Topological rigidity for non-aspherical manifolds by M. Kreck and W. L"uck September 11, 2005 Abstract The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f :N ! M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N ! M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for Sk x Sd, for sphere bundles over aspherical closed manifolds of dimension 3 and for 3-manifolds with torsionfree fundamental groups. We show that this rigidity is inherited under connected sums in dimensions 5. We also classify manifolds of dimension 5 or 6 whose fundamental group is the one of a surface and whose second homotopy group is trivial. Key words: Topological rigidity, Borel Conjecture, classification of low- dimensional topological manifolds. Mathematics Subject Classification 2000: 57N99, 57R67. 0. Introduction and Statement of Results In this paper we study the question which non-aspherical oriented closed con- nected topological manifolds are topological rigid. Recall that the Borel Con- jecture predicts that every aspherical closed topological manifold is topologic* *al rigid in the sense that every homotopy equivalence of such manifolds is homo- topic to a homeomorphism. We focus on the following two problems which we will describe next. We say that two maps f, g :X ! Y of path-connected spaces induce the same map on the fundamental groups up to conjugation if for one (and hence all base points) x 2 X there exists a path w from f(x) to g(x) such that for the group isomorphism tw :ss1(Y, f(x)) ! ss1(Y, g(y)) which sends the class of a lo* *op v to the class of the loop w- * v * w we get ss1(g, x) = tw O ss1(f, x). Homoto* *pic maps induce the same map on the fundamental groups up to conjugation. 1 Convention 0.1. Manifold will always mean connected oriented closed topo- logical manifold unless stated explicitly differently. Definition 0.2 (Borel-manifold). A manifold M is called a Borel manifold if for any orientation preserving homotopy equivalence f :N ! M of manifolds there exists an orientation preserving homeomorphism h: N ! M such that f and h induce the same map on the fundamental groups up to conjugation. It is called a strong Borel manifold if every orientation preserving homotopy equivalence f :N ! M of manifolds is homotopic to a homeomorphism h: N ! M. Remark 0.3 (Relation to the Borel Conjecture). If M is aspherical, two homotopy equivalences f, g :N ! M are homotopic if and only if they induce the same map on the fundamental groups up to conjugation. Hence an aspher- ical manifold M is a Borel manifold if and only if every homotopy equivalence f :N ! M of manifolds is homotopic to a homeomorphism. This is the precise statement of the Borel Conjecture for aspherical manifolds. Hence the Borel Conjecture can be rephrased as the statement that every aspherical manifold M is a Borel manifold, or equivalently, is a strong Borel manifold. More infor- mation on the Borel Conjecture can be found for instance in [8], [9], [10], [11* *], [13], [14], [15], [23], [25], [26]. Remark 0.4 (Relation to the Poincar'e Conjecture). The statement that Sn is a strong Borel manifold is equivalent to the Poincar'e Conjecture that every manifold which is homotopy equivalent to a sphere Sn is homeomorphic to Sn. This follows from the fact that there are exactly two homotopy classes of self-homotopy equivalences Sn ! Sn which both have homeomorphisms as representatives. In particular Sn is a Borel manifold if and only if it is a st* *rong Borel manifold. Problem 0.5 (Classification of Borel manifolds). Which manifolds are Borel manifolds? In the light of both the Borel Conjecture and the Poincar'e Conjecture, it is natural to consider the class of manifolds M, whose universal covering fM is homotopy-equivalent to a wedge of k-spheres Sk for some 2 k 1. We call such a manifold a generalized topological space forms. If k 6= 1, this condition is equivalent to saying that the reduced integral homology vanishes except in dimension k and it is a direct sum of copies of Z in dimension k. If k = 1, then this condition is equivalent to saying that M is an aspherical manifold. A simp* *ly- connected generalized topological space form is the same as a homotopy sphere. More generally, a generalized topological space form with finite fundamental group, is the same as a spherical topological space form. If G acts freely and cocompactly and properly discontinuously on Sk x Rm-k , then M = Sk x Rm-k =G is a generalized topological space form. If M and N are m-dimensional aspherical manifolds, then M#N is a generalized topological space form. If M is aspherical, then for each k the manifold M xSk is a generalized space form, * *or more generally, all Sk-bundles over M with k > 1 are generalized space forms. 2 Most results in this paper concern generalized space forms M. One can try to attack the question whether M is Borel by computing its structure set Stop(M). It consists of equivalence classes of orientation preserving homotopy equivalen* *ces N ! M with a manifold N as source, where two such homotopy equivalences f0: N0 ! M and f1: N1 ! M are equivalent if there exists a homeomorphism g :N0 ! N1 with f1 O g ' f0. The group ho-autss(M) of homotopy classes of self equivalences inducing the identity on ss1 up to conjugation acts on this set by composition. A manifold M is strongly Borel if and only if Stop(M) consists of one element. A manifold M is Borel if and only if ho-autss(M) acts transitively on Stop(M) . In general it is very hard to compute the structure set. But if the Farrell-Jones Conjecture for ss1(M) holds, then one often can do this. More precisely we mean the Farrell-Jones Conjecture for K- and L- theory for the group G. In all relevant cases G will be torsionfree. Hence this phrase will mean that Wh (G) and eKn(ZG) vanish for n 0 and that the assembly map Hn(BG; L) ! Ln(ZG) is bijective for all n 2 Z, where L is the (non-connective periodic) L-theory spectrum and Ln(ZG) is the n-th quadratic L-group of ZG. (We can ignore the decoration by the Rothenberg sequences and the assumption that Wh (G) and eKn(ZG) vanish for n 0.) More information about the Farrell-Jones Conjecture can be found for instance in [12], [23] and [26]. For example the Farrell-Jones Conjecture holds for Z and the fundamen- tal group of surfaces of genus 1. Combining this with the construction of certain self-equivalences, we obtain the following result concerning generalized topological space forms. Theorem 0.6 (Sphere bundles over surfaces). Let K be S1 or a 2-dimen- sional manifold different from S2. Let Sd ! E ! K be a fiber bundle over K such that E is orientable and d 3. Then E is a Borel manifold. It is a strong Borel manifold if and only if K = S1. This gives examples of Borel manifolds in all dimensions > 3, which are neither homotopy spheres nor aspherical. In dimension 3 the existence of Borel manifolds is related to the Poincar'e Conjecture and to Thurston's Geometrization Conjecture. Results of Wald- hausen and Turaev imply: Theorem 0.7 (Dimension 3). Suppose that Thurston's Geometrization Con- jecture for irreducible 3-manifolds with infinite fundamental group and the 3- dimensional Poincar'e Conjecture are true. Then every 3-manifold with torsion- free fundamental group is a strong Borel manifold. Using the Kurosh theorem and the prime decomposition of 3-manifolds one can even show that if the assumptions of this theorem are fulfilled then the fundamental group determines the homeomorphism type, a close analogy be- tween surfaces and 3-manifolds (although the latter case is of course much more complicated). 