffPLTOP: BP L ! BF is denoted by F=P L. In particular, we have a fibration (2.5) T OP=P L ---a! F=P L -- b-! F=T OP. S Finally, F=T OP = Fn=T OPn where Fn=T OPn denotes the fiber of the fibration BT OPn ! BFn, and F=T OP has the direct limit topol- ogy with respect to the filtration {Fn=T OPn}. The same holds for other öh mogeneous spaces" F=P L, T OP=P L, etc. Notice that, because of well-known results of Milnor [34 ], all these öh mogeneous spaces" have the homotopy type of CW -spaces. Fur- thermore, all the spaces BO, BP L, BT OP, BF, F=P L, T OP=P L, etc. are infinite loop spaces and, in particular, H-spaces, see [4]. We mention also the following useful fact. 2.5. Theorem. Let Z denote on of the symbols O, P L, F . The above described map BZn ! BZn+1 induces an isomorphism of homotopy groups in dimensions n - 1 and an epimorphism in dimension n. Proof. For Z = O and Z = F it is well known, see e.g [5], for Z = P L __ it can be found in [20 ]. |__| PL STRUCTURES 19 2.6. Remark. Let Gn denote the topological monoid of homotopy equivalences Sn-1 ! Sn-1 . Then the classifying space BGn of Gn classifies Sn-1 -fibrations. Every h 2 T OPn induces a map Rn \ {0} ! Rn \ {0} which, in turn, yields a self-map ßh : Sn-1 ! Sn-1 , ßh(x) = h(x)=||h(x)||. So, we have a map T OPn ! Gn which, in turn, induces a map BT OPn -! BGn of classifying spaces. In the language of bundles, this map converts a topological Rn-bundle into a spherical fibration via deletion of the section. We can also consider the space BG by tending n to 1. In particular, we have the spaces G=P L and G=T OP . There is an obvious forgetful map Fn ! Gn, and it turns out that the induced map BF ! BG (as n ! 1) is a homotopy equivalence. see e.g. [31 , Chapter 3]. In particular, F=P L ' G=P L and F=T OP ' G=T OP . 3. Structures on manifolds and bundles A PL atlas on a topological manifold is an atlas such that all the transition maps are PL ones. We define a PL manifold as a topological manifold with a maximal PL atlas. Furthermore, given two PL man- ifolds M and N, we say that a homeomorphism H : M ! N a PL isomorphism if h is a PL map. (One can prove that in this case h-1 is a PL map as well, [24 ].) 3.1. Definition. (a) We define a @PL -manifold to be a topological manifold whose boundary @M is a PL manifold. In particular, every closed topological manifold is a @PL -manifold. Furthermore, every PL manifold can be canonically regarded as a @PL -manifold. (b) Let M be a @PL -manifold. A PL structuralization of M is a homeomorphism h : V ! M such that V is a PL manifold and h : @V ! @M induces a PL isomorphism @V ! @M of boundaries (or, equivalently, PL isomorphism of corresponding collars). Two PL struc- turalizations hi : Vi ! X, i = 0, 1 are concordant if there exist a PL isomorphism ' : V0 ! V1 and a homeomorphism H : V0 x I ! M x I such that h0 = H|V x {0} and H|V0 x {1} = h1' and, moreover, H : @V0 x I ! @M x I coincides with h0 x 1I. Any concordance class of PL structuralizations is called a PL structure on M. We denote by TPL (M) the set of all PL structures on X. 20 YULI B. RUDYAK (c) If M itself is a PL manifold then TPL (M) contains the distin- guished element: the concordance class of 1M . We call it the trivial element of TPL (M). 3.2. Remarks. 1. Clearly, every PL structuralization of M equips M with a certain PL atlas. Conversely, if we equip M with a certain PL atlas then the identity map can be regarded as a PL structuralization of M. 2. Clearly, if M itself is a PL manifold then the concordance class of any PL isomorphism is the trivial element of TPL (M). 3. Recall that two homeomorphism h0, h1 : X ! Y are isotopic if there exists a homeomorphism H : X x I ! Y x I (isotopy) such that p2H : X x I ! Y x I ! I coincides with p2 : X x I ! I. Given A X, we say that h0 and H1 are isotopic relA if there exists an isotopy H such that H(a, t) = h0(a) for every a 2 A and every t 2 I. In particular, if two PL structuralization h0, h1 : V ! M are isotopic rel@V then they are concordant. 4. Given two PL structuralizations hi : Vi ! M, i = 0, 1, they are not necessarily concordant if V0 and V1 are PL isomorphic. We are not able to give examples here, but we do it later, see Remark 3.10(2) and Example 22.3. 3.3. Definition (cf. [5, 48]).Given a topological Rn-bundle ,, a PL structuralization of , is a morphism ' : , ! flnPL of topological Rn- bundles. We say that two PL structuralizations '0, '1 : , ! flnPL are concordant if there exists a morphism : , x 1I ! flnPL of topological Rn-bundles such that |, x 1{i}= 'i, i = 0, 1. Let f : X ! BT OPn classify a topological Rn-bundle ,, and let g : X ! BP Ln be an ffPLTOP(n)-lifting of f. Then we get a morphism (defined uniquely up to concordance in view of 2.2) , ~=f*flnTOP = g*ff(n)*flnTOP = g*flnPL-classif--!flNPL, ff(n) := ffPLTOP(n). Clearly, this morphism , ! flnPL is a PL structuralization of ,. It is easy to see that in this way we have a correspondence (3.1) [Liftff(n)f] -! {PL structures on ,}. 3.4. Theorem. The correspondence (3.1) is a bijection. PL STRUCTURES 21 Proof. This can be proved similarly to [48 , Theorem IV.2.3], cf. also_ [5, Chapter II]. |__| Consider now the map ~f: X -- f-! BT OPn BT OP and the map ff : BP L ! BT OP as in (2.4). Then every ff(n)-lifting of f is the ff-lifting of ~f. So, we have a correspondence (3.2) u, : {PL structures on ,} -! [Liftff(n)f] -! [Liftff~f] where the first map is the inverse to (3.1). Furthermore, there is a canonical map v, : {PL structures on ,} -! {PL structures on , `1}, and these maps respect the maps u,, i.e. u, `1 = v,u,. So, we have the map (3.3) lim {PL structures on , `n} -! [Liftff~f] n!1 where lim means the direct limit of the sequence of sets. 3.5. Proposition. If X is a finite CW -space then the map (3.3) is a bijection. Proof. The surjectivity follows since every compact subset of BT OP is contained in some BT OPn. Similarly, every map X x I ! BP L passes __ through some BP Ln, and therefore the injectivity holds. |__| Furthermore, if , itself is a PL bundle then, by Theorem 1.3, there is a bijection [Liftff~f] ~=[X, T OP=P L]. Thus, in this case the bijection (3.3)turns into the bijection (3.4) lim { PL structures on , `n} -! [X, T OP=P L]. n!1 3.6. Definition. Let M be a @PL -manifold. A homotopy PL struc- turalization of M is a homotopy equivalence h : V ! X such that V is a PL manifold and h : @V ! @M is a PL isomorphism. Two homotopy PL structuralizations hi : Vi ! X, i = 0, 1 are equivalent if there exists a PL isomorphism ' : V0 ! V1 and a homotopy H : V0 x I ! M such that h0 = H|V x {0} and H|V0 x {1} = h1' and, moreover, H|V x {t} : @V0 ! @M coincides with h0. Any equivalence class of homotopy PL structuralizations is called a homotopy PL structure on X. We denote by SPL (X) the set of all homotopy PL structures on X. If M itself is a PL manifold, we define the trivial element of SPL (M) as the equivalence class of 1M : M ! M. 22 YULI B. RUDYAK 3.7. Definition. Given an (Sn, *)-fibration , over X, a homotopy PL structuralization of , is an (Sn, *)-morphism ' : , ! (flnPL)o. We say that two PL structuralizations '0, '1 : , ! (flnPL)o are equivalent if there exists a morphism : , x 1I ! (flnPL)o of (Sn, *)-fibrations such that |, x 1{i}= 'i, i = 0, 1. Every such an equivalence class is called a homotopy PL structure on ,. Now, similarly to (3.4), for a finite CW -space X we have a bijection (3.5) lim {homotopy PL structures on , `n} -! [X, F=P L]. n!1 However, here we can say more. 3.8. Proposition. The sequence {{homotopy PL structures on , `n}}1n=1 stabilizes. In particular, the map {homotopy PL structures on , `n} ! [F=P L] is a bijection if dim , >> dim X __ Proof. This follows from 2.5. |__| Summarizing, for every PL RN -bundle , we have a commutative diagram {PL structure on , } -- - ! [X, T OP=P L] ? ? ? ? y a*y {homotopy PL structure on ,o } -- - ! [X, F=P L] Here the map a in (2.5) induces the map a* : [X, T OP=P L] ! [X, F=P L]. The left vertical arrow converts a morphism of RN -bundles into a morphism of (SN , *)-bundles and regards the last one as a mor- phism of (SN , *)-fibrations. For a finite CW -space X, the horizontal arrows turn into bijections if we stabilize the picture. i.e. pass to the limit as in (3.4). Furthermore, the bottom arrow is an isomorphism if N >> dim X. 3.9. Remark. Actually, following the proof of the Main Theorem, one can prove that T OPm =P Lm = K(Z=2, 3) for m 5, see [28 , Essay IV, x9]. So, an obvious analog of 2.5 holds for T OP also, and therefore the top map of the above diagram is a bijection for N large enough. But, of course, we are not allowed to use these arguments here. PL STRUCTURES 23 3.10. Remark. 1. We can also consider smooth (=differentiable C1 ) structures on topological manifolds. To do this, we must replace the words "PL" in Definition 3.1 by the word "smooth". The related set of smooth concordance classes is denoted by TD (M). The set SD (M) is defined in a similar way, we leave it to the reader. Moreover, recall that every smooth manifold can be canonically con- verted into a PL manifold (the Cairns-Hirsch-Whitehead Theorem, see e.g [22 ]). So, we can define the set PD (M) of smooth structures on a PL manifold M. To do this, we must modify definition 3.1 as follows: M is a PL manifold with a compatible smooth boundary, Viare smooth manifolds, hi and H are PL isomorphisms. 2. We can now construct an example of two smooth structural- izations hi : V ! Sn, i = 1, 2 which are not concordant. First, notice that there is a bijective correspondence between SD (Sn) and the Kervaire-Milnor group n of homotopy spheres, [25 ]. Indeed, n consists of equivalence classes of oriented homotopy spheres: two ori- ented homotopy spheres are equivalent if they are orientably diffeomor- phic (=h-cobordant). Now, given a homotopy smooth structuralization h : n ! Sn, we orient n so that h has degree 1. In this way we get a well-defined map u : SD (Sn) ! n. Conversely, given a homotopy sphere n, consider a homotopy equivalence h : n ! Sn of degree 1. In this way we get a well-defined map n ! SD (Sn) which is inverse to u. Notice that, because of the Smale Theorem, every smooth homotopy sphere n, n 5, possesses a smooth function with just two critical points. Thus, SD (Sn) = TD (Sn) = PD (Sn) for n 5. Kervaire and Milnor [25 ] proved that 7 = Z=28, i.e., because of what we said above, SD (S7) = TD (S7) = PD (S7) consists of 28 elements. On the other hand, there are only 15 smooth manifolds which are homeomorphic (and PL isomorphic, and homotopy equivalent) to S7 but mutually non-diffeomorphic. Indeed, if an oriented smooth 7- dimensional manifold is homeomorphic to S7 then bounds a par- allelizable manifold W , [25 ]. We equip W an orientation which is compatible with and set oe(W ) a( ) = _______ mod 28 8 where oe(W ) is the signature of W . Kervaire and Milnor [25 ] proved that the correspondence 7 ! Z=28, 7! a(W ) is a well-defined bijection. 