jfdmath.sitehost.iu.eduManifolds Homotopy Equivalent to Certain Torus Bundles over Lens Spaces JAMES F. DAVIS Indiana University WOLFGANG LÜCK Universität Bonn Abstract We compute

Manifolds Homotopy Equivalentto Certain Torus Bundles

over Lens Spaces

JAMES F. DAVISIndiana University

WOLFGANG LÜCKUniversität Bonn

Abstract

We compute the topological simple structure set of closed manifolds that occuras total spaces of flat bundles over lens spaces S l=.Z=p/ with fiber T n for anodd prime p and l � 3 provided that the induced Z=p-action on �1.T n/ D Zn

is free outside the origin. To the best of our knowledge this is the first compu-tation of the structure set of a topological manifold whose fundamental group isnot obtained from torsionfree and finite groups using amalgamated and HNN-extensions. We give a collection of classical surgery invariants such as splittingobstructions and �-invariants that decide whether a simple homotopy equiva-lence from a closed topological manifold to M is homotopic to a homeomor-phism. © 2020 Wiley Periodicals LLC

Contents

0. Introduction 21. Preliminaries about the Group Zn Ì Z=p 42. Preliminaries about the Farrell-Jones Conjecture 43. Algebraic K-theory of the Group Ring 64. Algebraic L-theory of the Group Ring 75. Structure Sets 186. Periodic Simple Structure Set of BP for a Finite p-group 197. Periodic Simple Structure Set of B� 228. Periodic Simple Structure Set of M 249. Geometric Simple Structure Set of M 33

10. Invariants for Detecting the Structure Set of M 3711. Appendix: Open Questions 47Bibliography 48

Communications on Pure and Applied Mathematics, 0001–0050 (PREPRINT)© 2020 Wiley Periodicals LLC

2 J. F. DAVIS AND W. LÜCK

0 Introduction0.1 Flat Torus Bundles over Lens Spaces

Throughout this paper we will consider the following setup and notation:� Let p be an odd prime.� Let �W Z=p ! Aut.Zn/ D GLn.Z/ be a group homomorphism so that the

induced action of Z=p on Zn � f0g is free.� The homomorphism � defines an action of Z=p on the torus Tn D Rn=Zn.

If we want to emphasize the Z=p-action, we write Tn� D BZn� .

� Fix a free action of Z=p on a sphere Sl for an odd integer l � 3. We referto the orbit space Ll WD Sl=.Z=p/ as a lens space.� Define a closed .nC l/-manifold

M WD Tn� �Z=p Sl :

� The fundamental group of M nCl is the semidirect product denoted by

� WD Zn �� Z=p:

We computed the K-theory of the C �-algebra of � in [13].An example of such an action � is given by the regular representation ZŒZ=p�

modulo the ideal generated by the norm element, in which case we have �W Z=p !Aut.Zp�1/.

The action of Z=p on Zn�f0g is free if and only if the fixed point set .Tn/Z=p isfinite. Equip the torus and the sphere with the standard orientations; this determinesan orientation on M . Since the Z=p-action on Sl is free, there is a fiber bundleTn� ! M nCl ! Ll . It is worth noting that Tn, L1, and Tn

� �Z=p S1 are

models for BZn, BZ=p, and B� , respectively. Notice that our assumptions implythat dim.M/ D nC l � 5.

Next we summarize our main results.

0.2 Geometric Topological Simple Structure Set of M

We will prove the following theorem in Theorem 9.2 (5).

THEOREM 0.1 (Geometric topological simple structure set of M ). As an abeliangroup we have

S geo;s.M/ Š Zpk.p�1/=2

˚

n�1MiD0

Ln�i .Z/ri ;

where the natural number k is determined by the equality n D k.p � 1/ and thenumbers ri are defined in (4.6).

In particular, the structure set is infinite.To our knowledge this is the first computation of the structure set of a topo-

logical manifold whose fundamental group is not obtained from torsionfree andfinite groups using amalgamated and HNN-extensions. The computation is rather

TORUS BUNDLES OVER LENS SPACES 3

involved and based on the Farrell-Jones conjecture. We also compute the peri-odic structure sets of B� and of M and prove detection results for these structuresets. The notion of a structure set is recalled in Section 5. Its study is motivatedby the question of determining the homeomorphism classes of closed manifoldshomotopy equivalent to M .

0.3 Homotopy Equivalence Versus HomeomorphismThe following result gives a criterion when a simple homotopy equivalence of

closed manifolds withM as target is homotopic to a homeomorphism. It is provedin Section 10.