3 Recently Perelman has announced a proof of Thurston's Geometrization Conjecture but details are still checked by the experts. Given the analogy between the classification of surfaces and 3-manifolds with torsionfree fundamental group, it is natural to study in analogy to sphere bundles over surfaces sphere bundles over 3-manifolds. Our result here is: Theorem 0.8 (Sphere bundles over 3-manifolds). Let K be an aspherical 3-dimensional manifold. Suppose that the~Farrell-Jones Conjecture for K- and L-theory holds for ss1(K). Let Sd ! E -=! K be a fiber bundle over K with orientable E such that d 4 or such that d = 2, 3 and there is a map i: K ! E with p O i ' idK. Then (a)E is strongly Borel if and only if H1(K; Z=2) = 0; (b)If d = 3 mod 4 and d 7, then K x Sd is Borel; (c)If d = 0 mod 4 and d 8 and H1(K; Z=2) 6= 0, then K xSd is not Borel. The following result shows that if the fundamental groups of two d-dimen- sional Borel manifolds M and N contain no 2-torsion, then the connected sum M#N is a Borel manifold. Here we assume that d > 4. Theorem 0.9 (Connected sums). Let M and N be manifolds of the same dimension n 5 such that neither ss1(M) nor ss1(N) contains elements of or- der 2. If both M and N are (strongly) Borel, then the same is true for their connected sum M#N. Remark 0.10. If M and N are aspherical Borel manifolds of dimension 6= 4 then M#N is a generalized topological space form, which is Borel. Combining the previous results, we have found infinitely many non-aspherical and non-simply connected Borel manifolds in each dimension 6= 4. The proof is in all cases based on a determination of the structure set and by providing enough self equivalences following the scheme described above. The main reason why these proofs do not work at present in dimension 4 is that for the fundamental groups under consideration it is not known whether they are good in the sense of Freedman. For this reason one has to look at other classes of 4-manifolds where also the determination of the structure set is kno* *wn but it is not clear how to construct enough self equivalences to guarantee a tr* *an- sitive action. However, if ss1(M) is cyclic and M is a spin manifold, one can u* *se other methods to show that the homotopy type determines the homeomorphism type (respecting the identification of fundamental groups). Previously known Borel 4 manifolds are flat 4-manifolds, where the Borel Conjecture was proven. Theorem 0.11 (Dimension 4). (a) Let M be a 4-manifold with Spin struc- ture such that its fundamental group is finite cyclic. Then M is Borel. If M is simply connected and Borel, then it has a Spin structure. 4 (b)Let N be a flat smooth Riemannian 4-manifold or be S1 x S3. Then N is strongly Borel. If M is a simply connected 4-manifold with Spin-structure, then M#N is Borel. By Theorem 0.11(b) we have provided infinitely many non-aspherical and non-simply connected Borel manifolds in dimension 4. Except for S1 x S3 and the flat manifolds, these manifolds are not generalized topological space forms. We have seen that under some mild restrictions the connected sum of two Borel manifolds is a Borel manifold. It is natural to ask the corresponding question for the cartesian product of Borel manifolds. If M and N are aspherica* *l, then M x N is aspherical and so Borel, if the Borel Conjecture holds. But if the manifolds are not aspherical Borel manifolds, the picture becomes complicated. An interesting test case is provided by the product of two spheres, where we give a complete answer in terms of the unstable Arf invariant. Let frk,k+dbe the bordism set of smooth k-dimensional manifolds M with an embedding i: M ! Rk+d together with an (unstable) framing of the normal bundle (i). If d > k, this is the same as the bordism group frkof stably fram* *ed smooth k-dimensional manifolds since any k-dimensional smooth manifold M admits an embedding into Rk+d and a stable framing on (M, Rk+d) is the same as an unstable framing for d > k. The Arf invariant Arf(M) 2 Z=2 of a stably framed manifold M is the Arf invariant of the surgery problem associated to any degree one map M ! Sdim(M) with the obvious bundle data coming from the stable framing. It induces a homomorphism of abelian groups Arfk : frk! Z=2. (0.12) If g :Sk xSd ! Sk is a map, define its Arf invariant Arf(g) 2 Z=2 to be the Arf invariant of the stably framed manifold __g-1({o}) for any map __g:Sk x Sd ! Sk which_is homotopic to g and transverse to {o} Sd. Here the stable framing of g-1 ({o}) is given by the standard stable framing of the normal bundle of SkxSd and the trivialization of the normal bundle of __g-1({o}) Sk x Sd coming from transversality. Theorem 0.13 (Products of two spheres). Consider k, d 2 Z with k, d 1. Then: (a)Suppose that k + d 6= 3. Then Sk x Sd is a strong Borel manifold if and only if both k and d are odd; (b)For d 6= 2 the manifolds S1 x Sd is Borel, and S1 x S2 is strongly Borel if and only if the 3-dimensional Poincar'e Conjecture is true; (c)The manifold S2 x S2 is Borel but not strongly Borel; (d)Suppose k, d > 1 and k + d 4. Then the manifold Sk x Sd is Borel if and only if the following conditions are satisfied: (i) Neither k nor d is divisible by 4; 5 (ii)If k = 2 mod 4, then there is a map gk: Sk x Sd ! Sk such that its Arf invariant Arf(gk) is non-trivial and its restriction to Sk x {o} * *is an orientation preserving homotopy equivalence Sk x {o} ! Sk; (iii)If d = 2 mod 4, then there is a map gd: Sk x Sd ! Sd such that its Arf invariant Arf(gd) is non-trivial and its restriction to {o} x Sd * *is an orientation preserving homotopy equivalence {o} x Sd ! Sd. Remark 0.14 (Relation to the Arf-invariant-one-problem). The condi- tion (dii)appearing in Theorem 0.13 (d)implies that the (stable) Arf invariant homomorphism Arfk of (0.12)is surjective. The problem whether Arfk is sur- jective is the famous Arf-invariant-one-problem (see [3]). The map frk! Z=2 is known to be trivial unless 2k + 2 is of the shape 2l for some l 2 Z (see [3]* *). Hence a necessary condition for Sk x Sd to be Borel is that k is odd or that 2k + 2 is of the shape 2l for some l 2 Z and analogously for d. Suppose that the unstable Arf-invariant-map Arfk,k+d: frk,k+d! Z=2. (0.15) is surjective. Then condition (dii)is automatically satisfied by the following argument. Choose a framed manifold M Sk+d with Arfk,k+d([M]) = 1. By the Pontrjagin-Thom construction we obtain a map g0k:Sk+d ! Sk which is transversal to {o} Sk and satisfies g-1({o}) = M. Now define the desired map gk to be the composition of g0kwith an appropriate collapse map Sk x Sd ! Sk+d. So the surjectivity of the unstable Arf invariant map (0.15)implies condition (dii). Of course the surjectivity of (0.15)is in general a stronger condition than the surjectivity of (0.12). The unstable Arf invariant map (0.15) is known to be surjective if k = 2 and d 1. Hence Sk x Sd is Borel if k = 2 and d 2 or if k 2 and d = 2. Now we discuss the following question. How complicated can the homotopy type of Borel manifolds be? In the situation of the Borel and Poincar'e Conjec- tures the homotopy type is determined by the fundamental group and - in the case of homotopy spheres - by the homology groups. Most of our results con- cerned generalized topological space forms (or connected sums of these), spaces which are "close neighbours" of aspherical manifolds resp. homotopy spheres. Besides the products of spheres we have given results concerning other classes of manifolds only in dimension 4. If we concentrate on manifolds with torsion- free fundamental groups (the lens spaces show that even very simple manifolds are in general not Borel for cyclic fundamental groups (see for instance [6, x * *29 and x 31], [25, Section 2.4])), these results in dimension 4 for non-aspherical manifolds concern manifolds with fundamental group Z. Here the fundamental group and the intersection form on ss2, which is a homotopy invariant, deter- mines the homeomorphism type. The following classes of manifolds are natural generalizations of these manifolds. Problem 0.16 (Classification of certain low-dimensional manifolds). Classify up to orientation preserving homotopy equivalence, homeomorphism (or 6 diffeomorphism in the smooth case) all manifolds in dimension 1 k < n 6 for which ss = ss1(M) is non-trivial and is isomorphic to ss1(K) for a manifold K of dimension k 2 with ss1(K) 6= {1} and whose second homotopy group ss2(M) vanishes. Remark 0.17 (Simply-connected case). We have excluded in Problem 0.16 the case ss1(K) = {1} since then a complete answer to this problem is already known. Namely, if M is a 2-connected n-dimensional manifold, then n 3, it is homotopy equivalent to S3 for n = 3 and it is homeomorphic to Sn for n = 4, 5. If n = 6 and M is a 2-connected smooth manifold, then its oriented homotopy type and its oriented diffeomorphism are determined by the intersection from on H3(M) (see Wall [36]). This also applies to the topological category by the work of Kirby-Siebenmann [20]. The following results give an almost complete answer to this problem. Theorem 0.18. (Manifolds appearing in Problem 0.16 of dimension 5). Let M and K be as in Problem 