24 YULI B. RUDYAK However, if a( 1) = -a( 2) then 1 and 2 are diffeomorphic: namely, 2 is just the 1 with the opposite orientation. So, there are only 15 smooth manifolds which are homeomorphic (and homotopy equivalent, and PL isomorphic) to S7 but mutually non-diffeomorphic. In terms of structures, it can be expressed as follows. Let æ : Sn ! Sn be a diffeomorphism of degree -1. Then the smooth structural- izations h : 7 ! S7 and æh : 7 ! S7 are not concordant, if a( 7) 6= 0, 14. For convenience of references, we fix here the following theorem of Smale [50 ]. Actually, Smale proved it for smooth manifolds, a good proof can also be found in Milnor [36 ]. However, the proof can be transmitted to the PL case, see Stallings [52 , 8.3, Theorem A]. 3.11. Theorem. Let M be a closed PL manifold which is homotopy equivalent to the sphere Sn, n 5. Then M is PL isomorphic to Sn. __ |__| 4. From manifolds to bundles Recall that, for every topological manifold Mn , its tangent bundle øM and (stable) normal bundle M are defined. Here øM is a topological Rn-bundle, and we can regard N as a topological RN -bundle with N >> n. Furthermore, if M is a PL manifold then øM and M turns into PL bundles in a canonical way, see [28 , 48]. 4.1. Construction. Consider a PL manifold M and a PL structural- ization h : V ! M. Let g = h-1 : M ! V . Since g is a homeo- morphism, it yields a topological morphism ~ : M ! V , and so we have the correcting topological morphism c(~) : M ! g* V . Now, the morphism c(~) * classif N M --- ! g V --- - ! flPL is a PL structuralization of M . It is easy to see that in this way we have the correspondence jTOP : TPL (M) -! {PL structures on M } -! [M, T OP=P L] where the last map comes from 3.4. Moreover, it is clear that, in fact, jTOP passes through the map [(M, @M), (T OP=P L, *)] ! [M, T OP=P L]. So, we can and shall regard jTOP as the map jTOP : TPL (M) -! [(M, @M), (T OP=P L, *)]. PL STRUCTURES 25 4.2. Remark. We constructed the map jTOP using PL structuraliza- tions of M . However, we can also construct the map jTOP by consid- ering other bundles related to M. For example, consider the tangent bundle øM . The topological morphism øM --- ! g*øV -classif--!flnPL can be regarded as a PL structuralization of øM , and so we have an- other way of constructing of jTOP . Moreover, we can also consider the topological morphism `N+n = øM NM -- - ! g*øV NM classif---!flN+nPL and regard it as a PL structuralization of `N+n , etc. One can prove that all these constructions are equivalent. Now we construct a map jF : SPL (M) ! [M, F=P L], a öh motopy analogueö f jTOP . This construction is more delicate, and we treat only the case of closed manifolds here. So, let M be a connected closed PL manifold. 4.3. Definition. Given an (Sn, *)-fibration , = {E ! B} with a section s : B ! E, we define its Thom space T , as the quotient space E=s(B), Given a topological RN -bundle j, we define the Thom space T j as T j := T (jo). Given a morphism ' : , ! j of (Sn, *)-fibrations, we define T ' : T , ! T j to be the unique map such that the diagram E --- ! E0 ? ? ? ? y y T' T , --- ! T j commutes. Here E0 is the total space of j. 4.4. Definition. A pointed space X is called reducible if there is a pointed map f : Sm ! X such that f* : Hei(Sm ) ! Hei(X) is an isomorphism for i m. Every such map f (as well as its homotopy class or its stable homotopy class) is called a reduction for X. We embed M in RN+n , N >> n, and let M , dim M = N be a normal bundle of this embedding. Recall that M is a PL bundle E ! M whose total space E is PL isomorphic to a (tubular) neighbourhood U of M in RN+n . We choose such isomorphism and denote it by ' : U ! E. One can prove that, for N large enough, the normal bundle always exists, [20 , 29, 30]. 26 YULI B. RUDYAK 4.5. Construction-Definition. Let T M be the Thom space of M . Then there is a unique map _ : RN+n =(RN+n \ U) ! T such that _|U = '. We define the collapse map ' : SN+n ! T M (the Browder-Novikov map) to be the composition quotientN+n N+n N+n N+n _ ' : SN+n --- - ! S =(S \ U) = R =(R \ U) --- ! T . See [5, II.2.11] or [48 , 7.15] for details. It is well known and easy to see that ' is a reduction for T , see Corollary 11.7 below. It turns out that, for N large enough, the normal bundle of a given embedding M ! RN+n is unique. For detailed definitions and proofs, see [20 , 29, 30]. The uniqueness gives us the following important fact. Let 0 = {E0 ! M} be another normal bundle and '0 : U0 ! E0 be another PL isomorphism. Let ' : SN+n ! T and '0: SN+n ! T 0 be the corresponding Browder-Novikov maps. Then there is a morphism ! 0 of PL bundles which carries ' to a map homotopic to '0. 4.6. Theorem. Consider a PL RN -bundle j over M such that T j is reducible. Let ff 2 ßN+n (T j) be an arbitrary reduction for T j. Then there exist an FN -equivalence ~ : oM ! jo such that (T ~)*(') = ff, and such a ~ is unique up to fiberwise homotopy over M. __ Proof. We postpone it to the next Chapter, see 11.11. |__| 4.7. Construction-Definition. Given a homotopy equivalence h : V ! M of closed connected PL manifolds, let V be a normal bundle of a certain embedding V RN+n , and let u 2 ßN+n (T V ) be the homotopy class of a collapsing map SN+n ! T V . Let g : M ! V be homotopy inverse to h and set j = g* V . The adjoint to g morphism ' := Ig, V: j ! V yields the map T ' : T j ! T V . It is easy to see that T ' is a homo- topy equivalence, and so there exists a unique ff 2 ßN+n (T j) with (T ')*(ff) = u. Since u is a reduction for T V , we conclude that ff is a reduction for T j. By Theorem 4.6, we get an F -equivalence ~ : oM ! jo with (T ~)*(') = ff. Now, the morphism ~ o classif N ( M )o-- - ! j --- - !flF PL STRUCTURES 27 is a homotopy PL structuralization of M . Because of the uniqueness of the normal bundle, the concordance class of this structuralization is well defined. So, in this way we have the correspondence jF : SPL (M) -! {homotopy PL structures on M } ~=[M, F=P L] where the last bijection comes from 3.8. The map jF is called the normal invariant, and its value on a homo- topy PL structure is called the normal invariant of this structure. Notice that there is a commutative diagram jTOP TPL (M) --- ! [M, T OP=P L] ? ? (4.1) fi?y ?ya* jF SPL (M) --- ! [M, F=P L] where fi regards a PL structuralization as the homotopy PL structural- ization. 5. Homotopy PL structures on T kx Dn Below T kdenotes the k-dimensional torus. 5.1. Theorem. Assume that k + n 5. If x 2 SPL (T kx Sn) can_be_ represented by a homeomorphism M ! T kx Sn then jF (x) = 0. |__| This is a special case of the Sullivan Normal Invariant Homeomor- phism Theorem. We prove 5.1 (in fact, a little bit general result) in the next chapter. We also prove the Sullivan Theorem in full generality in Chapter 3, section 2* *1. We must do this repetition since the proof in Chapter 3 uses 5.1. 5.2. Construction-Definition. Let x 2 TPL (M) be represented by a map h : V ! M, and let p : fM ! M be a covering. Then we have a commutative diagram Ve -- eh-!Mf ? ? q?y ?yp V -- h-! M where q is the induced covering. Since ehis defined uniquely up to deck transformations, the concordance class of ehis well defined. So, we have a well-defined map p* : TPL (M) ! TPL (fM ) 28 YULI B. RUDYAK where p*(x) is the concordance class of eh. Similarly, one can construct a map p* : SPL (M) ! SPL (fM ). If p* is a finite covering, we say that a class p*(x) 2 SPL (fM ) finitely covers the class x. 5.3. Theorem. Let k + n 5 Then the following holds: (i) if n > 3 then the set SPL (T kx Dn) consists of precisely one (trivial) element; (ii) if n < 3 then every element from SPL (T kx Dn) can be finitely covered by the trivial element; (iii) the set SPL (T kxD3) contains at most one element which cannot be finitely covered by the trivial element. Some words about the proof. First, we mention the proof given by Wall, [62 ] and [63 , Section 15 A]. Wall proved the bijection w : SPL (T kx Dn) ! H3-n (T k). Moreover, he also proved that finite cov- erings respect this bijection, i.e. if p : T kx Dn ! T kx Dn is a finite covering then there is the commutative diagram SPL (T kx Dn) --w-! H3-n (T k; Z=2) x x p*?? ??p* SPL (T kx Dn) --w-! H3-n (T k; Z=2) . Certainly, this result implies all the claims(i)-(iii). Walls proof uses difficult algebraic calculations. Another proof of the theorem can be found in [23 , Theorem C]. Minding the complaint of Novikov concerning Sullivan's results (see Prologue), we must mention that Hsiang and Shaneson [23 ] use a Sulli- van's result. Namely, they consider the so-called surgery exact sequence -- @-! S k n jF k n PL (S x T ) --- ! [S x T , F=P L]-- - ! . . . and write (page 42, Section 10): By [44], every homomorphism h : M ! Sk x T n, k = n 5, represents an element in the image of @. Here the item [44] of the citation is our bibliographical item [56 ]. So, in fact, Hsiang and Shaneson use Theorem 5.1. As I already said, we __ prove 5.1 in next Chapter and thus fix the proof. |__| PL STRUCTURES 29 6. The product structure theorem, or from bundles to manifolds Let M be an n-dimensional @PL -manifold. Then every PL struc- turalization h : V ! M yields a PL structuralization h x 1 : V x Rk ! M x Rk. Thus, we have a well-defined map e : TPL (M) ! TPL (M x Rk). 