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. Hence Lsn+1(j) is
bijective if and only if Hn+1(c; L<1>) is surjective. This shows that M is stro*
*ngly
Borel if and only if Lsn+1(j) is bijective.
(b) Suppose that M is strongly Borel and a rational homology sphere. Since
then Stop(M) = {0}, the map Hn+1(M; Ls<1>) ! Lsn+1(Zss) is surjective. Since
this map factorizes as
s<1>)
Hn+1(M; Ls<1>) Hn+1(f;L---------!Hn+1(Bss; Ls<1>) ! Lsn+1(Zss)
for the classifying map f :M ! Bss and the latter map is rationally injective by
assumption, the homomorphism Hn+1(f; Ls<1>) is rationally surjective. Given
a CW -complex X we have the isomorphism given by the Chern character
M ~=
Hn+1-4i(M; Q) -! Hn+1(M; Ls<1>).
i 1
Hence the map Hn+1-4i(f; Q): Hn+1-4i(M; Q) ! Hn+1-4i(Bss; Q) is surjective_
for i 1. |__|
References
[1]J. F. Adams. On the groups J(X). IV. Topology, 5:21-71, 1966.
[2]A. Bak. K-theory of forms. Princeton University Press, Princeton, N.J.,
1981.
[3]W. Browder. The Kervaire invariant of framed manifolds and its general-
ization. Ann. of Math. (2), 90:157-186, 1969.
[4]S. E. Cappell. A splitting theorem for manifolds. Invent. Math., 33(2):69-
170, 1976.
[5]S. Chang and S. Weinberger. On invariants of Hirzebruch and Cheeger-
Gromov. Geom. Topol., 7:311-319 (electronic), 2003.
[6]M. M. Cohen. A course in simple-homotopy theory. Springer-Verlag, New
York, 1973. Graduate Texts in Mathematics, Vol. 10.
32
[7]A. Dold. Relations between ordinary and extraordinary homology. Colloq.
alg. topology, Aarhus 1962, 2-9, 1962.
[8]F. T. Farrell. Lectures on surgical methods in rigidity. Published for the
Tata Institute of Fundamental Research, Bombay, 1996.
[9]F. T. Farrell. The Borel conjecture. In F. T. Farrell, L. G"ottsche,
and W. L"uck, editors, High dimensional manifold theory, number 9 in
ICTP Lecture Notes, pages 225-298. Abdus Salam International Centre
for Theoretical Physics, Trieste, 2002. Proceedings of the summer school
"High dimensional manifold theory" in Trieste May/June 2001, Number 1.
http://www.ictp.trieste.it/"pub_off/lectures/vol9.html.
[10]F. T. Farrell and L. E. Jones. A topological analogue of Mostow's rigidity
theorem. J. Amer. Math. Soc., 2(2):257-370, 1989.
[11]F. T. Farrell and L. E. Jones. Rigidity and other topological aspects of
compact nonpositively curved manifolds. Bull. Amer. Math. Soc. (N.S.),
22(1):59-64, 1990.
[12]F. T. Farrell and L. E. Jones. Isomorphism conjectures in algebraic K-
theory. J. Amer. Math. Soc., 6(2):249-297, 1993.
[13]F. T. Farrell and L. E. Jones. Topological rigidity for compact non-
positively curved manifolds. In Differential geometry: Riemannian geome-
try (Los Angeles, CA, 1990), pages 229-274. Amer. Math. Soc., Providence,
RI, 1993.
[14]F. T. Farrell and L. E. Jones. Rigidity for aspherical manifolds with ss1
GL m (R). Asian J. Math., 2(2):215-262, 1998.
[15]S. C. Ferry, A. A. Ranicki, and J. Rosenberg. A history and survey of
the Novikov conjecture. In Novikov conjectures, index theorems and rigid-
ity, Vol. 1 (Oberwolfach, 1993), pages 7-66. Cambridge Univ. Press, Cam-
bridge, 1995.