0.16. Let f :M ! K = Bss be the classifying map for the ss-covering fM ! M. Suppose n 5. Then (a)Both the oriented homotopy type and the oriented homeomorphism type of M are determined by its second Stiefel-Whitney class. Namely, there is precisely one fiber bundle S3 ! E p-!K with structure group SO(4) whose second Stiefel-Whitney class agrees with the second~Stiefel-Whitney class of M under the isomorphism H2(f; Z=2): H2(K; Z=2) -=!H2(M; Z=2). There exists an orientation preserving homeomorphism g :M ! E such that p O g and f are homotopic. In particular the vanishing of the second Stiefel-Whitney class of M im- plies that there is an orientation preserving homomorphism g :M ! S3 x Bss such that prBssOg and f are homotopic; (b)The manifold M is never a strong Borel manifold but is always a Borel manifold. Theorem 0.19. (Manifolds appearing in Problem 0.16 of dimension 6). Let M and K be as in Problem 0.16. Suppose that n = 6. (a)The manifold M is never strongly Borel but always Borel; (b)Suppose that w2(M) = 0, or, equivalently, that M admits a Spin-structure. Then both the oriented homotopy type and oriented homeomorphism type of M is determined by the Zss-isomorphism class of the intersection form on H3(fM). Given a finitely generated stably free Zss-module together with a non- degenerate skew-symmetric Zss-form on it, it can be realized as the inter- section form of a 6-dimensional manifold having the properties described in Problem 0.16. 7 One expects that Borel manifolds are an exception. The following results which give necessary conditions for M to be Borel support this intuition. Theorem 0.20 (A necessary condition for sphere bundles over aspher- ical manifolds). Let Sd ! E -p!K be a fiber bundle such that E and K are manifolds and K is aspherical. Assume that there is a map i: K ! E with p O i ' idK. Suppose that d is odd and d 3 or suppose that d is even and k d - 2 for k = dim(K). Then a necessary condition for E to be a Borel manifold is that Hk-4i(K; Q) vanishes for all i 2 Z, i 1. Theorem 0.21 (A necessary condition for being a Borel manifold). Let M be a Borel manifold of dimension n with fundamental group ss. Let ho-autss(M) be the set of homotopy classes of orientation preserving self-homo- topy equivalences f :M ! M which induce up to conjugation the identity on the fundamental group. Let L(M)i 2 H4i(M; Q) be the i-th component of the L-class L(M) of M. L Then the subset of i2Z,i 1Hn-4i(M; Q) S := {f*(L(M) \ [M]) - L(M) \ [M] | [f] 2 ho-autss(M)} is an abelian subgroup and the Q-submodule generated by S must contain the kernel of the map induced by the classifying map c: M ! Bss M M c*: Hn-4i(M; Q) ! Hn-4i(Bss; Q). i2Z,i 1 i2Z,i 1 In particular for every i 1 with L(M)i = 0 the map c*: Hn-4i(M; Q) ! Hn-4i(Bss; Q) must be injective. We have mentioned that lens spaces are in general not Borel, an indication that torsion in the fundamental group makes Borel less likely. The following result shows that in dimension 4k + 3 torsion excludes Borel. Theorem 0.22 (Chang-Weinberger[5]). Let M4k+3 be a closed oriented manifold for k 1 whose fundamental group has torsion. Then there are in- finitely many pairwise not homeomorphic smooth manifolds which are homotopy equivalent to M (and even simply and tangentially homotopy equivalent to M) but not homeomorphic to M. Another natural class of manifolds are the homology spheres where surgery gives a necessary and sufficient condition for being strongly Borel: Theorem 0.23 (Homology spheres). Let M be a n-dimensional manifold of dimension n 5 with fundamental group ss = ss1(M). (a)Let M be an integral homology sphere. Then M is strongly Borel if and only if the inclusion j :Z ! Zss induces an isomorphism ~= s Lsn+1(j): Lsn+1(Z) -! Ln+1(Zss) on the simple L-groups Lsn+1; 8 (b)Suppose that M is a rational homology sphere and Borel. Suppose that ss satisfies the Novikov Conjecture in dimension (n + 1), i.e. the assembly map Hn+1(Bss; L) ! Ln+1(Zss) is rationally injective. Then Hn+1-4i(Bss; Q) = 0 for i 1 and n + 1 - 4i 6= 0. The paper is organized as follows: 1. On the Structure Set of Certain Topological Manifolds 2. Constructing Self-homotopy Equivalences 3 Sphere Bundles 4 Connected Sums 5 Dimension 3 6 Dimension 4 7. Products of Two Spheres 8. On the Homotopy Type of Certain Low-Dimensional Manifolds 9. On the Classification of Certain Low-Dimensional Manifolds 10. A Necessary Condition for Being a Borel Manifold 11 Integral Homology Spheres References We thank Andrew Ranicki for fruitful discussions about this paper. 1. On the Structure Set of Certain Topological Manifolds We begin with a fundamental criterion for Borel manifolds which follows directly from the definitions. Theorem 1.1 (Surgery criterion for Borel manifolds). A manifold M is a Borel manifold if and only if the action of the group of homotopy classes of se* *lf- homotopy equivalence M ! M which induce the identity on the fundamental group up to conjugation on the topological structure set Stop(M) is transitive, and M is a strong Borel manifold if and only Stop(M) consists of one element. Now we determine the topological structure set for certain manifolds. In the sequel we denote by L<1> the 1-connected cover of the quadratic L-theory spectrum L and by u: L<1> ! L the canonical map. We get ae oe 8) = Lq(Z)0 ,,ifqifq1 0= : Z=2 , ifq 2, q = 2 mod 4 0 otherwise Theorem 1.2. Let M be an n-dimensional manifold for n 5. Let K be an aspherical k-dimensional manifold with fundamental group ss. Suppose that the Farrell-Jones Conjecture for algebraic K- and L-theory holds for ss. Let 9 f :M ! K be a 2-connected map. Suppose that we can choose a map i: K ! M such that f O i is homotopic to the identity. (a)The homomorphism Hm (idK; u): Hm (K; L<1>) ! Hm (K; L) is bijective for m k + 1 and injective for k = m; (b)The exact topological surgery sequence for M yields the short split-exact sequence 0 ! Stop(M) oen--!Hn(M; L<1>) Hn(f;L<1>)-------!Hn(K; L<1>) ! 0. In particular we get an isomorphism __oe top ~= n :S (M) -! Hn(i: K ! M; L<1>). Proof. (a)Let E be the homotopy fiber of u. Hence we have a fibration of spectra E ! L<1> u-!L which induces a long exact sequence . .!.Hm+1 (K; L<1>) ! Hm+1 (K; L) ! Hm (K; E) ! Hm (K; L<1>) ! Hm (K; L) ! . ... Since ssq(E) = 0 for q 0, an easy spectral sequence argument shows that Hm (K; E) = 0 for m k. Hence the map Hm (idK; u): Hm (K; L<1>) ! Hm (K; L) is bijective for m k + 1 and injective for k = m. (b) There is an exact sequence of abelian groups called algebraic surgery exact sequence [32, Definition 15.19 on page 169]. . .o.en+1---!Hn+1(X; L<1>) An+1---!Ln+1(Zss1(X)) @n+1---! Stop(X) oen--!Hn(X; L<1>) An--!Ln(Zss1(X)) @n-!. . .(1.3) which is defined for every simplicial connected complex X and natural in X. It agrees for X a n-dimensional manifold for n 5 with the Sullivan-Wall geometric exact surgery sequence [32, Theorem 18.5 on page 198]. Notice that by assumption Wh i(ss1(K)) vanishes for i 1 so that we can ignore any decorations 10 in the sequel. The following diagram commutes for all m Hm (M; L<1>) -Am---!Lm (Zss) ? ? Hm (f;L<1>)?y id?y~= Hm (K; L<1>) -Am---!Lm (Zss) ? ? Hm (idK;u)?y id?y~= Hm (K; L) -Am---!