6.1. Theorem (The Product Structure Theorem). For every n 5 and every k 0, the map e : TPL (M) ! TPL (M x Rk) is a bijection. In particular, if TPL (M x Rk) 6= ; then TPL (M) 6= ;. Concerning the proof. I did not find a proof which is essentially better then the original one. So, I refer the reader to the original source [28 ]. I want also to mention here that the proof of Theorem 6.1 uses the Theorem 5.3 for n = 0. For another approach to the proof of Theorem __ 6.1, see [13 , Remark 5.3]. |__| 6.2. Corollary (The Classification Theorem). If dim M 5 and M admits a PL structure, then the map jTOP : TPL (M) ! [(M, @M), (T OP=P L, *)] is a bijection. Proof. We construct a map (6.1) oe : [(M, @M), (T OP=P L, *)] ! TPL (M) which is inverse to jTOP . For simplicity of notations, we consider the case of M closed. Take an element a 2 [M, T OP=P L] and, using 3.4, interpret it as a concordance class of a morphism ' : `NM ! flNPL of topological RN -bundles (cf. Remark 4.2). The morphism ' yields a correcting isomorphism `NM ! b*flNPL of topological RN -bundles over M, where b : M ! BP L is the base of the morphism '. So, we have the commutative diagram M x RN --h-! W ? ? ? ? y y M _______M where h is a fiberwise homeomorphism and W ! M is a PL RN -bundle b*flNPL. In particular, W is PL manifold. Regarding h-1W ! M x RN as a PL structuralization of M x RN , we conclude that, by the Product 30 YULI B. RUDYAK Structure Theorem 6.1, h-1 is concordant to a map g x 1 for some PL structuralization g : V ! M. We define oe(a) 2 TPL (M) to be the concordance class of g. One can check that oe is a well-defined map __ which is inverse to jTOP . Cf. [28 , Essay IV]. |__| 6.3. Corollary (The Existence Theorem). A topological manifold M with dim M 5 admits a PL structure if and only if the tangent bundle to M admits a PL structure. Proof. Only claim "if" needs a proof. Let ø = {ß : D ! M} be the tangent bundle of M, and let = {r : E ! M} be a stable normal bundle of M, dim = N. Then E is homeomorphic to an open subset of RN+n , and therefore we can (and shall) regard E as a PL manifold. Since ø is a PL bundle, we conclude that r*ø is a PL bundle over E. In particular, the total space M x RN+n of r*ø turns out to be a PL manifold. Now, because of the Product Structure Theorem 6.1, M __ admits a PL structure. |__| Let f : M ! BT OP classify the stable tangent bundle of a closed topological manifold M, dim M 5. 6.4. Corollary. The following conditions are equivalent: (i) M admits a PL structure; (ii) ø admits a PL structure; (iii) there exists k such that ø `k admits a PL structure; (iv) the map f admits an ffPLTOP-lifting to BP L. Proof. It suffices to prove that (iv) =) (iii) =) (i). The implication (iii) =) (i) can be proved similarly to 6.3. Furthermore, since M is compact, we conclude that f(M) BT OPm for some m. So, if (iv) __ holds then f lifts to BP Lm , i.e. ø `m-k admits a PL structure. |__| 6.5. Remark. It follows from 1.3, 6.3 and 6.2 that the set TPL (M) of PL structures on M is in a bijective correspondence with the set of fiber homotopy classes of ffPLTOP-liftings of f. (We leave it to the reader to extend the result on @PL -manifolds.) 6.6. Remark. It is well known that jF is not a bijection in general. The "kernel"' and öc kernelö f jF can be described in terms of so- called Wall groups, [63 ]. (For M simply-connected, see also Theorem 13.2.) On the other hand, the bijectivity of jTOP (the Classification Theorem) follows from the Product Structure Theorem. So, informally speaking, kernel and cokernel of jF play the role of obstructions to splitting of structures. It seems interesting to develop and clarify these naive arguments. PL STRUCTURES 31 6.7. Remark. Since tangent and normal bundles of smooth manifolds turn out to be vector bundles, one can construct a map k : PD (M) ! [M, P L=O] which is an obvious analog of jTOP (here we assume M to be closed). Moreover, the obvious analog of the Product Structure Theorem (as well as of the Classification and Existence Theorems) holds without any dimensional restriction. In particular, k is a bijection for every smooth manifold, [22 ]. It is well known (although difficult to prove) that ßi(P L=O) = 0 for i 6. (See [48 , IV.4.27(iv)] for the references.) Thus, every PL manifold M of dimension 7 admits a smooth structure, and this structure is unique if dim M 6. 7. Non-contractibility of T OP=P L 7.1. Theorem (Freedman's Example). There exists a closed topolog- ical 4-dimensional manifold F with w1(F ) = 0 = w2(F ) and such that the signature of F is equal to 8. Here wi denotes the i-th Stiefel-Whitney class. __ Proof. See [14 ] or the original work [15 ]. |__| 7.2. Theorem (Rokhlin Signature Theorem). Let M be a closed 4- dimensional PL manifold with w1(M) = 0 = w2(M). Then the signa- ture of M is divisible by 16. Proof. See [26 , 37] or the original work [44 ]. In fact, Rokhlin proved the result for smooth manifolds, but the proof works for PL manifolds as well. On the other hand, in view of 6.7, there is no difference between_ smooth and PL manifolds in dimension 4. |__| 7.3. Corollary. The topological manifolds F and F x T k, k 1 do not admit any PL structure. Proof. The claim about F follows from 7.2. Suppose that F x T k has a PL structure. Then F x Rk has a PL structure. So, because of the Product Structure Theorem 6.1, F x R has a PL structure. Hence, by 6.7, it possesses a smooth structure. Then the projection p2 : F x R ! R can be C0-approximated by a map f : F x R ! R which coincides with p2 on F x (-1, 0) and is smooth on F x (1, 1). Take a regular value a 2 (0, 1) of f (which exists because of the Sard Theorem) and set W = f-1 (a). Then W is a smooth manifold (by the Implicit Function Theorem), and it is easy to see that w1(W ) = 0 = 32 YULI B. RUDYAK w2(W ) (because it holds for both manifolds R and F x R). On the other hand, both manifolds F and W cut the üt be" F x R. So, they are (topologically) bordant, and therefore W has signature 8. But this contradicts the Rokhlin Theorem 7.2. __ |__| 7.4. Corollary. The space T OP=P L is not contractible. Proof. Indeed, suppose that TOP/PL is contractible. Then every map X ! BT OP lifts to BP L, and so, by 6.3, every topological manifold of dimension greater than 4 admits a PL structure. But this contradicts __ 7.3. |__| 7.5. Remark. Kirby and Siebenmann [27 , 28] constructed the original example of a topological manifold which does not admit a PL structure. Again, the Rokhlin Theorem 7.2 is one of the main ingredients of the proof. 8. Homotopy groups of T OP=P L Let M be a compact topological manifold equipped with a metric æ. Then the spacefHiof homeomorphisms gets a metric d with d(f, g) = sup{x 2 M fiæ(f(x), g(x))}. 8.1. Theorem. The space H is locally contractible. __ Proof. See [7, 11]. |__| 8.2. Corollary. There exists " > 0 such that every homeomorphism __ h 2 H with d(H, 1M ) < " is isotopic to 1M . |__| 8.3. Construction. We regard the torus T k as a commutative Lie group (multiplicative) and equip it with the invariant metric æ. Con- sider the map p~ : T k! T k, p~(a) = a~, ~ 2 N. Then p~ is a ~k-sheeted covering. It is also clear that all the deck transformations of the cov- ering torus are isometries. 8.4. Lemma. Let h : T kx Dn ! T kx Dn is a self-homeomorphism which is homotopic rel@(T kx Dn) to the identity. Then there exists a commutative diagram eh T kx Dn --- ! T kx Dn ? ? p~?y ?yp~ T kx Dn --h-! T kx Dn where the lifting ehof h is isotopic rel@(T kx Dn) to the identity. PL STRUCTURES 33 Proof. (Cf. [28 , Essay V].) First, consider the case n = 0. Without loss of generality we can assume that h(e) = e where e is the neutral element of T k. Consider a covering p~ : T k! T kas in 8.3 and take a covering eh: T k! T k, p~eh= ehp~ of h such that eh(e) = e. In order to distinguish the domain and the range of p~, we denote the domain of p~ by eTand the range of p~ by T . Since all the deck transformations of eTare isometries, we conclude that the diameter of each of (isometric) fundamental domain tends to zero as ~ ! 1. Furthermore, since h is homotopic to 1T , we conclude that every point of the lattice L := p-1~(e) is fixed under eh. Given " > 0, choose ffi such that æ(eh(x), eh(y)) < "=2 whenever æ(x, y) < ffi. Furthermore, choose ~ so large that the diameter of any closed fundamental domain is less then min {"=2, ffi}. Now, given x 2 eT, choose a 2 L such that a and x belong to the same closed fundamental domain. Now, " " æ(x, eh(x) æ(x, a) + æ(a, eh(x)) = æ(x, a) + æ(eh(a), eh(x)) < __+ __= ". 2 2 So, for every " > 0 there exists ~ such that d(eh, 1Te) < ". Thus, by 8.2, ehis isotopic to 1Tefor ~ large enough. The proof for n > 0 is similar but a bit more technical. Let D'' Dn be the disk centered at 0 and having the radius j. We can always as- sume that h coincides with identity outside of T kx D''. Now, asserting as for n = 0, take a covering p~ as above and choose ~ and j so small that the diameter of every fundamental domain in Te x D''is small enough. Then eh: eTx D''! eTx D'' is isotopic to the identity (and ehcoincides with identity outside eTxD''). This isotopy is not an isotopy rel eTx @D''. Nevertheless, we can easily extend it to the whole Te x Dn so that this extended isotopy is an isotopy rel@(Tex Dn). If you want formulae, do the following. Given a = (b, c) 2 eTx D'', set |a| = |c|. Consider an isotopy ' : eTxD''xI ! eTxD''xI, '(a, 0) = a, '(a, 1) = eh(a), a 2 eTxD''. Define _ : eTx D''x I ! eTx D''= xI by setting æ '(a, t) if|a| j, _(a, t) = |a|-1 '(a, ____''-1t)if|a| j. __ Then _ is the desired isotopy rel@(Tex Dn). |__| 34 YULI B. RUDYAK 8.5. Corollary. Let fi : TPL (T kxDn) ! SPL (T kxDn) be the forgetful map as in (4.1). If fi(x) = fi(y) then there exists a finite covering_ p : T kx Dn ! T kx Dn such that p*(x) = p*(y). |__| Consider the map p*2 _ : ßn(T OP=P L) = [(Dn, @Dn), (T OP=P L, *)] --! jTOP k n [(T kx Dn, @(T kx Dn)), T OP=P L] --- ! TPL (T x D ) where oe is the map from (6.1). 8.6. Lemma. The map _ is injective. Moreover, if p*_(x) = p*_(y) for some finite covering p : T kx Dn ! T kx Dn then x = y. In particular, if p*_(x) is the trivial element of TPL (T kx Dn) then x = 0. Proof. The injectivity of _ follows from the injectivity of p*2and oe. Furthermore, for every finite covering p : T kx Dn ! T kx Dn we have the commutative diagram _ k n ßn(T OP L=P L) -- - ! TPL (T x D ) fl x fl ? * fl ? p _ k n ßn(T OP L=P L) -- - ! TPL (T x D ) Therefore x = y whenever p*_(x) = p*_(y). Finally, if p*_(x) is trivial_ element then p*_(x) = p*_(0), and thus x = 0. |__| Consider the map _ k n fi k n ' : ßn(T OP=P L) -- - ! TPL ((T x D ) --- ! SPL ((T x D ) where fi is the forgetful map described in (4.1). 