[16]M. H. Freedman. The disk theorem for four-dimensional manifolds. In
Proceedings of the International Congress of Mathematicians, Vol. 1, 2
(Warsaw, 1983), pages 647-663, Warsaw, 1984. PWN.
[17]M. H. Freedman and F. Quinn. Topology of 4-manifolds. Princeton Uni-
versity Press, Princeton, NJ, 1990.
[18]I. Hambleton and M. Kreck. Cancellation, elliptic surfaces and the topology
of certain four-manifolds. J. Reine Angew. Math., 444:79-100, 1993.
[19]J. Hempel. 3-Manifolds. Princeton University Press, Princeton, N. J., 1976.
Ann. of Math. Studies, No. 86.
33
[20]R. C. Kirby and L. C. Siebenmann. Foundational essays on topological man-
ifolds, smoothings, and triangulations. Princeton University Press, Prince-
ton, N.J., 1977. With notes by J. Milnor and M. F. Atiyah, Annals of
Mathematics Studies, No. 88.
[21]M. Kreck. Surgery and duality. Ann. of Math. (2), 149(3):707-754, 1999.
[22]M. Kreck. Cancellation for stable diffeomorphisms. preprint, 2004.
[23]M. Kreck and W. L"uck. The Novikov Conjecture: Geometry and Algebra,
volume 33 of Oberwolfach Seminars. Birkh"auser, 2005.
[24]E. Leichtnam, W. L"uck, and M. Kreck. On the cut-and-paste property of
higher signatures of a closed oriented manifold. Topology, 41(4):725-744,
2002.
[25]W. L"uck. A basic introduction to surgery theory. In F. T. Farrell,
L. G"ottsche, and W. L"uck, editors, High dimensional manifold theory, num-
ber 9 in ICTP Lecture Notes, pages 1-224. Abdus Salam International Cen-
tre for Theoretical Physics, Trieste, 2002. Proceedings of the summer school
"High dimensional manifold theory" in Trieste May/June 2001, Number 1.
http://www.ictp.trieste.it/"pub_off/lectures/vol9.html.
[26]W. L"uck and H. Reich. The Baum-Connes and the Farrell-Jones Conjec-
tures in K- and L-Theory. In Handbook of K-theory, volume 2, pages 703
- 842. Springer-Verlag, Berlin, 2005.
[27]J. Milnor. On simply connected 4-manifolds. In Symposium internacional
de topolog'ia algebraica International symposi um on algebraic topology,
pages 122-128. Universidad Nacional Aut'onoma de M'exico and UNESCO,
Mexico City, 1958.
[28]A. Ranicki. On the manifold structure set. in preparation, 2005.
[29]A. A. Ranicki. Algebraic L-theory. I. Foundations. Proc. London Math.
Soc. (3), 27:101-125, 1973.
[30]A. A. Ranicki. The algebraic theory of surgery. I. Foundations. Proc.
London Math. Soc. (3), 40(1):87-192, 1980.
[31]A. A. Ranicki. Exact sequences in the algebraic theory of surgery. Princeton
University Press, Princeton, N.J., 1981.
[32]A. A. Ranicki. Algebraic L-theory and topological manifolds. Cambridge
University Press, Cambridge, 1992.
[33]J.-P. Serre. Groupes d'homotopie et classes de groupes ab'eliens. Ann. of
Math. (2), 58:258-294, 1953.
34
[34]V. G. Turaev. Homeomorphisms of geometric three-dimensional manifolds.
Mat. Zametki, 43(4):533-542, 575, 1988. translation in Math. Notes 43
(1988), no. 3-4, 307-312.
[35]F. Waldhausen. Algebraic K-theory of topological spaces. I. In Algebraic
and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stan-
ford, Calif., 1976), Part 1, pages 35-60. Amer. Math. Soc., Providence,
R.I., 1978.
[36]C. T. C. Wall. Classification of (n - 1)-connected 2n-manifolds. Ann. of
Math. (2), 75:163-189, 1962.
[37]C. T. C. Wall. Surgery on compact manifolds. American Mathematical
Society, Providence, RI, second edition, 1999. Edited and with a foreword
by A. A. Ranicki.
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