~=Lm (Zss) The existence of the map i with f O i ' idK ensures that Hm (f; L<1>) is surjec- tive. Now the claim follows from assertion (a)and the exact sequence_(1.3)for X = M. |__| Theorem 1.4. Let K be an aspherical k-dimensional manifold with fundamen- tal group ss. Suppose that the Farrell-Jones Conjecture for algebraic K- and L-theory holds for ss. Let d be an integer with d 2 and d + k 5. Consider a fiber bundle Sd ! E p-!K such that E is oriented. Suppose that there exists a map i: K ! E with p O i = id. Then i: K ! E is an embedding of topological manifolds and we obtain an isomorphism of abelian groups ~= Stop(E) -! Hk(K; L<1>). It sends under the identification of Hk(K; L<1>) with the set N (K) ~=[K, G=T O* *P ] of normal surgery problems with K as reference space an element f :M ! E to the following surgery problem: By changing f up to homotopy we can arrange that f is transverse to i: K ! E. Let g :N = f-1 (i(K)) ! K be the map of manifolds of degree 1 induced by f and i-1 :i(K) ! K. By transversality we obtain a bundle map __g:_(N, M) ! (i) covering g. Choose a vector bundle , ! M and a bundle map f: (M) ! , covering f :M ! E. Then g is covered by the bundle map __g __f| * N : (N) = (N, M) (M)|N ! (i) i ,, and these data give the desired surgery problem with target K. Proof. See [37, Chapter 11], [32, pages 257-260]. |___| Theorem 1.5. Let K be an aspherical k-dimensional manifold with fundamen- tal group ss. Consider a fiber bundle Sd ! E -p!K with d 1. Suppose that the Farrell-Jones Conjecture for algebraic K- and L-theory holds for ss. Then: (a)If k = 2 and d 3, then Stop(E) ~=L2(Z) ~=Z=2; 11 (b)If k = 3 and d 4 or if k = 3, d = 2, 3 and there is a map i: K ! E with p O i = idK, then Stop(E) ~=H1(K; L2(Z)) ~=H1(K; Z=2); (c)If k = 1 and d 3, then Stop(E) = 0; Proof. We first prove the claim in the case k + d 5. Since K is k-dimensional and Sd is (d - 1)-connected, we can find a map i: K ! E such that p O i is homotopic to idK provided d > k. By the homotopy lifting property we can arrange that p O i is idK. By assumption such a map i exists also in the remaining case k = 3 and d = 2, 3. Now the claim follows from Theorem 1.4 and an easy computation with the Atiyah-Hirzebruch spectral sequence. Thus we have proven assertions (a)and (b)and for assertion (c)only the case k = 1 and d = 3 remains open. Then ss1(E) is Z which is a good fundamental group in the sense of Freedman [16]. Hence topological surgery works also in this dimension 4 and the same argument which gives the claim for k = 1 and k + d 5 works_ also for k = 1 and k + d = 4. |__| 2. Constructing Self-homotopy Equivalences In this section we describe a certain construction of self-homotopy equivalence* *s. It will be used to show that the action of the group of orientation preserving homotopy equivalences E ! E which induce the identity on the fundamental group up to conjugation on Stop(E) is transitive for certain manifolds E. Suppose we are given the following data: o Let K and E be manifolds with dim(K) = k and dim(E) = k + d for k, d 2; o An embedding iK :K ! E; o A map OE: Sd ! E which is transversal to iK :K ! E and the intersection of the images im(OE) and im(iK ) consists of precisely one point e0 2 E; o Let M be a manifold together with an embedding iM :M ! Sk+d and a framing of the normal bundle ~(iM ); o k + d 5 or E is simply connected. Fix an embedding jK :Dk ! K with iK O jK (0) = e0. Since Dk is con- tractible, we can choose a disk bundle map __j Dk x Dd ----!K D (iK ) ? ? pr?y ?y Dk ----!j K K 12 which is fiberwise a homeomorphism. Choose an embedding jM :Dk ! M. Since Dk is contractible, we can choose a disk bundle map ___j Dk x Dd - ---!MD (M, Sk+d) ? ? pr?y ?y Dk - ---!j M M which is fiberwise_a homeomorphism. Using a tubular neighborhood, we_will_ also regard jK as an embedding Dk x Dd ! E extending jK :Dk ! E and jM as an embedding Dk x Dd ! Sk+d extending jM :Dk ! Sk+d. In the sequel let jSd: Dd ! Sd be the obvious embedding given by the lower hemisphere. Recall that OE and iK are transversal to one another and the intersection of their images consists of the point e0. We can assume without loss of_generality that OE O jSd(0) = e0 = jK (0) holds. Now we can thicken OE * *to a map OE:Dk x Sd ! E such that the composite idxjSd k d _OE Dk x Dd -----! D x S -! E ___ agrees_with the embedding jK: Dk x Dd ! E and the intersection_of_the image of OEand of D (iK ) considered as subset of E is the image of jK: Dk xDd ! E. The Pontrjagin-Thom construction applied to iM : M ! Sd-k together with the given framing on M yields a map PT :Sk+d ! Sd such that PT is transversal to iSd(0) 2 Sd, the preimage of iSd(0) 2 Sd is just M and the bundle map given by transversality (iM ) ! ({iSd(0)} Sd) is just the given framing._ We can arrange by shrinking Dd that the composition of PT with jM :Dk x Dd ! Sk+d agrees with the composite jSd d Dk x Dd pr-!Dd --! S . Choose_a map c: Sk+d ! Dk such that its composite with the embedding jM :Dk x Dd ! Sk+d is the projection pr:Dk x Dd ! Dk, it is transversal to 0 2 Dk and the preimage of 0 is an embedded Sd. Such a map can be constructed from the map Sk+d = Sd * Sk-1 ! {o} * Sk-1 = Dk. Now consider the composite _OE ff: Sk+d cxPT----!Dk x Sd -! E. ___ Its composition_with the embedding jM :Dk x Dd ! Sk+d agrees with the embedding jK :Dk x Dd ! E. It is transverse to iK :K ! E such that the preimage of both iK (K) and of iK O jK (Dk) is M and the bundle map given by transversality from (M, Sk+d) ! i*K (i) is compatible up to isotopy with the given framing of (M; Sk+d) and some framing of the bundle i*K (i) over Dk. In the sequel_we_consider the connected_sum E#Sn+k with respect to the two embeddings jM :Dk x Dd ! Sk+d and jK: Dk x Dd ! E. By construction 13 the identity id:E ! E and the map ff: Sk+d ! E fit together and yield a map id#ff: E#Sk+d ! E. We claim that this map is a homotopy_equivalence. Choose_a point x 2 E which not contained in the images of jK: DkxDd ! E and of OE:DkxSd ! E. Then the preimage of x under id#ff is x and the map id#ff induces the identity on a neighborhood_of x. This implies that id#ff has degree one. The inclusions of E - im(jK ) into both E and E#Sk+d_are_(k + d - 1)-connected. Since id#ff induces the identity on E - im(jK ), the homomorphisms ssj(id#ff) are bijective for j k + d - 2. By assumption we have k + d 5 or we have k + d = 4 and ss1(E) = {1}. Now we conclude from Poincar'e duality that id#ff: E#Sk+d ! E is a homotopy equivalence. Obviously we can find a homeomorphism fi :E ! E#Sk+d_such_that the composite ff O fi is the identity outside the image of jK: Dk x Dd ! E. The map id#ff: E#Sk+d ! E is transversal to iK :K ! E. The preimage of iK is the connected sum K#M, which is taken with respect to the embeddings jM :Dk ! M and jK :Dk ! K. This map has degree one and is covered by bundle data due to transversality. The resulting normal map with target E agrees with the connected sum of the normal map id:K ! K and the normal map M ! Sk given by the collapse map of degree one and the bundle data coming from the given framing on M. Now additionally suppose that the given map OE: Sd ! E is an embed- ding. (It is automatically a local embedding near the intersection point with K_by transversality but a priori not a global embedding). Then also the map OE:Dk x Sd ! E can be chosen to be an embedding. It is not hard to check that the map id#ff: E#Sk+d ! E is transversal to OE and the corresponding~ surgery problem is given by a homeomorphism (id#ff)-1(ff(Sd)) =-!Sd covered by an isomorphism of the normal bundles. In particular this surgery problem represents the trivial element in N (Sd). Next we explain the maps in the following diagram frk,k+d-o---!Stop(E) ? ? a?y ?yb (2.1) N (Sk) ----!csN (Sk) K Recall that we denote by frk,dthe set of bordism classes of k-dimensional man- ifolds M together with an embedding M Rk+d and an (unstable) framing of its normal bundle (M Rk+d). The map o is given by the construction above which assigns to [M] 2 frk,k+dthe element in the structure set given by ff: E#Sk+d ! E. The map a sends a framed k-dimensional manifold M to the normal map given by the collapse map c: M ! Sk covered by stable bundle map from (M) to the trivial bundle over Sk given by the framing. The map csK is given by taking the connected sum of a surgery problem with target Sk with the one given by the identity idK:K ! K. The map b sends the class of a homotopy equivalence f :M ! E to the surgery problem with underlying map 14 f-1 (iK (K)) ! K after making f transversal to iK :K ! E. We have shown Theorem 2.2. (a) The diagram 2.1 commutes; (b)Each element in the image of o is represented by a self-homotopy equiva- lence E ! E which induces for some embedded disk Dn+k E the identity id:E - Dn+k ! E - Dn+k and in particular induces up to conjugation the identity on the fundamental groups; (c)Suppose additionally that the given map OE: Sd ! E is an embedding. Let b0:Stop(E) ! N (Sd) be the map given by making a homotopy equivalence f :M ! E transversal to OE. Then the composite 0 frk,k+do-!Stop(E) b-!N (Sd) is trivial; 3. Sphere Bundles In this section we prove Theorem 0.6 and Theorem 0.8. Proof. We begin with Theorem 0.6. It follows from Theorem 1.5, Theorem 2.2 (a) and Theorem 1.1 since the 2-dimensional torus with an appropriate framing yields an element fr2,2+dwhose image under the Arf invariant map fr2,2+d!