8.7. Theorem (The Reduction Theorem). The map ' is injective. Moreover, if p*'(x) = p*'(y) for some finite covering p : T kx Dn ! T kx Dn then x = y. In particular, if p*'(x) is the trivial element of TPL (T kx Dn) then x = 0. We call it the Reduction Theorem because it reduces the calculation of the group ßi(T OP=P L) to the calculation of the sets SPL (T kx Dn). Proof. If '(x) = '(y) then fi_(x) = fi_(y). Hence, by Corollary 8.5, there exists a finite covering ß : T kx Dn ! T kx Dn such that ß*_(x) = ß*_(y). So, by Lemma 8.6, x = y, i.e. ' is injective. PL STRUCTURES 35 Now, suppose that p*'(x) = p*'(y) for some finite covering p : T kx Dn ! T kx Dn. Then fi*p*_(x) = fi*p*_(y). Now, by Corollary 8.5, there exists a finite covering q : T kx Dn ! T kx Dn such that q*p*_(x) = q*p*_(y), i.e. (pq)*_(x) = (pq)*_(y). Thus, by_ Lemma 8.6, x = y. |__| 8.8. Corollary (The Main Theorem). ßi(T OP=P L) = 0 for i 6= 3. Furthermore, ß3(T OP=P L) = Z=2. Thus, T OP=P L = K(Z=2.3). Proof. The equality ßi(T OP=P L) = 0 for i 6= 3 follows from The- orem 5.3(i,ii) and Theorem 8.7. Furthermore, again because of 5.3 and 8.7, we conclude that ß3(T OP=P L) has at most two elements. In other words, T OP=P L = K(ß, 3) where ß = Z=2 or ß = 0. Fi- nally, by Corollary 7.4, the space T OP=P L is not contractible. Thus,_ T OP=P L = K(Z=2, 3). |__| 8.9. Remark. Notice that, for i > 5, the set TPL (Si) consists of just one element by the Smale Theorem 3.11. Because of this, the equality ßi(T OP=P L) = 0, i > 5 follows from Theorem 6.2. However, for i small we need Theorem 5.3. (Moreover, the proof of Theorem 6.1 uses 5.3 for n = 0.) From now on and till the end of the section we fix a closed topological manifold M and let f : M ! BT OP denote the classifying map for the stable tangent bundle of M. 8.10. Construction-Definition. Let - 2 H4(BT OP ; ß3(T OP=P L)) = H4(BT OP ; Z=2) be the characteristic class of the fibration i := {ffPLTOP: BP L ! BT OP }, (see e.g. [39 , 48 ] for the definition of the characteristic class of the fibration). We define the Kirby-Siebenmann class -(M) 2 H4(M; Z=2) of M by setting -(M) = f*-. Clearly, the class -(M) can also be described as the characteristic class of the T OP=P L-fibration f*i over M. 8.11. Corollary. The manifold M admits a PL structure if and only if -(M) = 0. In particular, if H4(M; Z=2) = 0 then M admits a PL structure. Furthermore, if M admits a PL structure then the set of all PL structure on M is in a bijective correspondence with H3(M; Z=2). 36 YULI B. RUDYAK Proof. By 6.4, M admits a PL structure if and only if f admits an ffPLTOP-lifting. BP L ? ? y f M -- - ! BT OP By the Main Theorem 8.8, the fiber of the fibration ffPLTOP: BP L -! BT OP is the Eilenberg-Mac Lane space K(Z=2, 3). Thus, because of the obstruction theory, f lifts to BP L if and only if f*- = 0. Finally, by 6.2 and 8.8, we have the bijections TPL (M) ~=[M, T OP=P L] ~=[M, K(Z=2, 3] = H3(M; Z=2) __ provided TPL (M) 6= ;. |__| 9. Do it We recommend that the reader return to the introduction and look again the graph of our proof of the Main Theorem. PL STRUCTURES 37 Chapter 2. Tools Here we prove two important results which we used without proofs in Chapter 1. First, we prove Theorem 4.6. Notice that Browder [5] proved the Theorem for M simply-connected. In fact, his proof works for every orientable M. Here we follow Browder's proof, the only essential modification is that we use the duality theorem 11.6 for arbitrary M while Browder [5] uses it for M simply-connected. Another goal of this chapter is to prove Theorem 5.1. In fact, we prove here a little bit more general result, Theorem 17.1. The proof uses the Sullivan's result on the homotopy type of F=P L. Notice that Madsen and Milgram [31 ] gave a detailed proof of those Sullivan result. 10. Stable equivalences of spherical bundles We denote by oek = oekX the trivial Sk-bundle over X with a fixed section. In another words, oek = (`k)o. Given a sectioned spherical bundle , over a finite CW -space X, let aut, denote the group of fiberwise homotopy classes of self-equivalences , ! , over X, where we assume that all homotopies preserve the section. 10.1. Proposition. There is a natural bijection aut oek = [X, Fk]. Proof. Because of the exponential law, every map X ! Fk yields a section-preserving map X x Sk ! X x Sk over X, and vice versa. Cf. __ [5, Prop. I.4.7]. |__| Consider the map ~ : Fk x Fk ! F2k, ~(a, b) = a ^ b : S2k = Sk ^ Sk ! Sk ^ Sk = S2k where we regard a, b 2 Fk as pointed maps Sk ! Sk. Let T : FkxFk ! Fk x Fk be the transpose map, T (a, b) = (b, a). 10.2. Lemma. The maps ~ : Fk x Fk ! F2k and ~T : Fk x Fk ! F2k, k > 0 are homotopic. Proof. Consider the map ø : S2k = Sk ^ Sk ! Sk ^ Sk = S2k, ø (u, v) = (v, u) and notice that, for every a, b 2 Fk, we have (~OT )(a, b) = ø O~(a, b)Oø. 38 YULI B. RUDYAK First, consider the case of k odd. Then there is a pointed homotopy ht between ø and 1S2k. Now, the pointed homotopy htO~(a, b)Oht is a pointed homotopy between (~OT )(a, b) and ~(a, b) which yields a homotopy between ~T and ~. Now consider the case of k even. We regard S2k as R2k [ 1 with R2k = {(x1, . .,.x2k)} and define ø 0, ø 00: S2k ! S2k by setting ø 0(x1, x2, x3, . .,.x2k)=(x2, x1, x3, . .,.x2k), ø 00(x1, . .,.x2k-2x2k-1x2k)= (x1, . .,.x2k-1, x2k-2, x2k), (i.e. ø 0permutes the first two coordinates and ø 00permutes the last two coordinates). Since k is even, we conclude that ø 0' ø ' ø 00. Furthermore, ø 00ø 0' 1S2k. If we fix such pointed homotopies then we get the pointed homotopies (~OT )(a, b) = ø O~(a, b)ø ' ø 00O~(a, b)ø 0= ø 00O(a ^ b)ø 0 = ø 00O(a ^ 1)O(1 ^ b)ø 0= (a ^ ø 00)O(ø 0^ b) = (a ^ 1)O(ø 00ø 0)O(1 ^ b) ' a ^ b = ~(a, b) __ which yield the homotopy ~OT ' ~. |__| 10.3. Corollary. Let ', _ : oek ! oek be two automorphisms of oek. Then the automorphisms 'y_ and _ y' of oe2k are fiberwise homotopic. __ |__| Given two spherical bundles , and j over X, consider the bundle , ^ j over X x X. We denote by : X ! X x X the diagonal and consider the -adjoint bundle morphism J := I ,,^'': , y j ! , ^ j. 10.4. Proposition. For every automorphism ' : j ! j the diagram , y j-- J-! , ^ j ? ? 1y'?y ?y1^' , y j-- J-! , ^ j __ commutes |__| 10.5. Corollary. The diagram (1y1)^' , y j y j--J-! (, y j) ^ j--- - ! (, y j) ^ j fl fl fl fl fl fl (1y')^1 , y j y j--J-! (, y j) ^ j--- - ! (, y j) ^ j __ commutes up to homotopy. |__| PL STRUCTURES 39 11. Proof of Theorem 4.6 We need some preliminaries on stable duality [51 ]. Given a pointed map f : X ! Y , let Sf : SX ! SY denote the (reduced) suspension over f. So, we have a well-defined map S : [X, Y ]o ! [SX, SY ]o. 11.1. Proposition. Suppose that ßi(Y ) = 0 for i < n and that X is a CW -space with dim X < 2n - 1. Then the map S : [X, Y ]o ! [SX, SY ]o is a bijection. Proof. This is the well-known Freudenthal Suspension theorem, see e.g __ [58 ] |__| Given two pointed spaces X, Y , we define {X, Y } to be the direct limit of the sequence [X, Y ]o--S-! [SX, SY ]o --S-! . .-.-- ! [SnX, SnY ]o -- S-! . . .. In particular, we have the obvious maps (Y, *)(X,*)-! [X, Y ]o -! {X, Y }. The image of a pointed map f : X ! Y in {X, Y } is called the stable homotopy class of f. The standard notation for this one is {f}, but, as usual, in most cases we will not distinguish f, [f] and {f}. It is well known that, for n 2, the set [SnX, SnY ]o has a natural structure of the abelian group, and the corresponding maps S are ho- momorphisms, [58 ]. So, {X, Y } turns out to be a group. Furthermore, by Theorem 11.1, if X is a finite CW -space then the map [SN X, SN Y ]o ! {SN X, SN Y } is a bijection for N large enough. 11.2. Definition. A map f : Sd ! A^A? is called a (stable) d-duality if the maps uE : {A, E} ! {S, E ^ A? }, uE (') = (' ^ 1A? )u and uE : {A? , E} ! {S, A ^ E}, uE (') = (1A ^ ')u are isomorphisms. 11.3. Proposition. Let u : Sd ! A ^ A? be a d-duality between two finite CW -spaces. Then, for every i and ß, u yields an isomorphism Hi(u; ß) : eHi(A? ; ß) ! eHd-i(A, ß). 40 YULI B. RUDYAK Proof. Recall that Hn (A? ; ß) = [A, k(ß, n)]o = [SN A, K(ß, N + n)]o where K(ß, i) is the Eilenberg-Mac Lane space. Because of Theorem 11.1, the last group coincides with {SN A, K(ß, N + n)} for N large enough, and therefore Hn (A? ; ß) = {SN A, K(ß, N + n)} for N large enough . Furthermore, let "n : SK(ß, n) ! K(ß, n + 1) be the map which is adjoint to the standard homotopy equivalence K(ß, n) ! K(ß, n+1), see e.g. [58 ]. Whitehead [64 ] noticed that Hen(A; ß) = lim[SN+n , K(ß, N) ^ A]o. -! Here lim-!is the direct limit of the sequence [SN+n , K(ß, N) ^ A]o -! [SN+n+1 , SK(ß, N) ^ A]o --"*-! [SN+n+1 , K(ß, N) ^ A]o (see [19 , Ch 18] or [48 , II.3.24] for greater details). Since "n is an n-equivalence, and because of Theorem 11.1, we conclude that Hen(A; ß) = [SN+n , K(ß, N) ^ A] for N large enough . So, again because of Theorem 11.1, eHn(A; ß) = {SN+n , K(ß, N) ^ A} for N large enough. Now, consider a d-duality u : Sd ! A ^ A? . Fix i and choose N large enough such that Hei(A? ; ß) = {SN A? , K(ß, N + i)}, eHd-i(A; ß) = {SN+d , K(ß, N + i) ^ A}. By suspending the domain and the range, we get a duality (denoted also by u) u : SN+d ! A ^ SN A? . This duality yields the desired isomorphism Hi(u; ß) : = uK(i,N+i) : eHi(A? ; ß) = {SN A? , K(ß, N + i)} ! {SN+d , K(ß, N + i) ^ A} = eHd-i(A; ß) __ |__| PL STRUCTURES 41 11.4. Definition. Dualizing 4.4, we say that a pointed map a : A ! Sk (or its stable homotopy class a 2 {A, Sk}) is a coreduction if the induced map a* : eHi(Sk) ! eHi(A) is an isomorphism for i k. 11.5. Proposition. Let u : Sd ! A ^ A? be a d-duality between two finite CW -spaces, and let k d. A class ff 2 {A? , Sk} is a coreduction k d-k if and only if the class fi := uS ff 2 {S , A} is a reduction. Proof. Let Hi(u) : eHi(A? ) ! eHd-i(A) be the isomorphism as in 11.3. Notice that the standard homeomorphism v : Sd ! Sk ^ Sd-k is a d-duality. It is easy to see that the diagram Hei(A? ) -Hi(u)--!eHd-i(A) x x ff*?? ??fi* Hei(Sk) -Hi(v)--!eHd-i(Sd-k) commutes. In particular, the left vertical arrow is an isomorphism if __ and only if the right one is. |__| Consider a closed connected n-dimensional PL manifold M and em- bed it in RN+n+k with N large enough. Let ' : SN+n+k ! T N+k be a collapse map as in 4.5, and let J : ( N+k )o = ( N )o y oek -! ( N )o ^ oek be the morphism as in 10.4. 11.6. Theorem. The map SN+n+k -- '-! T N+k --TJ-! T N ^ oek is an (N + n + k)-duality map. Proof. This is actually proved in [10 ]. For greater detail, see [48 , V.2.3(i)] (where in the proof the reference 2.8(a) must be replaced by__ 2.8(b)). |__| 11.7. Corollary. The collapse map ' : SN+n ! T N is a reduction. Proof. Recall that T oek = (M x Sk)=M = Sk(M+ ). Consider a sur- jective map e : M+ ! S0 and define " = Ske : T oek ! Sk. Since the map Sk' : SkSN+n = SN+n+k ! T N+k = SkT 42 YULI B. RUDYAK can be written as k' N+k N k 1^" N k k SN+n+k -S-! T = T ^ T oe --! T ^ S = S T , where the composition of first two maps is the duality from 11.6, we conclude that ' is dual to " with respect to duality (11.2). Clearly, "_is a coreduction. Thus, the result follows from 11.5. |__| For technical reasons, it will be convenient for us to consider the duality (11.1) SN+n+2k ! T N+2k TJ-!T N+k ^ T oek. This duality yields an isomorphism (11.2) k k k N+n+2k k N+k N+n+k N+k D := uS : {T oe , S } ! {S , S ^ T } = {S , T }. 11.8. Proposition. For every automorphism ' : oek ! oek the follow- ing diagram commutes up to homotopy: T(1y')^1 N+k k SN+n+2k -- '-! T N+2k --TJ-! T N+k ^ T oek--- - - !T ^ T oe fl fl fl fl fl fl fl fl fl 1^T' N+k k SN+n+2k -- - ! T N+2k --TJ-! T N+k ^ T oek --- ! T ^ T oe __ Proof. This follows from 10.5 . |__| Every automorphism ' : oek ! oek yields a homotopy equivalence T (1 y ') : T N+k = T ( N y oek)---! T ( N y oek) = T N+k and hence an isomorphism T (1 y ')* : {SN+n+k , T N+k } ! {SN+n+k , T N+k }. So, we have the aut oek-action a : aut oek x {SN+n+k , T N+k }! {SN+n+k , T N+k }, a (', ff) = T (1 y ')*(ff). Similarly, every automorphism ' of oek induces a homotopy equivalence T oek ! T oek, and therefore we have the action aff: aut oek x {T oek, Sk} ! {T oek, Sk}. 11.9. Theorem. The diagram aut oek x {T oek, Sk} --aoe-! {T oek, Sk} ? ? 1xD ?y ?yD aut oek x {SN+n+k , T N+k }--a-! {SN+n+k , T N+k } PL STRUCTURES 43 commutes. __ Proof. This follows from 11.8 and the definition of D, a and aff. |__| Because of Theorem 11.1, for k large enough we have {T oek, Sk} = ßk(T oek) and {SN+n+k , T N+k } = ßN+n+k (T N+k ). Then we can rewrite the diagram from Theorem 11.9 as aoe aut oek x ßk T oek --- ! ßk T oek ? ? (11.3) 1xD?y ?yD a autoek x ßN+n+k T N+k --- ! ßN+n+k T N+k Let R 2 ßN+n+k (T N+k ) be the set of reductions, and let C 2 ßk(T oek) be the set of coreductions. Then, clearly, a (R) R and aff(C) C. Therefore, in view of Proposition 11.5, the diagram (11.3) yields the commutative diagram aut oek x C --aoe-!C ? ? (11.4) 1xD ?y ?yD aut oek x R --a-! R 11.10. Theorem. For every ff, fi 2 C there exists an automorphism ' of oek such that aff(', ff) = fi. Moreover, this ' is unique up to fiberwise homotopy. In other words, the action aff: aut oek x C ! C is free and transitive. Proof. Recall that T oek = (M xSk)=M. So, for every m 2 M, a pointed map f : T oek ! Sk yields a pointed map fm : Skm ! Sk where Skm is the fiber over m. Furthermore, f represents a coreduction if and only if all maps fm belong to Fk. In other words, every coreduction for T oek yields a homotopy class M ! Fk. Moreover, it is easy to see that, in view of Proposition 10.1, the action affcoincides with the map [M, Fk] x [M, Fk] ! [M, Fk] __ induced by the product in Fk, and the result follows. |__| Since D is an isomorphism, Theorem 11.10 yields the following corol- lary. 11.11. Corollary. The action a : aut oek x R ! R is free and tran- __ sitive. |__| 44 YULI B. RUDYAK Now we can finish the proof of Theorem 4.6. Assuming dim j = N +k to be large enough, we conclude that o and jo are homotopy equivalent over M, see Atiyah [2, Prop. 3.5]. (Notice that Atiyah works with non-sectioned bundles, but there is no problem to adapt the proof for sectioned ones.) Choose any such FN+k -equivalence ' : jo ! o and consider the induced homotopy equivalence T ' : T j ! T . Clearly, the composition T' fi : SN+n+k -- ff-!T j --- ! T is a reduction. So, by 11.11, there exists an FN+k -equivalence ~ : o ! jo over M with (T ~)*(fi) = '. Now, we define ~ : o ! jo to be the fiber homotopy inverse to ~'. (The existence of an inverse equivalence can be proved following Dold [9], cf. [32 ]). Clearly, ~*' = ff. This proves the existence of the required equivalence ~. Furthermore, if there exists another equivalence ~0 : jo ! o, then ~0O~-1(') = ', and so ~ and ~0 are homotopic over M. This proves the__ uniqueness of ~. Thus, Theorem 4.6 is proved. |__| 12. Normal maps and F=P L Throughout the section we fix a closed PL manifold Mn . 12.1. Definition ([5, 31]). A normal map at M is a commutative di- agram of PL maps bb E -- - ! E0 ? ? ? ? y y V -- b-! M where V = {E ! V } is the normal PL RN -bundle over a closed PL manifold V n, , = {E0 ! M} is a PL RN -bundle over M and, finally, bbinduces a PL isomorphism of fibers and preserves the sections. In other words, a normal map is a bundle morphism ' : V ! ,. A normal map is called reducible if the map collapse T' SN+n -- - - !T V --- ! T , is a reduction. Because of the Thom Isomorphism Theorem, a normal map is re- ducible whenever b is a map of degree 1. (One can prove that , is orientable if V and M are.) PL STRUCTURES 45 12.2. Construction-Definition. Given a map (homotopy class) f : M ! F=P L, we represent it by an FN -morphism ' : oM ! (flNPL)o with N large enough, see 3.5 and/or 3.8. We denote the base of ' by b and set , = b*flNPL. Then the correcting FN -morphism oM ! ,o yields a commutative diagram g 0 o Uo -- - ! U ? ? (12.1) q?y ?yp M _______ M where M = {q : U ! M}, , = {p : U0 ! M} and Uo, U0oare fiberwise one-point compactifications of U and U0, respectively. We consider M as the zero section of ,, M U0 and deform g to a map t : Uo ! U0o which is transversal to M. Set V = t-1(M). We can always assume that V U. So, we get a morphism of PL RN -bundles bb E -- - ! E0 ? ? (12.2) ? ? y y V -- b-! M where {E0 ! M} is a normal bundle of M in U0 and {E ! V } is a normal bundle of V in U. Here b = t|V . Clearly, , := {E0 ! M} is the bundle of M in U0. Furthermore, since U is a total space of a normal bundle, the normal bundle of U is trivial. Thus, {E0 ! M} is the normal bundle V of V . In other words, the diagram (12.2) is a normal map at M. We say that the normal map (12.2) is associated with a map (homotopy class) f : M ! F=P L. Clearly, there any many normal maps associated with a given map f : M ! F=P L. 12.3. Construction-Definition. Let (12.3) ' : V ! , be a reducible normal map at M and assume that dim V is large. Consider a collapse map (homotopy class) ' : SN+n ! T M as in 4.5. Since the map collapse T' ff : SN+n --- - ! T V -- - ! T , is a reduction, there exists, by Theorem 4.6, a unique FN -morphism ~ : oM ! ,o with ~*(') = ff. Now, the morphism ~* o classif N o oM --- ! , --- ! (flPL ) 46 YULI B. RUDYAK is a homotopy PL structuralization of M . Thus, it gives us a map f' : M ! F=P L. 12.4. Proposition. The normal map (12.3) is associated with the__ map f' : M ! F=P L. |__| Recall that a closed manifold is called almost parallelizable if it be- comes parallelizable after deleting of a point. Notice that every al- most parallelizable manifold is orientable (e.g., because its first Stiefel- Whitney class is equal to zero). 12.5. Proposition. Let V m be an almost parallelizable PL manifold. Then every map b : V m ! Sm of degree 1 is the base of a reducible normal map at Sm . fiP Proof. We regard Sm = {(x1, . .,.xm+1 ) fi x2i= 1} as the union of two discs, Sm = D+ [ D- , where D+ = {x 2 Sm |xm+1 0}, D- = {x 2 Sm |xm+1 0}. We can always assume (deforming b if necessary) that there is a small closed disk D0 in V such that b+ := b|D0 : D0 ! D+ is a PL isomor- phism. We set W = V \(Int D0). Since W is parallelizable, there exists a morphism '- : V |W ! `D- of PL bundles such that b|W : W ! D- is the base of '. Furthermore, since b+ is a PL isomorphism, there ex- ists a morphism '+ : V |D0 ! `|D+ over b+ such that '+ and '- coincide over b|@W : @W ! Sm-1 . Together '+ and '- give us a morphism ' : V ! , where , is a PL bundle over Sm . Clearly, ' is a __ normal map with the base b, and it is reducible because deg b = 1. |__| 13. The Sullivan map s : [M, F=P L] ! PdimM We define the groups Pi by setting 8 < Z if i = 4k, Pi = Z=2 if i = 4k + 2, : 0 if i = 2k + 1 where k 2 N. Given a closed connected n-dimensional PL manifold M (which is assumed to be orientable for n = 4k), we define a map (13.1) s : [M, F=P L] ! Pn PL STRUCTURES 47 as follows. Given a homotopy class f : M ! F=P L, consider a normal map (12.2) bb E -- - ! E0 ? ? ? ? y y V -- b-! M associated with f. For n = 4k, let _ be the symmetric bilinear intersection form on Ker {b* : H2k(Z; Q) ! H2k(M; Q)}. We define s(u) = ff(_)_8where oe(_) is the signature of _. It is well known that oe(_) is divisible by 8, (see e.g. [5]), and so s(u) 2 Z. Also, it is easy to see that oe(Z) - oe(M) oe(u) = ______________ 8 where oe(M), oe(Z) is the signature of the manifold M, Z, respectively. For n = 4k + 2, we define s(u) to be the Kervaire invariant of the normal map (b,bb). The routine arguments show that s is well-defined, i.e. it does not depend on the choice of the associated normal map. See [5, Ch. III, x4] or [40 ] for details. In particular, if b is a homotopy equivalence then s(u) = 0. One can prove that every map s is a homomorphism, where the abelian group structure on [M, F=P L] is given by an H-space structure on F=P L. Given a map f : M ! F=P L, it is useful to introduce the notation s(M, f) := s([f]) where [f] is the homotopy class of f. 13.1. Theorem. (i) The map s : [S4i, F=P L] ! Z is surjective for all i > 1, (ii) The map s : [S4i-2, F=P L] ! Z=2 is surjective for all i > 0 . (iii) The image of the map s : [S4, F=P L] ! Z is the subgroup of index 2. Proof. (i) For every k > 1 Milnor constructed a parallelizable 4k- dimensional smooth manifold W 4kof signature 8 and such that @W is a homotopy sphere, see [5, V.2.9]. Since, by Theorem 3.11, every homotopy sphere of dimension 5 is PL isomorphic to the standard one, we can form a closed PL manifold V := W [S4k-1D4k 48 YULI B. RUDYAK of the signature 8. Because of Proposition 12.5, there exists a reducible normal map with the base V 4k! S4k. Because of Proposition 12.4, this normal map is associated with a certain map (homotopy class) f : S4k ! F=P L. Thus, s(S4k, f) = 1. (ii) The proof is similar to that of (i), but we must use (4k+2)- dimensional parallelizable Kervaire manifolds W , @W = S4k+1 of the Kervaire invariant one, see [5, V.2.11]. (iii) The Kummer algebraic surface [26 ] gives us an example of 4- dimensional almost parallelizable smooth manifold of the signature 16. So, Im s 2Z. Now suppose that there exists f : S4 ! F=P L with s(S4, f) = Z. Then there exists a normal map with the base V 4! S4 and such that V has signature 8. Since normal bundle of V is induced from a bundle over S4, we conclude that w1(V ) = 0 = w2(V ). But this contradicts__ the Rokhlin Theorem 7.2. |__| 13.2. Theorem. For every closed simply-connected PL manifold M of dimension 5, the sequence jF s 0 --- ! SPL (M) --- ! [M, F=P L] --- ! PdimM is exact, i.e. jF is injective and Im jF = s-1(0). Proof. See [5, II.4.10 and II.4.11]. Notice that the map ! in loc. cit is the zero map because, by Theorem 3.11, every homotopy sphere of __ dimension 5 is PL isomorphic to the standard sphere. |__| 13.3. Corollary. We have ß4i(F=P L) = Z, ß4i-2(F=P L) = Z=2, and ß2i-1(F=P L) = 0 for every i > 0. Moreover, the map s : [Sk, F=P L] ! Pk is an isomorphism for k 6= 4 and has the form Z = ß4(F=P L) -- s-! P4 = Z, a 7! 