_Z=2 is non-trivial. |__| Proof. Next we prove Theorem 0.8. Assertion (a)follows from Theorem 1.5 (b) and Theorem 1.1. To prove Assertion (b)and Assertion (c)we use the modified surgery theory from [21], to which we refer for notation. Let f :M ! K x Sd be a homotopy equivalence. The normal (d - 2) - type of M is K x BT OP , where BT OP is the (d-1)-connected cover of the classifying space of topological vector bundles BT OP . The reason is that the restriction of the normal bundle to the (d - 1)-skeleton, which is homotopy equivalent to K is determined by w2( (M)). But the Stiefel-Whitney classes are homotopy invariants, and so w2( (M)) = 0. If the restriction of the normal bundle to the (d - 1)-skeleton is trivial, we obtain a normal (d - 1)-smoothing of M in K x BT OP by choosing a map from M to K inducing f* on ss1 up to conjugation and by choosing a framing on the (d-1)-skeleton. Again we use that the (d-1)-skeleton is homotopy equivalent to K. Thus a framing on the restriction of the normal bundle to K, considered as the (d-1)-skeleton of M, together with f*: ss1(M) ! ss1(K) determine a normal (d - 1)-smoothing of M in K x BT OP . Since d 7, we conclude that (d - 1) is larger than half the dimension of K x Sd. Thus by [21, Theorem 3] and the remark before it and by [21, Theorem 4] and the remark before it the obstruction for replacing a normal bordism between K x Sd and M considered as elements of TOPd+3(K) by a s-cobordism takes 15 values in the Wall group Ld+4(ss1(K)). Here we recall that since we assume the Farrell-Jones Conjecture for K-theory we don't need to take the Whitehead torsion into account. From Theorem 1.2 we know that the L-group acts trivially on the structure set which implies that the obstruction in our situation vanish* *es since the action in our situation factors through the structure set. Summarizing these considerations we see that M is homeomorphic to K xSd inducing f* on ss1 up to conjugation if and only if after choosing the framing * *on K, considered as the (d - 1)-skeleton of M, appropriately the two manifolds are bordant in TOPd+3(K). The different choices of a lift of the normal Gauss map of K x Sd to BT OP correspond to the choice of framings on K and so K x Sd is null-bordant for all choices of lifts. This implies the follow* *ing criterion which we will use below: M is homeomorphic to K x Sd inducing f* on ss1 up to conjugation if and only if M for one (and then for all) topological framings on K, considered as the (d-1)- skeleton of M , together with the map to K given by f* on ss1 represents the zero class in TOPd+3(K). Next we determine the bordism groups TOPkfor d k d + 3. If N together with a lift of the normal Gauss map represents an element in this group we can make it highly connected by surgery. If k is odd we can even pass to a homotopy sphere which by the topological Poincar'e Conjecture is null-bordant. Thus TOPk= 0 for k odd (in our range). If k is even, the obstruction for passing to a homotopy sphere is the signature, if k = 0 mod 4 and the Arf invariant, if k = 2 mod 4. Since there exist almost parallelizable manifolds with signature resp. Arf invariant non-trivial, the bordism groups TOPk are Z, classified by the signature, if k = 0 mod 4 and Z=2 detected by the Arf invariant, if k = 2 mod 4. Now we are ready to prove assertion (b). The Atiyah-Hirzebruch spectral sequence implies for d = 3 mod 4 TOPd+3(K) ~= TOPd+3 H2(K; Z) ~=Z=2 H2(K; Z). Here the first component is determined by the Arf invariant. For the detection * *of second component we note that H2(K; Z) is isomorphic to Z, and so we can pass to Q-coefficients. But then the second component is determined by the higher signatures. Since the Farrell-Jones Conjecture implies the Novikov Conjecture, the higher signatures of M ! K agree with the ones of K x Sd ! K and hence vanish. For the Arf invariant we note that we one can interpret it as an Arf invariant of a quadratic from given by a Wu-orientation [3, Theorem 3.2] and so it is also a homotopy invariant. It vanishes for K x Sd ! K and hence also for M ! K. Thus M ! K is null-bordant in TOPd+3(K). Hence an application of the criterion above finishes the proof of assertion (b). To prove assertion (c)we again use the Atiyah-Hirzebruch spectral sequence to show TOPd+3(K) ~=H1(K; Z=2) H3(K; Z). We suppose that for d = 0 mod 4 and d 8 we have H1(K; Z=2) 6= 0. Let g :S1 ! K be a map representing a non-trivial element in H1(K; Z=2). We 16 consider the connected sum of K xSd and S1xA, where A is the framed highly connected topological manifolds with Arf invariant 1 (obtained from plumbing two disk bundles of the tangent bundle of the sphere). So we get normal degree one map idKxSd #(g O p1): (K x Sd)#(S1 x A) ! K x Sd. After composition with the projection K x Sd ! K we obtain an element in TOPd+3(K) = H1(K; Z=2) H3(K; Z). The element is non-trivial since its component H1(K; Z=2) is the element represented g. This follows from the product structure of the Atiyah-Hirzebruch spectra sequence. By the following sequence of surgeries we replace this map by a homotopy equivalence f :M ! K x Sd. Since ss1((K x Sd)#(S1 x A)) ~=ss1(K x Sd) * Z, we can do one 1-dimensional surgery to make the map an isomorphism with out changing the homology groups of the universal coverings up to the middle dimension. Since A is (d=2 + 1)-connected and Hd=2+1(A) is Z Z, the map is already highly connected and the kernel in the dimension (d=2 + 1) on the level of the universal coverings is a free Z[ss1(K x Sd)]-module of rank two. Thus by two further surgeries we can obtain the desired homotopy equivalence f :M ! K x Sd. It represents after composition with the projection K x Sd ! K a non-trivial class in TOPd+3(K). By the criterion mentioned above there is no homeomorphism from M to K x Sd inducing the same map on ss1 up to __ conjugation. Thus in this situation K x Sd is not Borel. |__| 4. Connected Sums In this section we prove Theorem 0.9. Proof. The main ingredient is the result of Cappell [4, Theorem 0.3] that un- der our assumptions for every homotopy equivalence f :N ! M1#M2 there are n-dimensional manifolds N0 and N1 together with orientation preserving homotopy equivalences f0: N0 ! M0 and f1: N1 ! M1 and an orientation preserving homeomorphism h: N0#N1 ! N such that f O h is homotopic to __ f0#f1. Now the claim follows from Theorem 1.1. |__| 5. Dimension 3 Next we prove Theorem 0.7. Proof. If M and N are prime Haken 3-manifolds, then every homotopy equiv- alence ss1(M) ! ss1(N) is homotopic to a homeomorphism. This is a result of Waldhausen (see for instance [19, Lemma 10.1 and Corollary 13.7]). Tu- raev [34] has extended this result to showing that a simple homotopy equiv- alence between 3-manifolds with torsionfree fundamental group is homotopic to a homeomorphism provided that Thurston's Geometrization Conjecture for irreducible 3-manifolds with infinite fundamental group and the 3-dimensional 17 Poincar'e Conjecture are true. This statement remains true if one replaces sim- ple homotopy equivalence by homotopy equivalence. This follows from the fact explained below that the Whitehead group of the fundamental group of a 3- manifold vanishes provided that Thurston's Geometrization Conjecture for ir- reducible 3-manifolds with infinite fundamental group is true. The vanishing of the Whitehead group is proved for Haken manifolds in Waldhausen [35, Theorem 19.3]. In order to prove it for prime 3-manifolds it remains to treat closed hyperbolic manifolds and closed Seifert manifolds. These cases are consequences of [12, Theorem 2.1 on page 263 and Proposition 2.3]. Now apply the fact that the Whitehead group of a free amalgamated product__ is the direct sum of the Whitehead groups. |__| Every 3-manifold is a generalized topological space form by the following argument. Suppose that ss1(M) is finite. Then the universal covering is a closed simply connected 3-manifold and hence homotopy equivalent to S3. Suppose that ss1(M) is infinite. Then the universal covering is a non-compact 3-manifold and hence homotopy equivalent to a 2-dimensional CW -complex. This implies that the second homology group of fM is a subgroup of a free abelian group, namely the second chain module of the cellular chain complex of fM, and hence free as abelian group and that all other homology groups of fM are trivial. 