2a for k = 4. Proof. First, if k > 4 then, because of the Smale Theorem 3.11, SPL (Sk) is the one-point set. Now the result follows from 13.2 and 13.1. If k 4 then ßk(P L=O) = 0, cf. Remark 6.7. So, ßk(F=P L) = ßk(F=O). Moreover, the forgetful map ßk(BO) ! ßk(BF ) coincides with the Whitehead J-homomorphism. So, we have the long exact sequence . .!.ßk(F=O) ! ßk(BO) --J-! ßk(BF ) ! ßk-1(F=O) ! . ... PL STRUCTURES 49 For k 5 all the groups ßk(BO) and ßk(BF ) are known (notice that ßk(BF ) is the stable homotopy group ßk+N-1 (SN )), and it is also known that J is an epimorphism for k = 1, 2, 4, 5, see e.g. [1]. Thus,_ ßk(F=O) ~=Pi for k 4. The last claim follows from 13.1. |__| 14. The homotopy type of F=P L[2] As usual, given a space X and an abelian group ß, we do not dis- tinguish elements of Hn (X; ß) and maps (homotopy classes) X ! K(ß, n). For example, regarding a Steenrod cohomology operation Sq2 as an element Sq2 2 Hk+2 (K(Z=2, k); Z=2), we can treat it as a map Sq2 : K(Z=2, k) ! K(Z=2, k + 2). Given a prime p, let Z[p] be the subring of Q consisting of all irre- ducible fractions with denominators relatively prime to p, and let Z[1=p] be the subgroup of Q consisting of the fractions m=pk, m 2 Z. Given a simply-connected space X, we denote by X[p] and X[1=p] the Z[p]- and Z[1=p]-localization of X, respectively. Furthermore, we denote by X[0] the Q -localization of X. For the definitions, see [21 ]. Consider the short exact sequence j* 0 --- ! Z[2] --2-! Z[2] --- ! Z=2 --- ! 0 where 2 over the arrow means multiplication by 2 and æ is the modulo 2 reduction. This exact sequence yields the Bockstein exact sequence j n . . .-! Hn (X; Z[2]) --2-!Hn (X; Z[2]) --- ! H (X; Z=2) (14.1) --ffi-!Hn+1 (X; Z[2]) -! . ... Put X = K(Z=2, n) and consider the fundamental class ' 2 Hn (K(Z=2, n); Z=2). Then we have the class ffi := ffi(') 2 Hn+1 (K(Z=2, n), Z[2]). Accord- ing to what we said above, we regard ffi as a map ffi : K(Z=2, n) ! K(Z[2], n + 1). 14.1. Proposition (Sullivan [56 , 57]). For every i > 0 there are co- homology classes K4i2 H4i(F=P L; Z[2]), K4i-22 H4i-2(F=P L; Z=2) such that s(M4i, f) = for every closed oriented PL manifold M, and s(N4i-2, f) = . 50 YULI B. RUDYAK for every closed manifold N. Here [M] 2 H4i(M) is the orientation of M, [N]2 2 H4i-2(N; Z=2) is the modulo 2 fundamental class of N, and <-, -> is the Kronecker pairing. Proof. Let MSO*(-) denote the oriented bordism theory, see e.g [48 ]. Recall that if two maps f : M4i ! F=P L and g : N4i ! F=P L are bordant (as oriented singular manifolds) then s(M, f) = s(N, g). Thus, s defines a homomorphism es: MSO4i(F=P L) ! Z. It is well known that the Steenrod-Thom map t : MSO*(-) Z[2] ! H*(-; Z[2]) splits, i.e. there is a natural map v : H*(-; Z[2]) ! MSO*(-) Z[2] such that tv = 1 (a theorem of Wall [61 ], see also [55 , 48 , 3]. In particular, we have a natural homomorphism bs: H4i(F=P L; Z[2])-- v-! MSO4i(F=P L) Z[2] ---es! Z. Since the evaluation map ev : H*(X; Z[2]) ! Hom (H*(X; Z[2]), Z[2]), (ev(x)(y) = is surjective for all X, there exists a class K4i2 H4i(F=P L; Z[2]) such that ev(K4i) = bs. Now s(M, f) = bs(f*[M]) = = . So, we constructed the desired classes K4i. The construction of classes K4i-2is similar. Let MO*(-) denoted the non-oriented bordism theory. Then the map s yields a homomorphism es: MO4i-2(F=P L) ! Z=2. Furthermore, there exists a natural map H*(-; Z=2) ! MO*(-) which splits the Steenrod-Thom homomorphism, and so we have a homomor- phism bs: H4i-2(F=P L; Z=2) --- ! MO4i-2(F=P L) Z[2] --es-!Z=2 with bs(f*([M]2) = s(M, f). Now we can complete the proof similarly __ to the case of classes K4i. |__| We set Y (14.2) := (K(Z[2], 4i) x K(Z=2, 4i - 2)). i>1 PL STRUCTURES 51 Together the classes K4i : F=P L ! K(Z[2], 4i), i > 1 and K4i-2 : F=P L ! K(Z[=2, 4i - 2), i > 1 yield a map K : F=P L ! . 14.2. Lemma. The map (14.3) K[2] : F=P L[2] ! induced an isomorphism of homotopy groups in dimensions 5. __ Proof. This follows from Theorem 13.1 and Corollary 13.3. |__| 14.3. Proposition. The Postnikov invariant - 2 H5(K(Z=2, 3), Z[2]) of F=P L[2] is non-zero. Moreover, - = ffiSq2 2 H5(K(Z=2, 3); Z[2]). Proof. Let h : ß4(F=P L) ! H4(F=P L) be the Hurewicz homomor- phism. Suppose that - = 0. Then H4(F=P L; Z[2]) = Z[2] H4(K(Z=2, 2; Z[2], and therefore the homomorphism H4(F=P L; Z[2]) --ev-! Hom (H4(F=P L) Z[2], Z[2]) --h*-! Hom (ß 4(F=P L) Z[2], Z[2]) must be surjective. But this contradicts 13.1(ii). Thus, - 6= 0. Furthermore, since F=P L is an infinite loop space, - has the form N a for some a 2 HN+5 (K(Z=2, N +3), Z[2]). But, for general reasons, for N large enough the last group consists of elements of the order 2. Thus, - has the order 2. It is easy to see that H5(K(Z=2, 3), Z[2] = __ Z=4 = {x} with 2x = ffiSq2, see [48 ]. Thus, - = ffiSq2. |__| 14.4. Remark. It is interesting to have a geometrical description of the class z 2 H4(F=P L) with = 1. Let j denote the canonical complex line bundle over CP2. One can prove that 24j is fiberwise homotopy trivial. So, there exist a map f : CP2 ! F=P L such that the map f CP2 --- ! F=P L --- ! BP L classifies 24j. Since p1(j) = 1, we conclude that p1(24j) = 24, and therefore L1(24j) = 8 (here L1 denotes the first Hirzebruch class), see [38 ]. Thus, s(CP2, f) = 8=8 = 1, and therefore = 1. 52 YULI B. RUDYAK Let Y be the homotopy fiber of the map ffiSq2 : K(Z=2, 2) ! K(Z[2], 5). In other words, there is a fibration p (14.4) K(Z[2], 4) --i-! Y --- ! K(Z=2, 2) with the characteristic class ffiSq2. Because of Proposition 14.3, the space Y is the Postnikov 4-stage of F=P L[2]. In particular, we have a map _ : F=P L[2] ! Y which induces an isomorphism of homotopy groups in dimension 4. Together with the map K[2] from 14.3, this map yields a map OE : F=P L[2] ! Y x . 14.5. Theorem. The map (14.5) OE : F=P L[2] ' Y x is a homotopy equivalence. __ Proof. This follows from 14.3 and what we said about _. |__| 14.6. Lemma. Let X be a finite CW -space such that the group H*(X) is torsion free. Then the group [X, F=P L[1=2]] is torsion free. Proof. It suffices to prove that [X, F=P L[p]] is torsion free for every odd prime p. Notice that F=P L[p] is an infinite loop space since F=P L is. So, there exists a connected p-local spectrum E such that eE0(Y ) = [Y, F=P L[p]] = [Y, F=P L] Z[p]. Moreover, E-i(pt) = ßi(E) = ßi(F=P L) Z[p], So, because of the isomorphism Ee0(X) ~= [X, F=P L[p]], it suffices to prove that E*(X) is torsion free. Consider the Atiyah-Hirzebruch spectral sequence for E*(X). Its initial term is torsion free because E*(pt) is torsion free. Hence, the spectral sequence degenerates, and thus the group E*(X) __ is torsion free. |__| 14.7. Proposition. Let X be a finite CW -space such that the group H*(X) is torsion free. Let f : X ! F=P L be a map such that f*K4n = 0 and f*K4n+2 = 0 for every n 1. Then f is null-homotopic. PL STRUCTURES 53 Proof. Consider the commutative square F=P L --l1-! F=P L[2] ? ? l2?y ?yl3 F=P L[1=2] --l4-! F=P L[0] where the horizontal maps are the Z[2]-localizations and the verti- cal maps are the Z[1=2]-localizations. Because of 14.5, [X, F=P L] is a finitely generated abelian group, and so it suffices to prove that both l1Of and l2Of are null-homotopic. First, we remark that l2Of is null-homotopic whenever l1Of is. Indeed, since H*(X) is torsion free, the group [X, F=P L[1=2]] is torsion free by 14.6. Now, if l1Of is null-homotopic then l3Ol1Of is null-homotopic, and hence l4Ol2Of is null-homotopic. Thus, l2Of is null-homotopic since [X, F=P L[1=2]] is torsion free. So, it remains to prove that l1Of is null-homotopic. Clearly, the equalities f*K4i= 0 and f*K4i-2= 0, i > 1, imply that the map p2 X --- ! F=P L --l1-! F=P L[2] ' Y x --- ! is null-homotopic. So, it remains to prove that the map f l1 p1 g : X -- - ! F=P L -- - ! F=P L[2] ' Y x -- - ! Y is null-homotopic. It is easy to see that H4(Y ; Z[2]) = Z[2]. Let u 2 H4(Y ; Z[2]) be a free generator of the free Z[2]-module H4(Y ; Z[2]). The fibration (14.4) gives us the following diagram with the exact row: p* 2 H4(X; Z[2]) --i*-! [X, Y ] --- ! H (X; Z=2) ? ? yu* H4(X; Z[2]) Notice that u*i* : Z[2] ! Z[2] is the multiplication by 2" where " is an invertible element of the ring Z[2]. Since f*K2 = 0, we conclude that p*(g) = 0, and so g = i*(a) for some a 2 H4(X; Z[2]). Now, 0 = u*(g) = u*(i*a) = 2a". __ But H*(X; Z[2]) is torsion free, and thus a = 0. |__| 54 YULI B. RUDYAK For completeness, we mention also that F=P L[1=2] ' BO[1=2]. So, there is a Cartesian square (see [31 , 57]) F=P L -- - ! x Y ? ? ? ? y y ph Q BO[1=2] -- - ! K(Q, 4i) where ph is the Pontryagin character. 15. Splitting theorems 15.1. Definition. Let An+k and W n+k be two connected PL mani- folds (without boundaries), and let Mn be a closed submanifold of A. We say that a map g : W n+k! An+k splits along Mn if there exists a homotopy gt : W n+k! An+k , t 2 I such that: (i) g0 = g; (ii) there is a compact subset K of W such that gt|W \ K = g|W \ K for every t 2 I; (iii) the map g1 is transversal to M; (iv) the map b := (g1)|g-11(M) : g-11(M) ! M is a homotopy equiv- alence. An important special case is when An+k = Mn x Bk for some con- nected manifold Bk. In this case we can regard M as submanifold M x {b0}, b0 2 B of A and say that g : W ! A splits along M if it splits along M x {b0}. Clearly, this does not depend on the choice of {b0}, i.e. g splits along M x {b0} if and only if it splits along M x {b} with any other b 2 B. Recall that a map f is called proper if f-1 (C) is compact whenever C is compact. A map f : X ! Y is called a proper homotopy equivalence if there exists a map g : Y ! X and the homotopies F : gf ' 1X , G; fg ' 1Y such that all the four maps f, g, F : X x I ! X and G : Y x I ! Y are proper. 