6. Dimension 4 Here we prove Theorem 0.11. Proof. (a)Hambleton-Kreck [18, Theorem C] show that the homeomorphism type (including a reference map M ! Bss1(M)) is determined for a 4-manifold with Spin structure and finite cyclic fundamental group by the intersection form on M. Hence such a manifold is Borel. Here we use the result taken from [17, 10.2B] that for a 4-manifold M with Spin structure its signature is divisible by 16 and its Kirby Siebenmann invari* *ant can be read off from the signature by ks(M) = sign(M)=8 mod 2 and hence is an invariant of its oriented homotopy type. Now suppose that M is simply connected and admits no Spin structure. Then there exists another simply connected 4-manifold *M with the same in- tersection form but different Kirby Siebenmann invariant (see [17, 10.1]. In particular M and *M are not homeomorphic but they are oriented homotopy equivalent by [27]. (b) The Borel Conjecture is true for a flat smooth Riemannian 4-manifold (see [9, page 263]. Hence such a manifold is strongly Borel. The manifold S1 x S3 is strongly Borel by Theorem 0.13 (a). The claim about the connected sum M#N follows from the version of The- orem 0.9 for dimension 4 whose proof goes through in dimension 4 since the fundamental group of M#N is good in the sense of Freedman [16]. 18 |___| 7. Products of Two Spheres In this section we give the proof of Theorem 0.13 Proof. (a)If k + d = 2, then the property strongly Borel follows from classical results. If k = 1 and d 3 or k 3 and d = 1, the claim follows from Theorem 1.5 and Theorem 1.1 since the Farrell-Jones Conjecture is known to be true for ss = Z. It remains to treat the case k, d 2 and k + d 4. Then Sk x Sd is simply-connected. The structure set can be computed by ~= a1 x a2: Stop(Sk x Sd)-! Lk(Z) Ld(Z), (7.1) where a1 and a2 respectively send the class of an orientation preserving ho- motopy equivalence f :M ! Sk x Sd to the surgery obstruction of the surgery problem with target Sk and Sd respectively which is obtained from f by making it transversal to Sk x {o} and {o} x Sd respectively. The proof of this claim is analogous to the one of Theorem 1.4 (see also [32, Example 20.4 on page 211]). Hence the structure set is trivial if and only if both k and d are odd. Now the claim follows from Theorem 1.1. Notice that we can apply surgery theory also to the 4-dimensional manifold S2 x S2 since its fundamental group is good in the sense of Freedman [16]. (b) This follows from Theorem 1.5 c and Theorem 1.1 in the case k 3. It remains to treat the case k = 2. Suppose that S1 x S2 is Borel. Let N be a homotopy 3-sphere. There exists an orientation preserving homotopy equivalence f :S1 x S2#N ! S1 x S2. Since S1xS2 is Borel by assumption, we can choose f to be a homeomorphism. By the uniqueness of the prime decomposition N must be homeomorphic to S3. Hence the 3-dimensional Poincar'e Conjecture is true. Now suppose that the 3-dimensional Poincar'e Conjecture is true. By the Sphere Theorem [19, Theorem 4.3], an irreducible (closed) 3-manifold is as- pherical if and only if it has infinite fundamental group. A prime 3-manifold is either irreducible or is homeomorphic to S1 x S2 [19, Lemma 3.13]. Hence for a prime 3-manifold M with infinite fundamental group the following state- ments are equivalent: i.) M is irreducible, ii.) M is aspheric) ! Hk+d({o}, L<1>) implies that the induced map on Stop(Sk x Sd) the identity. Hence f3 = f1O f2 is the desired map because of the formula which has been communicated to us by Andrew Ranicki [28] [f2 O f1] = (f2)*([f1]) + [f2] = [f1]) + [f2] and the fact the isomorphism ~= a1 x a2: Stop(Sk x Sd) -! Lk(Z) Ld(Z), is compatible with the abelian group structures. This finishes the proof_of Theorem 0.13. |__| 8. On the Homotopy Type of Certain Low-Di- mensional Manifolds We first compute the homology of the universal covering fM for a manifold appearing in Problem 0.16. Lemma 8.1. Let M and K be manifolds as described in Problem 0.16. Then n 4, if k = 1 and n 5, if k = 2. Moreover 8 < Z p = 0, n - k; Hp(fM; Z) ~=Zss : 0 1 p 2; 0 p n - 2, p 6= n - k, where Z carries the trivial ss-action. The Zss-module H3(fM) is finitely genera* *ted stably free if n = 6. Proof. Let f :M ! Bss be the classifying map for ss = ss1(M). In the sequel we identify ss = ss1(M) = ss1(K). The map is 3-connected because of ss2(M) = 0. Let ef:fM! eK be the induced ss-equivariant map on the universal coverings. The induced Zss-chain map C*(fe): C*(fM) ! C*(Ke) 21 is homological 3-connected by the Hurewicz Theorem. This implies that its mapping cone is chain homotopy equivalent to a Zss-chain complex whose chain modules are trivial in dimensions 3. Therefore we obtain isomorphisms ~= Hp(f; Z): Hp(M; Z) -! Hp(Bss; Z) and ~ Hp(fe; Z): Hp(fM; Z) -=!Hp(Ess; Z) for p 2 and the induced Zss-chain map Cn-*(fe): Cn-*(Ke) ! Cn-*(fM) induces isomorphism for n - p 2 ~= n-* Hp(Cn-*(fe)): Hp(Cn-*(Ke)) -! Hp(C (fM)). Obviously Hp(Ke; Z) = Hp(C*(Ke)) is Zss-isomorphic to the trivial Zss-module for p = 0 and is trivial for p 6= 0. Recall that we have the Poincar'e Zss-chain homotopy equivalences - \ [M]: Cn-*(fM) ! C*(fM); - \ [K]: Ck-*(Ke) ! C*(Ke). By Poincar'e duality applied to K we conclude that Hp(Cn-*(Ke)) is Zss-isomor- phic to Z for n - p = k and is trivial for n - p 6= k. Hence Hp(Cn-*(fM)) is Zs* *s- isomorphic to Z for n - p = k and is trivial for n - p 2 {0, 1, 2}, n - p 6= k.* * From Poincar'e duality applied to fM we conclude that Hp(fM; Z) is Zss-isomorphic to Z for p = n - k and is trivial for p 2 {n, n - 1, n - 2}, p 6= n - k. We already know that Hp(fM; Z) = Hp(Ke; Z) = 0 for p = 1, 2. This implies n - k 62 {1, 2} since Z is not trivial. Hence n 4 if k = 1 and n 5 if k = 2. It remains to show that the Zss-module H3(fM) is free if n = 6. Let cone(f*) be the mapping cone of C*(fe): C*(fM) ! C*(Ke). The composition C*(fe) O (- \ [M]) O C6-*(fe): C6-*(Ke) ! C*(Ke) is obviously zero. Hence the Zss-chain map (- \ [M]) O C6-*(fe): C6-*(Ke) ! C*(fM) induces in the obvious way a Zss-chain map C6-*(Ke) ! -1 cone(f*). Let D* be its mapping cone. It inherits from the structure of a symmetric 6-dimensional Poincar'e chain complex on C*(fM) the structure of a symmetric Poincar'e chain complex on D* with Poincar'e dimension 6. This follows from [29, Proposition 4.1 on page 141]. The homology of D* is zero in dimensions 22 different from 3 and H3(D*) is Zss-isomorphic to H3(fM). Let E* be the Zss- subchain complex of D* for which Ek = 0 for k 2, E3 = ker(d3: D3 ! D2) and Ek = Dk for k 4. Since 0 ! ker(d3) ! D3 d3-!D2 d2-!D1 d1-!D0 ! D-1 ! 0 is exact, E* is a finitely generated free Zss-chain complex. The inclusion i: E* ** ! D* is a homology equivalence and hence a Zss-chain homotopy equivalence. In particular we can pullback the structure of a symmetric Poincar'e chain complex of Poincar'e dimension 6 on D* to E*. Hence E6-* is a Zss-chain complex which is concentrated in dimensions -1, 0, 1, 2 and 3, whose homology is zero in dimensions different from 3 and for which H3(E6-*) is Zss-isomorphic to H3(fM). This implies that there is an exact sequence of Zss-modules 0 ! H3(fM) ! E*3! E*4! E*5! E*6! E*7! 0. Hence H3(fM) is a finitely generated stably free Zss-module. This finishes_the proof of Lemma 8.1. |__| Next we determine the homotopy type of M in the case n = 5. Theorem 8.2. Let M and K be as in Problem 0.16. Let f :M ! K = Bss be the classifying map for the ss-covering fM ! M. Suppose n 5. Then (a)Let k = 2. The homotopy type of M is determined by its second Stiefel- Whitney class. Namely, there is precisely one fiber bundle S3 ! E -p! K with structure group SO(4) whose second Stiefel-Whitney class agrees ~= under the isomorphism H2(f; Z=2): H2(K; Z=2) -! H2(M; Z=2) with the second Stiefel-Whitney class of M. There exists a homotopy equivalence g :M ! E such that p O g and f are homotopic; (b)Suppose that k = 2 and the second Stiefel-Whitney class of M vanishes, or that k = 1. Then there is a homotopy equivalence g :M ! Sn-k x Bss such that prBssOg and f are homotopic. Proof. Obviously the second assertion is a special case of the first one. The f* *irst one is proven as follows. We conclude from Lemma 8.1 that for n 5 the universal covering fM has the same homology as Sn-k and that ss acts trivially on the homology of fM. Recall that K is a model for Bss. There is a fibration F ! E -p!K together with a homotopy equivalence g :E ! M such that f O g ' p and F is homotopy equivalent to fM. Hence the fibration p: E ! K has Sn-k as fiber. Since the ss-action on the homology of fM is trivial, this spherical fibration is orienta* *ble, i.e. the fiber transport of this fibration ss = ss1(K) ! [F, F ] is trivial. Such fibrations over K are classified by maps u: K ! BSG(n - k), where BSG(n - k) is the classifying space of the monoid SG(n - k) of orientation preserving self-homotopy equivalences u: Sn-k ! Sn-k . We have n - k 3. 23 The space SG(n - k) is connected and hence the space BSG(n - k) is simply connected. Suppose that k = 1. Each map S1 ! BSG(n - k) is trivial up to homotopy and the claim follows. Now suppose k = 2. Then n 5 by Lemma 8.1 Hence n = 5 because of the assumption n 5. There is an obvious commutative diagram SO(3) ----! SO(4) ----! S3 ? ? ? J0?y J?y id?y 3S3 ----! SG(3) ----! S3 where the horizontal maps are fibrations and J comes from the obvious action of SO(4) on S3. The following diagram commutes ~= ss1(SO(3)) ----! ss1(SO) ? ? ss1(J0)?y ?yJ1 ss1( 3S3) = ss4(S3)----!~=sss1 where the horizontal maps are given by stabilization and are isomorphisms and J1 is the J-homomorphism. The abelian groups ss1(SO) and sss1are both iso- morphic to Z=2. The map J1 is bijective by [1]. Hence J :SO(4) ! SG(3) is a map of connected space which induced an isomorphism on the fundamental groups. This implies that BJ :BSO(4) ! BSG(3) is a map of simply connected spaces inducing an isomorphism on ss2. Let w2: BSG(3) ! K(Z=2, 2) be given by the second Stiefel-Whitney class. It and its composite w2 O BJ :BSO(4) ! K(Z=2, 2) are 3-connected. Since K is 2-dimensional, we conclude that ev- ery orientable fibration S3 ! E ! K is fiber homotopy equivalent to a fiber bundle S3 ! E ! K with structure group SO(4) and two such fiber bundles with structure group SO(4) over K are isomorphic if and only if their second Stiefel-Whitney classes agree. We have H2(f; Z=2)(w2(p)) = w2(M) since for a vector bundle , :E0 ! K we get a decomposition T E0|K ~=, T K and the Stiefel-Whitney classes of K are trivial. __ This finishes the proof of Theorem 8.2. |__| In dimension n = 6 we can at least compute the homology of the universal covering up to stable isomorphism. Lemma 8.3. Let M and K be as in Problem 0.16. Suppose n = 6. Let O(M) be the Euler characteristic. Then -O(M) + (1 + (-1)k) . O(K) 0 and we get 8 < Z p = 0, n - k; Hp(fM; Z) ~=Zss : 0 0 p 2; 0 p n - 2, p 6= n - k. 24 and for some s 0 k).O(K) s H3(fM; Z) Zsss ~=ZssZss-O(M)+(1+(-1) Zss . Proof. Because of Lemma 8.1 it suffices to show for the finitely generatedkstab* *ly free Zss-module H3(fM) that the classes of H3(fM) and Zss-O(M)+(1+(-1) ).O(K) agrees in K0(Zss). We obtain a finite projective resolution for the trivial Zss-module Z by C*(* *Ke) and get in eK0(Zss) [Z] = O(K) . [Zss]. Therefore every homology module Hp(fM) for p 6= 3 possesses a finite projective Zss-resolution and hence defines an element in K0(Zss). This automatically implies that the same is true for H3(fM). We compute in K0(Zss) X6 [H3(fM)] = - (-1)k . [Hk(fM)] + [Z] + (-1)6-k[Z] k=0 X6 = - (-1)k . [Ck(fM)] + (1 + (-1)k) . [Z] k=0 = -O(M) . [Zss] + (1 + (-1)k) . O(K) . [Zss] k = -O(M) + (1 + (-1) ) . O(K). [Zss]. Now Lemma 8.3 follows. |___| 9. On the Classification of Certain Low-Dimen- sional Manifolds Theorem 9.1. Let M and K be as in Problem 0.16. Suppose k = 2. Then n 5 and there is an isomorphism of abelian groups ~= Stop(M) -! Z=2. If f :M ! M is an orientation preserving self-homotopy equivalence and g :N ! M is an orientation preserving homotopy equivalence of manifolds, then we ob- tain in Stop(M) [g O f] = [g] + [f]. Proof. Notice for the sequel that the Farrell-Jones Conjecture for algebraic K- and L-theory holds for ss1(K) (see [13]). 25 The Atiyah-Hirzebruch spectral sequence yields two commutative diagrams with exact columns respectively rows 0 0 ?? ? y ?y H0(M; L6(Z)) H2(M; L4(Z))H0(f;L6(Z))-H2(f;L4(Z))----------------!~=H0(K; L6(Z* *)) H2(K; L4(Z)) ?? ? y ?y H6(M; L<1>) H6(f;L<1>)-------! H6(K; L<1>) ? ? e6?y e6?y H4(M; L2(Z)) -----! 0 ?? ? y ?y 0 0 and 0 ----! H1(M; L4(Z)) ----! H5(M; L<1>) --e5--!H3(M; L2(Z)) ----! 0 ? ? ? H1(f;L4(Z))?y~= H5(f;L<1>)?y 0?y 0 ----! H1(K; L4(Z)) ----! H5(K; L<1>) --e5--! 0 ----! 0 We conclude that the restriction of en :Hn(M; L<1>) ! Hn-2(M; L2(Z)) to the kernel of Hn(f; L<1>)): Hn(M; L<1>) ! Hn(K; L<1>) is bijective. We conclude from Theorem 1.2 that the following composition is an isomorphism for n = dim(M). Stop(M) oen--!Hn(M; L<1>) en-!Hn-2(M; L2(Z)). We have the following composition of isomorphisms 2(f;L2(Z)) -\[M] H2(K; L2(Z)) H--------!H2(M; L2(Z)) ----! Hn-2(M; L2(Z)). Since k = 2 by assumption, we get H2(K; L2(Z)) ~=L2(Z) ~=Z=2. If f :M ! M is an orientation preserving self-homotopy equivalence and g :N ! M orientation preserving homotopy equivalence of manifolds, then we obtain in Stop(M) the formlua which has been communicated to us by Andrew Ranicki [28] [f O g] = f*[g] + [f] Since any automorphism of the group Z=2 is the identity, this finishes_the proof of Theorem 9.1. |__| 26 Now we are ready to prove Theorem 0.18 and Theorem 0.19 (a). Proof. In view of Theorem 8.2, Theorem 9.1 and Theorem 1.1 it remains to show that there exists a homotopy equivalence f :M ! M inducing the up to conjugation the identity on the fundamental group which does not represent the trivial element id:M ! M in Stop(M). We want to conclude this from Theorem 2.2 (a) and (b) As soon as we have provided the necessary data, the claim follows from Theorem 2.2 (a)since the 2-dimensional torus with an appropriate framing yields an element fr2,2+dfor d 1 whose image under the Arf invariant map fr2,2+d! Z=2 is non-trivial. Since the dimension of K is less or equal to the connectivity of Sd, we can choose a map i: K ! Sd such that p O i is homotopic to the identity. By the homotopy lifting property we can arrange that p O i = idK. In particular we see that i: K ! M is an embedding. It remains to construct the map ff: Sd ! M. Let = (i) be the normal bundle of i: K ! M and D be the associated disk bundle. Let j1: Dd D M be the inclusion of the fiber of a point x 2 K. Obviously Dd intersects K transversally and the intersection consists of one point. It suffices to show that the map j0: Sd-1 = @Dd ! M - K given by restricting j1 to Sd-1 is nullhomotopic because then we can extend j0 to a map j2: Dd ! M - K and can define the desired map OE by j2 [j0j1: Sd = Dd [Sd-1Dd ! M. The map i: K ! M and the map p: M ! K induce isomorphisms on the fundamental groups. Fix a universal covering eK ! K. Its pullback with p: M ! K is the universal covering fM ! M. Obviously we get ss-equivariant maps ei:eK! fM and ep:fM! eK covering i and p and satisfying epO ei= ideK. Let ej1:Dd ! fM and ej0:Sd-1 ! fM be lifts of j1 and j0 with ej1|Sd-1 = ej0. The map ssd-1(fM - ei(Ke)) ! ssd-1(M - i(K)) is induced by a covering map and hence an isomorphism because of d - 1 2. It sends the class of ej0to the class of j0. Hence it suffices to show that the class of ej0in ssd-1(fM - ei(K* *e)) is zero. Its image under the map ssd-1(fM - ei(Ke)) ! ssd-1(ssd-1(fM)) is zero, a nullhomotopy for the image is given by ej1. Since fM is simply connected and its homology is trivial in dimensions d - 2 by Lemma 8.1, the space fM is (d - 2)-connected. Since the codimension of ei(Ke) in fM is d, the inclusion Mf -ei(Ke) ! fM is (d - 1)-connected and hence fM-ei(Ke) is (d - 2)-connected. Therefore the Hurewicz homomorphism ~= ssd-1(fM -ei(Ke)) -! Hd-1(fM -ei(Ke)) is an isomorphism. The image of the class of ej0under the Hurewicz homomor- phism is send to zero under the map Hd-1(fM - ei(Ke)) ! Hd-1(fM). Hence it remains to show that this map is injective. By the long exact sequence of the pair it suffices to prove that Hd(fM) ! Hd(fM, fM-ei(Ke)) is surjective. By Poincar'e duality we get a commutative diagram with isomorphisms as vertical maps 27 Hd(fM) ----! Hd(fM, fM-ei(Ke)) ? ? ~=?y ?y~= Hk(fM) ----! Hk(Ke) where the lower horizontal arrow is induced by eiand is split surjective becaus* *e_of peOei= ideK. This finishes the proofs of Theorem 0.18 and Theorem 0.19 (a). |_* *_| Next we give the proof of Theorem 0.19 (b). Proof. Let M be a closed topological oriented 6-manifold with ss1(M) ~=ss1(K), where K is a K(ss, 1)-manifold of dimension 2, and ss2(M) = 0. We want to prove that the Zss-isomorphism class of the intersection form on H3(fM) determines the homeomorphism type. The normal 2-type of M in the sense of Kreck [21] is K x BT opSpin pr2--!BT opSpin P-!BST OP, if M is