15.2. Theorem. Let Mn , n 5 be a closed connected n-dimensional PL manifold such that ß1(M) is a free abelian group. Then every proper homotopy equivalence h : W n+1! Mn x R splits along Mn . Proof. Because of the transversality theorem, there is a homotopy ht : W ! M x R which satisfies conditions (i)-(iii) of 15.1. We let f = h1. Because of a crucial theorem of Novikov [41 , Theorem 3], there PL STRUCTURES 55 is a sequence of interior surgeries of the inclusion f-1 (M) W in W such that the final result of these surgeries is a homotopy equivalence V W . (This is the place where we use the fact that ß1(M) is free.) Using the Pontryagin-Thom construction, we can realize this sequence of surgeries via a homotopy ft such that ft satisfies conditions (i)-(iii)_ of 15.1 and f-11(M) = V . |__| 15.3. Theorem. Let Mn be a manifold as in 15.2. Then every homo- topy equivalence W n+1! Mn x S1 splits along Mn . Proof. This follows from results of Farrell and Hsiang [12 , Theorem __ 2.1] since Wh (Zm ) = 0 for every m. |__| 15.4. Corollary. Let Mn be a manifold as in 15.2. Let T k denote the k-dimensional torus. Then every homotopy equivalence W n+k ! Mn x T ksplits along Mn . __ Proof. This follows from 15.3 by induction. |__| 15.5. Theorem. Let Mn be a manifold as in 15.2. Then every home- omorphism h : W n+k! Mn x Rk of a PL manifold W n+k splits along Mk. Proof. We use the Novikov's torus trick. The canonical inclusion T k-1x R Rk yields the inclusion M x T k-1x R M x Rk. We set W1 := h-1(M x T k-1x R). Notice that W1 is a smooth manifold since it is an open subset of W . Now, set u = h|W1 : W1 ! M x T k-1x R. Then, by 15.2, u splits along M x T k-1, i.e. there is a homotopy ut as in 15.1. We set f := u1, V := f-1 (M x T k-1) and g := f|V . Because of 15.4, g : V ! M x T k-1splits along M. Hence, f splits along M, and therefore u splits along M. Let __utbe the homotopy which realizes this splitting as in 15.1. Now, we define the homotopy ht : W ! M xRn by setting ht|W1 := __ut|W1 and ht|W \ W1 := h|W \ W1. Notice that {ht} is a well-defined and continuous family since the family {__ut} satisfies 15.1(ii). It is clear that ht satisfies the conditions (i)_(iii) of 15.1_and that h1 extends f on the whole W , i.e. h splits along M. |__| 15.6. Remark. The above used theorems of Novikov and Farrell- Hsiang were originally proved for smooth manifolds, but they are valid for PL manifolds as well, because there is an analog of the Thom Transversality Theorem for PL manifolds, [65 ]. 15.7. Lemma. Suppose that a map g : W ! A splits along a sub- manifold M of A. Let , = {E ! A} be a PL bundle over A, let 56 YULI B. RUDYAK g*, = {D ! W }, and let k : g*, ! , be the g-adjoint bundle mor- phism. Finally, let l : D ! E be the map of the total spaces induced by k. Then l splits over M. (Here we regard A as the zero section of ,, and so M turns out to be a submanifold of E). Proof. Let G x I ! A be a homotopy which realizes the splitting of g. Recall that g*, x I is equivalent to G*,. Now, the morphism IG,, g*, x I ~= G*, --- ! , __ gives us the homotopy D x I ! E which realizes a splitting of l. |__| 15.8. Lemma. Let M be a manifold as in 15.2. Consider two PL RN - bundles , = {U ! M} and j = {E ! M} over M and a topological morphism ' : , ! j over M of the form g U -- - ! E ? ? ? ? y y M _______M. Then there exists k such that the map g x 1 : U x Rk ! E x Rk splits along M, where M is regarded as the zero section of j. Proof. Take a PL Rm -bundle i such that j i = `N+m and let W be the total space of , i. Then the map ' 1 : , i ! j i = `N+m yields a map (15.1) : W ! M x RN+m of the total spaces. Because of Theorem 15.5, the map splits along M. Now, considering the morphism ' 1 1 : , i j ! j i j and passing to the total spaces, we get a map g x 1 : U x R2N+m ! E x R2N+m . In view of Lemma 15.7, this map splits over M because does. So,__ we can put k = 2N + m. |__| Now, let a : T OP=P L ! F=P L be a map as in (2.5). PL STRUCTURES 57 15.9. Theorem. Let M be as in 15.2. Then the composition [M, T OP=P L] --a*-! [M, F=P L] --s-! PdimM is trivial, i.e., sa*(v) = 0 for every v 2 [M, T OP=P L]. In other words, s(M, af) = 0 for every f : M ! T OP=P L. Proof. In view of 3.4, every element v 2 [M, T OP=P L] is a concordance class of a topological morphism ' : NM -! flNPL of PL RN -bundles. Passing from the class v 2 [M, T OP=P L] to the class a*v 2 [M, F=P L], we must consider the equivalence class of FN - morphism 'o : ( M )o ! (flNPL)o. Now, following the definition of the map s : [M, F=P L] ! PdimM , we get a commutative diagram g 0 o Uo -- - ! U ? ? (15.2) q?y ?yp M _______ M like (12.1). However, here we know that g is a homeomorphism. Thus, g(U) = U0, and so we get the diagram g 0 U --- ! U ? ? (15.3) q?y ?yp M _______M which is a topological morphism of PL bundles over M. We embed M in U0 as the zero section. By the definition of the map s, we conclude that s(M, a*v) = 0 if the map g : U ! U0 splits along M (because in this case the associated normal map is a map over a homotopy equivalence). Moreover, since, for every k, the topological morphisms ' and (' 1) N k N+k M `k --- ! flPL ` -- - ! flPL represent the same element of [M, T OP=P L], it suffices to prove that there exists k such that the map g x 1 : U x Rk ! U0 x Rk __ splits along M. But this follows from Lemma 15.8. |__| Now we show that the condition dim M 5 in 15.9 is not necessary. 15.10. Corollary. Let M be a closed connected PL manifold such that ß1(M) is a free abelian group. Then s(M, af) = 0 for every map f : M ! T OP=P L. 58 YULI B. RUDYAK Proof. Let CP 2denote the complex projective plane, and let p1 : M x CP 2! M be the projection on the first factor. Then s(M x CP 2, gp1) = s(M, g) for every g : M ! F=P L, see [5, Ch. III, x5]. In particular, for every map f; M ! T OP=P L we have s(M, af) = s(M x CP2, (af)p1) = s(M x CP2, a(fp1)) = 0 __ where the last equality follows from Theorem 15.9. |__| 16. Detecting families Recall the terminology: a singular smooth manifold in a space X is a map M ! X of a smooth manifold. Given a CW -space X, consider a connected closed smooth singular manifold fl : M ! X in X. Then, for every map f : X ! F=P L, the invariant s(M, ffl) 2 PdimM is defined. Clearly, if f is null-homotopic then s(M, ffl) = 0. 16.1. Definition. Let {fli : Mi ! X}i2I be a family of closed con- nected smooth singular manifolds in X. We say that the family {fli : Mi ! X} is a detecting family for X if, for every map f : X ! F=P L, the validity of all the equalities s(Mi, ffli) = 0, i 2 I implies that f is null-homotopic. Notice that F=P L is an H-space, and hence, for every detecting family {fli : Mi ! X}, the collection s(Mi, ffli) determine a map f : X ! F=P L uniquely up to homotopy. The concept of detecting family is related to Sullivan's "characteris- tic variety", but it is not precisely the same. If a family F of singular manifolds in X contains a detecting family, then F itself is a detect- ing family. On the contrary, the characteristic variety is in a sense "minimal" detecting family. 16.2. Theorem. Let X be a connected finite CW -space such that the group H*(X) is torsion free. Then X possesses a detecting family {fli : Mi ! X} such that each Mi is orientable. Proof. Since H*(X) is torsion free, every homology class in H*(X) can be realized by a closed connected smooth oriented singular manifold, see e.g. [8, 15.2] or [48 , 6.6 and 7.32]. Let {fli : Mi ! X} be a family of smooth oriented closed connected singular manifolds such that the elements (fli)*[Mi] generate all the groups H2k(X). PL STRUCTURES 59 We prove that {fli : Mi ! X} is a detecting family. Consider a map f : X ! F=P L such that si(Mi, ffli) = 0 for all i. We must prove that f is null-homotopic. Because of 14.7, it suffices to prove that f*K4n = 0 and f*K4n-2 = 0 for every n 1. Furthermore, H*(X) = Hom (H*(X), Z) because H*(X) is torsion free. So, it suffices to prove that (16.1) = 0 for everyx 2 H4n(X) and (16.2) = 0 for everyx 2 H4n-2(X; Z=2). First, we prove (16.1). Since the classes (fli)*[Mi], dim Mi = 4n gener- ates the group H4n(X), it suffices to prove that = 0 whenever dim Mi = 4n . But, because of 14.1, for every 4n-dimensional Mi we have 0 = s(Mi, ffli) = <(ffli)*K4n, [Mi]>= . This completes the proof of the equality (16.1). Passing to the case n = 4k + 2, notice that the group H2k(X; Z=2) is generated by the elements (fli)*[Mi]2, dim Mi = 2k, since H*(X) is torsion free. Now the proof can be completed similarly to the case __ n = 4k. |__| 17. A special case of the theorem on the normal invariant of a homeomorphism 17.1. Theorem. Let M be closed connected PL manifold such that each of the group Hi(M) and ß1(M) is a free abelian group. Then jF (x) = 0 whenever x 2 SPL can be represented by a homeomorphism h : V ! M. Proof. The maps jTOP and jF from section 3 can be included in the following commutative diagram: jTOP TPL (M) --- ! [M, T OP=P L] ? ? ? ? (17.1) y ya* jF SPL (M) --- ! [M, F=P L] where the left arrow is the obvious forgetful map and a* is induced by a as in (2.5). Suppose that x can be represented by a homeomorphism h : V ! M. Consider a map f : M ! T OP=P L such that jTOP (h) is homotopy 60 YULI B. RUDYAK class of f. Then, clearly, the class jF (x) 2 [M, F=P L] is represented by the map f a M --- ! T OP=P L -- - ! G=P L. By 16.2, M possesses a detecting family {fli : Mi ! M}. We can assume (performing oriented surgeries of Mi if necessary) that ß1(Mi) is a subgroup of ß1(M), and so ß1(Mi) is a free abelian group. Hence, by 15.9 and 15.10, s(Mi, affli) = 0 for every i. But {fli : Mi ! M} is a detecting family, and therefore af is null-homotopic. Thus, jF (x))_=_ 0. |__| PL STRUCTURES 61 Chapter 3. Applications 18. Topological invariance of rational Pontryagin classes 18.1. Lemma. The homotopy groups ßi(P L=O) are finite. Thus, the space P L=O[0] is contractible. __ Proof. See [48 , IV.4.27(iv)]. |__| Recall that H*(BO; Q) = Q[p1, . .,.pi, . .].where pi, dim pi = 4i is the universal rational Pontryagin class, see e.g. [38 ]. (In fact, pi is the image of the integral Pontryagin class pi 2 H*(BO).) 18.2. Theorem. The forgetful map ff = ffOPL : BO ! BP L induces an isomorphism ff* : H*(BP L; Q) ! H*(BO; Q). Proof. It suffices to prove that ff[0] : BO[0] ! BP L[0] is a homotopy equivalence. But this holds because the homotopy fiber of ff[0] is the_ contractible space P L=O[0]. |__| It follows from 18.2 that H*(BP L; Q) == Q[p01, . .,.p0i, . .].where p0iare the cohomology classes determined by the condition ff*(p0i) = pi. Now, given a PL manifold M, we define its rational Pontryagin classes p0i(M) 2 H4i(M; Q) by setting p0i(M) = t*p0i where t : M ! BP L classifies the stable tangent bundle of M. Clearly, if we regard a smooth manifold as a PL manifold then pi(N) = p0i(N). 