a topological Spin-manifold and K x BT opSpin ExP---!BST OP x BST OP -! BST OP if M is not a topological Spin-manifold, where E is a vector bundle over K with w2(E) 6= 0. This normal 2-type is determined by ss = ss1(K) and w2 = w2(M) 2 Z=2 and so we denote it by B(ss, w2). Notice that (M; __) determines a class in the bordism group 6(B(ss, w2)). Now we want to apply [22, Corollary 3]. It says that if (M0; __) is another normal 2-smoothing in B(ss, w2), then M and M0 are homeomorphic, where the homeomorphism is compatible with the maps on ss1, if and only if the pairs determine the same class in 6(B(ss, w2)) and the intersection form together with quadratic refinement on K(ss3(M)) ! ss3(B(ss, w2)) = ss3(M), which by Lemma 8.1 is stably free and by Poincar'e duality unimodular, are isomorphic. Since 6= 0 we have to consider the quadratic refinement with values in Zss=. This is a quadratic refinement with respect to a form parameter in the sense of Bak [2]. Since ss has no element of order 2 and the quadratic form on ss3(M) takes values in Z[ss]= = 0, * *the quadratic form is determined by the intersection form. So it remains to show that the intersection form determines the bordism class in 6(B(ss, w2)). The Atiyah-Hirzebruch spectral sequence shows that 6(B(ss, w2)) ~=H2(K; Z)). If dim(K) = 1, then 6(B(ss, w2)) is trivial and the claim follows. Suppose dim(K) = 2. Then 6(B(ss, w2)) is isomorphic to Z. Consider the following composite 6(B(ss, w2)) ff-! 6(Bss) oe-!L6(Zss) where ff is induced by the obvious map B(ss, w2) ! Bss and the second map is given by the symmetric signature in the sense of Ranicki [30], [31]. The argume* *nt in the proof of Lemma 8.1 shows that the class fi O ff([M; __] which is given by 28 the chain complex of the associated ss-covering of M and the Poincar'e chain homotopy equivalence is the same as the class represented by the intersection form, since the two are obtained from one another by algebraic surgery. Hence the intersection from determines fi Off([M; __]. Since 6(B(ss, w2)) is torsion* *free, it remains to show that fi O ff is rationally injective. There is the following commutative diagram of Z-graded rational vector spaces as explained in [24, page 728] __D i *(Bss) *(*)Q n ----!~= KOn(Bss) Z Q ----! Kn(Bss) Z Q ? ? ? oe?y AR?y AC?y Ln(C*r(ss; R)) Z Qsign----!~=KOn(C*r(ss; R))--ZjQ--!Kn(C*r(ss)) Z Q The map AC is the assembly map appearing in the Baum Connes Conjecture and is known to be an isomorphism. The map i is a change of rings map and known to be rationally injective. The map oe factorizes as n n * *(Bss) *(*)Q n! L (Zss) Z Q ! L (Cr(ss; R)) Z Q. Hence it suffices to show that the composite 6(B(ss, w2)) ZQ ff-ZidQ----! 6(Bss) ZQ ! *(Bss) *(*)Q ! KO6(Bss) ZQ is injective. This follows from a spectral sequence argument. This shows that the intersection form determines the homeomorphism type. The next question is which unimodular forms ~ on stably free Z[ss1]-modules V can be realized as intersection forms of manifolds under consideration. Since, if (V, ~) can be realized, and (V, ~) splits off a hyperbolic form, i.e. (V, ~* *) = (V 0, ~0) ? H, then (V 0, ~0) can be realized by surgery on the hyperbolic plan* *e, the realization problem is reduced to the stable realization problem: Which elements in eL6(ss1(K)) can be realized by a stable homeomorphism class? If w2 = 0, the answer is: All. The reason is that we have a commutative diagram ~ TopSpin4 -=! L4({e}) ~=# x K # ~= TopSpin6(K) -! eL6(ss1(K)) If w2 6= 0 we don't know the answer. |___| 10. A Necessary Condition for Being a Borel Manifold Next we prove Theorem 0.21. 29 Proof. For every homology theory satisfying the disjoint union axiom and hence in particular for H*(-; L<1>) there is a natural Chern character (see Dold [7]). M ~= chn(X): Hn-4i(X; Q) -! Hn(X; L<1>) Z Q (10.1) i2Z,i 1 For an n-dimensional manifold M the composite -1 M Stop(M) oen--!Hn(M; L<1>) ! Hn(M; L<1>) ZQ chn(M)------! Hn-4i(M; Q) i2Z,i 1 sends the class of an orientation preserving homotopy equivalence f :N ! M to the element {f*(L(N)\[N])-L(M)\[M] | i 2 Z, i 1}. (see [32, Example 18.4 on page 198]). Now suppose that M is a Borel manifold. Then by Theorem 1.1 the oper- ation of the group of homotopy classes of self-homotopy equivalence M ! M which induce the identity on the fundamental group up to conjugation on the topological structure set Stop(M) is transitive. This implies that this group a* *cts also transitive on the image of the composite -1 M Stop(M) oen--!Hn(M; L<1>) ! Hn(M; L<1>) ZQ chn(M)------! Hn-4i(M; Q). i2Z,i 1 L This image is obviously an abelian subgroup of i2Z,i 1Hn-4i(N; Q) and agrees with the set S := {f*(L(M) \ [M]) - L(M) \ [M] | [f] 2 ho-autss(M)}. By the exactness of the surgery sequence the Q-submodule generated by S contains the kernel of the map induced by the classifying map c: M ! Bss M M c*: Hn-4i(M; Q) ! Hn-4i(Bss; Q). i2Z,i 1 i2Z,i 1 |___| Now we are ready to prove Theorem 0.20. Proof. We first show that the set of homotopy classes of orientation preserving homotopy equivalences f :E ! E which induce up to conjugation the identity on the fundamental group is finite. Since K is aspherical and p: E ! K induces an isomorphism on the fundamental groups because of d 2, the maps p and pOf are homotopic. By the homotopy lifting property we can assume that p O f = p holds. Hence it suffices to show that the set of fiber homotopy classes of fiber homotopy equivalences f :E ! E which cover the identity idK: K ! K and induce a map of degree one on the fibers is finite. By elementary obstruction theory this follows if the i-th homotopy group of the space SG(Sd) of self- maps Sd ! Sd of degree one is finite for i k. There is an obvious fibration 30 dSd ! SG(Sd) ! Sd. The long exact homotopy sequence yields the exact sequence ssi+d(Sd) ! ssi(SG(Sd)) ! ssi(Sd), where we take the obvious base points. If d is odd, ssj(Sd) is finite for all j 0, and, if d is even, ssj(Sd* *) is finite for j 2d - 2. This has been proved by Serre [33]. Hence ssi(SG(Sd)) is finite if i 0 and d is odd or if i d - 2. Since K is aspherical and there is a map i: K ! E with pOi ' idK, the kernel of the map c*: Hk+d-4i(E) ! Hk+d-4i(Bss1(E); Q) induced by the classifying map c = p: E ! Bss1(E) = K contains Hk-4i(K; Q). Now the claim follows from Theorem 0.21 because the abelian subgroup S of a Q-module appearing __ there is finite and hence trivial. |__| 11. Integral Homology Spheres In this section we prove Theorem 0.23 Proof. (a)Let c: M ! Sn the collapse map which is a map of degree one. Since M is by assumption a homology sphere, it induces an isomorphism on integral homology. By the Atiyah Hirzebruch spectral sequence it induces isomorphisms Hp(c; L<1>): Hp(M; L<1>) ! Hp(Sn; L<1>) for all p 2 Z. We obtain the following commutative diagram whose vertical arrows are parts of the long exact surgery sequence (1.3)where we here use the decoration s, i.e. we take the Whitehead torsion into account. Hn+1(M; L<1>) Hn+1(c;L<1>)---------!~=Hn+1(Sn; L<1>) ?? ? y ?y Lsn+1(ss1(c)) Lsn+1(Zss) --------! Lsn+1(Z) ?? ? y ?y top(c) Stop(M) -S----! Stop(Sn) ?? ? y ?y Hn(M; L<1>) Hn(c;L<1>)-------!~=Hn(Sn; L<1>) ?? ? y ?y s(ss1(c)) Lsn(Zss) -Ln-----! Lsn(Z) By the Poincar'e Conjecture Stop(Sn) is trivial. An easy diagram chase together with Theorem 1.1 shows that M is strongly Borel if and only if Hn+1(c; L<1>): Hn+1(M; L<1>) ! Lsn+1(Zss) 31 is surjective. Notice that the natural map Ls ! L is a homotopy equivalence since the Whitehead group of the trivial group is zero. The following diagram s<1>) Hn+1(c;Ls<1>) Hn+1({o}; Ls<1>)-Hn+1(i;L--------!~=Hn+1(M;-Ls<1>)--------!~=Hn+1(Sn; Ls<1>). ?? ? ? y~= ?y ?y~= Lsn+1(j) Lsn(ss1(c)) Lsn+1(Z) -----! Lsn+1(Zss) ------! Lsn+1(Z) commutes, where i: {o} ! M and j :Z ! Zss denote the obvious inclusions and all maps marked with ~=are isomorphisms by the Atiyah-Hirzebruch spectral sequence. Obviously Lsn(ss1(c)) O Lsn+1(j) is the identity. 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Matthias Kreck Mathematisches Institut, Universit"at Heidelberg Im Neuenheimer Feld 288, 69120 Heidelberg, Germany kreck@mathi.uni-heidelberg.de www.mathi.uni-heidelberg.de/~kreck/ Fax: 06221-545618 Wolfgang L"uck Fachbereich Mathematik, Universit"at M"unster Einsteinstr. 62, 48149 M"unster, Germany lueck@math.uni-muenster.de http://www.math.uni-muenster.de/u/lueck FAX: 49 251 8338370 35