18.3. Corollary (PL invariance of Pontryagin classes, [60 , 45]).Let f : M1 ! M2 be a PL isomorphism of smooth manifolds. Then __ f*pi(M2) = pi(M1). |__| 18.4. Theorem. The forgetful map ff = ffPLTOP: BP L ! BT OP in- duces an isomorphism ff* : H*(BT OP ; Q) ! H*(BP L; Q). Proof. This can be deduced from Theorem 8.8 just in the same manner_ as we deduced Theorem 18.2 from Theorem 18.1. |__| Now we introduce the universal classes p00i2 H4i(BT OP ; Q) by the equality (ffPLTOP)*p0i= p00i. 62 YULI B. RUDYAK Furthermore, given a topological manifold M, we set p00i(M) = t*p00i (where t classifies the stable tangent bundle of M) and get the following corollary. 18.5. Corollary (topological invariance of Pontryagin classes, [41 ]). Let f : M1 ! M2 be a homeomorphism of smooth manifolds. Then __ f*pi(M2) = pi(M1). |__| 19. The space F=T OP It turns out to be that, in view of the Product Structure Theorem, the Transversality Theorem holds for topological manifolds and bun- dles. I am not able to discuss it here, see [48 , IV.7.18] for the references. Since we have the topological transversality, we can define the maps s0: [M, F=T OP ] ! PdimM which are obvious analogs of maps s defined in (13.1). However, here we allow M to be a topological manifold. The following proposition demonstrates the main difference between F=P L and F=T OP . 19.1. Proposition. The map s0: ß4(F=T OP ) ! Z is a surjection. Proof. Notice that the Freedman manifold from Theorem 7.1 is almost parallelizable and has signature 8. Now the proof can be completed __ just as 13.1(i). |__| Recall that in (2.5)we described a fibration T OP=P L ---a! F=P L -- b-! F=T OP. 19.2. Theorem. For i 6= 4 the map b : F=P L ! F=T OP induces an isomorphism b* : ßi(F=P L) ! ßi(F=T OP ). The map b* : Z = ßi(F=P L) ! Z = ßi(F=T OP ) is the multiplication by 2. Proof. Recall that T OP=P L = K(Z=2, 3). So, the homotopy exact sequence of the fibration T OP=P L -- a-! F=P L -- b-! F=T OP PL STRUCTURES 63 yields an isomorphism b* : ßi(F=T OP ) ~= ßi(F=P L) for i 6= 4. Fur- thermore, we have the commutative diagram 0 _______ß4(T OP=P L) ? a*?y Z _______ ß4(F=P L) -- s-! Z ? fl b*?y flfl 0 ß4(F=T OP ) -- s-! Z ? ? y ß3(T OP=P L) _______Z=2 ? ? y ß3(F=P L) _______ 0 where the middle line is a short exact sequence. So, ß4(F=T OP ) = Z or ß4(F T OP ) = Z Z=2. By 13.1, Im s is the subgroup 2Z of Z, while s0 is a surjection by 19.2. Thus, ß4(F=T OP ) = Z and b* is the__ multiplication by 2. |__| Now, following 14.1, we can introduce the classes K04i2 H4i(F=T OP, Z[2]) and K04i-22 H4i-2(F=T OP, Z=2) such that s0(M4i, f) = and s0(N4i-2, f) = . However, here M and N are assumed to be topological (i,e, not neces- sarily PL) manifolds. Together these classes yield the map Y K0 : F=T OP - ! (K(Z[2], 4i) x K(Z=2, 4i - 2)). i>0 19.3. Theorem. The map Y K0[2] : K0 : F=T OP [2] -! (K(Z[2], 4i) x K(Z=2, 4i - 2)) i>0 is a homotopy equivalence. Proof. Together 13.1 and 19.1 imply that the homomorphisms s0 : ß2i(F=T OP ) ! P2iare surjective. Now, in view of 19.2, all the homo-__ morphisms s0 are isomorphisms, and the result follows. |__| 64 YULI B. RUDYAK So, the only difference between F=P L and F=T OP is that F=T OP [2] has trivial Postnikov invariants while F=P L[2] has just one non-trivial Postnikov invariant ffiSq2 2 H5(K(Z=2, 2); Z[2]). 20. The map a : T OP=P L ! F=P L 20.1. Proposition. The map a : T OP=P L ! F=P L is essential. Proof. For general reasons, the fibration T OP=P L -- a-! F=P L -- - ! F=T OP yields a fibration (F=T OP ) -- u-! T OP=P L --a-! F=P L. If a is inessential then there exists a map v : T OP=P L ! (F=T OP ) with uv ' 1. But this is impossible because ß3(T OP=P L) = Z=2 while ß3( (F=T OP )) = ß4(F=T OP ) = Z. Take an arbitrary map f : X ! T OP=P L. Let ` : F=P L ! F=P L[2] denote the localization map. 20.2. Proposition. The following three conditions are equivalent: (i) the map f a X --- ! T OP=P L -- - ! F=P L is essential; (ii) the map f a ` X -- - ! T OP=P L --- ! F=P L -- - ! F=P L[2] is essential; (iii) the map f a ` projection X --- ! T OP=P L --- ! F=P L --- ! F=P L[2] --- - - !Y is essential. Proof. It suffices to prove that (i) ) (ii) )(iii). To prove the first implication, recall that a map u : X ! F=P L is inessential if both localized maps X --u-! F=P L ! F=P L[2], X -- u-! F=P L ! F=P L[1=2] are inessential. Now, (i) ) (ii) holds since T OP=P L[1=2] is con- tractible. To prove the second implication, notice that a map v : X ! F=P L[2] is inessential if both maps (we use notation as in 14.5) X --v-! F=P L[2] --K- ! , X -- v-! F=P L[2] --- ! Y PL STRUCTURES 65 are inessential. So, it suffices to prove that the map `af X --- ! F=P L[2] --- ! is inessential. This holds, in turn, because the map T OP=P L ! F=P L ! F=T OP is inessential and the diagram K[2] F=P L[2] --- ! _______ ? fl ? fl y b[2] fl K0[2]Q proj F=T OP [2] --- ! i>0(K(Z=2, 4i - 2) x K(Z[2], 4i))-- - ! __ commutes. |__| Consider the fibration K(Z[2], 4) ---i! Y --- ! K(Z=2.2) as in (14.4). 20.3. Lemma. For every space X, the homomorphism H4(X; Z[2]) = [X, K(Z[2], 4)] --i*-! [X, Y ] is injective. Moreover, i* is an isomorphism if H2(X; Z=2) = 0. Proof. The fibration (14.4) yields the exact sequence (see e.g. [39 ]) ffiSq24 i* 2 (20.1) H1(X; Z=2) --! H (X; Z[2]) -! [X, Y ] ! H (X; Z=2) __ where ffiSq2(x) 0 (because ffiSq2(x) = 0 whenever deg x = 1). |__| Let g : T OP=P L ! Y be the composition proj T OP=P L --a-! F=P L -- `-! F=P L[2] --- ! Y. Notice that g is essential because of 20.1 and 20.2. 20.4. Corollary. The map T OP=P L = K(Z=2, 3) --ffi-!K(Z[2], 4) ---i! Y is homotopic to g, i.e. g ' iffi. Proof. Since the sequence (20.1) is exact, the set [K(Z=2, 3), Y ] has precisely two elements. Since both maps g and iOffi are essential (the_ last one because of Lemma 20.3), we conclude that g ' iffi. |__| 66 YULI B. RUDYAK 20.5. Theorem. Given a map f : X ! T OP=P L, the map f a X --- ! T OP=P L -- - ! F=P L is essential if and only if the map f ffi X -- - ! T OP=P L = K(Z=2, 3) --- ! K(Z[2], 5) is essential. Proof. af is essential 20.2()gf is essential (20.4)iffif is essential (20.3)ff* *if_ is essential. |__| 21. The theorem on the normal invariant of a homeomorphism 21.1. Lemma. Let X be a finite CW -space such that Hn(X) is 2- torsion free. Then the homomorphism ffi : Hn (X; Z=2) ! Hn+1 (X; Z[2]) is zero. Proof. Because of the exactness of the sequence (14.1) Hn (X; Z=2) -- ffi-!Hn+1 (X; Z[2])-- 2-! Hn+1 (X; Z[2]), it suffices to prove that Hn+1 (X; Z[2]) is 2-torsion free. Since Hn(X) is 2-torsion free, we conclude that Ext (Hn(X), Z[2]) = 0. Thus, Hn+1 (X; Z[2]) = Hom (Hn+1(X; Z[2]) Ext(Hn(X); Z[2]) = Hom (Hn+1(X; Z[2]), __ and the result follows. |__| 21.2. Theorem. Let M be a closed PL manifold such that H3(M) is 2-torsion free. Then the normal invariant of any homeomorphism h : V ! M is trivial. Proof. Since h is a homeomorphism, the normal invariant jF (h) turns out to be the homotopy class of a map f a M --- ! T OP=P L --- ! F=P L where the homotopy class of f is jTOP (h). Because of 20.2 and 20.3, it suffices to prove that the map f ffi M --- ! T OP=P L = K(Z=2, 3) --- ! K(Z[2], 4) __ is inessential. But this follows from Lemma 21.1. |__| PL STRUCTURES 67 21.3. Corollary. Let M, dim M 5 be a closed simply-connected PL manifold such that H3(M) is 2-torsion free. Then every homeomor- phism h : V ! M is homotopic to a PL isomorphism. In particular, V and M are PL isomorphic. __ Proof. This follows from 13.2 and 21.2. |__| 21.4. Remark. Rourke [46 ] suggested another proof of 21.2, using the technique of simplicial sets. 22. A counterexample to the Hauptvermutung, and other examples 22.1. Example. Two manifolds which are homeomorphic but not PL isomorphic. Let RPn denote the real projective space of dimension n. 22.2. Lemma. For every homotopy equivalence h : RP5 ! RP5 we have jF (h) = 0. Proof. We triangulate RP4 and take the induced triangulation of the covering space S4. Take the corresponding triangulation of the sus- pension SS4 = S5. Let r : S5 ! S5 be the reflection with respect to the equator S4. Since r is an antipodal map, it yields a map f : RP5 ! RP5. Clearly, f is a map of degree -1. It follows from the obstruction theory that every homotopy equiva- lence RP5 ! RP5 is homotopic either to f or to the identity map. (For the homotopy classification of maps RPn ! RPn, see [18 ].) Since f is_ a PL isomorphism, the lemma follows. |__| Recall that jTOP : TPL (RP5) ! [RP5, T OP=P L] is a bijection. Con- sider a homeomorphism k : M ! RP5 such that jTOP (k) 6= 0 2 [RP5, T OP=P L] = H3(RP5; Z=2) = Z=2. Notice that ffi : Z=2 = H3(RP5; Z=2) ! H4(RP5; Z=2) = Z=2 is an isomorphism, and hence ffi(jTOP (k)) 6= 0. So, by Theorem 20.5, a*jTOP (k) 6= 0. In view of commutativity of the diagram (17.1), jF (k) = a*jTOP (k), i.e. jF (k) 6= 0. Thus, in view of Lemma 22.2, M is not PL isomorphic to RP5. 22.3. Example. A homeomorphism h : S3 x Sn ! S3 x Sn, n > 3 which is homotopic to a PL isomorphism but is not concordant to a PL isomorphism. 68 YULI B. RUDYAK Take an arbitrary homeomorphism f : V ! S3 x Sn. Then jF (f) is trivial by Theorem 21.2. Thus, by 13.2, f is homotopic to a PL isomorphism. In particular, V is PL isomorphic to S3 x Sn. Now, we refine the situation and take a homeomorphism h : S3 x Sn ! S3 x Sn such that jTOP (h) 6= 0 2 TPL (S3 x Sn) = H3(S3 x Sn; Z=2) = Z=2. So, h is not concordant to the identity map, and therefore h is not concordant to a PL isomorphism. But, as we have already seen, h is homotopic to a PL isomorphism. Notice that the maps h and the identity map have the same domain while they are not concordant. So, this example serves also the Remark 3.2(3). 22.4. Example. A topological manifold which does not admit any PL structure. Already constructed F x T n, see 7.3. 22.5. Example. A topological manifold which is homeomorphic to a polyhedron but does not admit any PL structure. Let M be a closed topological 7-dimensional manifold, and let -(M) 2 H4(M; Z=2) be the Kirby-Siebenmann invariant described in 8.10. Let ffi : H4(M; Z=2) ! H5(M; Z) be the Bockstein homomorphism. Scharlemann [49 ] proved that if ffi-(M) = 0 and if a quadruple suspension over a certain 3-dimensional homology sphere is homeomorphic to S7, then M admits a simplicial triangulation. Cannon [6] proved that a double suspension over every 3- dimensional homology sphere is homeomorphic to S5. So, if -(M) 6= 0 while ffi-(M) = 0 then M is homeomorphic to a polyhedron but does not admits a PL structure. Take M = F x T 3. 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Max-Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany E-mail address: rudyak@mpim-bonn.mpg.de Mathematisches Institut Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany E-mail address: rudyak@mathi.uni-heidelberg.de