THEOREM 0.2 (Simple homotopy equivalence versus homeomorphism). Let hWN !M be a simple homotopy equivalence with a closed topological manifold Nas source and the manifold M of Section 0.1 as target. Then h is homotopic to ahomeomorphism if and only only if the following conditions are satisfied:

� Vanishing of splitting obstructions:Let hW N ! Tn�Sl be obtained from h by pulling back the Z=p-coveringTn � Sl ! M . Consider any nonempty subset J � f1; 2; : : : ; ng. LetT J� � � Tn�Sl be the obvious jJ j-dimensional submanifold. By makingh transversal to T J � � , we obtain a normal map .xh/�1.T J � � / !T J � � that defines a surgery obstruction in LjJ j.Z/.

This obstruction has to be zero.� Equality of �-invariants:

Consider any subgroup P � � of order p. Let P 0 be the image of Punder the abelianization map prW � ! �ab. Let MP ! M be the covercorresponding to the subgroup pr�1.P 0/. Let hP W NP ! MP be the cor-responding covering simple homotopy equivalence. Then we must have theequality of �-invariants in zR.P 0/.�1/

.nClC1/=2

Œ1=p�:

�.NP ! BP 0/ D �.MP ! BP 0/;

Here is a detailed outline of the paper. Using the Farrell-Jones conjecture, itis straightforward to compute K1 and K0 of the group ring Z� (see Section 3).Computing the algebraic L-theory (Section 4) is more difficult; we use homolog-ical computations from our previous paper [13] and a result of Land and Niko-laus [21]. This result generalizes a result of Sullivan [33, 34] comparing L-theoryspectra with topological K-theory spectra after inverting 2. As with much of therest of the paper, we compute the algebraic L-theory at p and away from p.

Our goal is to compute and detect the geometric structure set of M . However,it is much easier to compute the periodic structure set (also known as the algebraicstructure set) of the classifying space B� . Indeed, as a simple application of theFarrell-Jones conjecture, we show in Section 7M

P2.P/

Sper;sm .BP /

Š�! S

per;sm .B�/:


In Section 6 we use equivariant KO-homology to show for any odd order p-group G, that S

per;sm .BG/ is a finitely generated free ZŒ1=p�-module. Section 8 is

the heart of the paper, where we compute the periodic structure set of M , workingat p and away from p. In Section 9 we give the computation of the geometricstructure set of M , as well as detection by algebraic topological invariants. InSection 10, we detect the structure set by the geometric invariants given in Theo-rem 0.2: splitting obstructions and the �-invariant. Finally, in Section 11 we men-tion some basic questions that we did not answer, in the hope that these questionsare accessible and will stimulate future work.

1 Preliminaries about the Group Zn Ì Z=p

In this section we collect various facts about � from [13, lemma 1.9].

LEMMA 1.1.(1) Let � D e2�i=p. There are nonzero ideals I1; : : : ; Ik of ZŒ�� and isomor-

phisms of ZŒZ=p�-modules

Zn Š I1 ˚ � � � ˚ Ik; Zn ˝Q Š Q.�/k :

Hence n D k.p � 1/.(2) Each nontrivial finite subgroup P of � is isomorphic to Z=p and its Weyl

group W�P WD N�P=P is trivial.(3) There are isomorphisms

H 1.Z=pIZn/Š�! cok.� � idW Zn ! Zn/ Š .Z=p/k

and a bijection

cok.� � idW Zn ! Zn/Š�!P WD f.P / j P � �; 1 < jP j <1g:

Here P is the set of conjugacy classes .P / of nontrivial subgroups of finiteorder. If we fix an element s 2 � of order p, the bijection sends the ele-ment xu 2 Zn=.1 � �/Zn to the conjugacy class of the subgroup of order pgenerated by us.

(4) We have jPj D pk .(5) There is a bijection from the Z=p-fixed set of the Z=p-space Tn

� WD Rn�=Zn�

to H 1.Z=pIZn�/. In particular, .Tn� /

Z=p consists of pk points.(6) Œ�; �� D im.� � idW Zn ! Zn/.(7) �=Œ�; �� Š cok.� � idW Zn ! Zn/˚ Z=p D .Z=p/kC1.

2 Preliminaries about the Farrell-Jones ConjectureTo classify high-dimensional manifolds one uses the surgery exact sequence.

One of the terms in the surgery exact sequence is the 4-periodic L-group L�.ZG/,where G is the fundamental group of the manifold under consideration. Althoughthe L-groups are algebraically defined, when G is infinite the computation of theL-groups is done by a mix of algebraic, topological, and geometric methods. This


is encoded in the Farrell-Jones conjecture, which can be stated in terms of an equi-variant homology theory in the sense of [23, sec. 1] as follows.

Let EW GROUPOIDS ! SPECTRA be a covariant functor from the categoryof small groupoids to the category of spectra, which is homotopy invariant; i.e.,it sends an equivalence of groupoids to a weak homotopy equivalence of spectra.Given a cellular map X ! Y of G-CW -complexes for a (discrete) group G,Davis-Lück [11] define

HGm .X ! Y IE/ WD �m

�mapG.G=�; cone.X ! Y // ^Or.G/ E.G=�/

�;

where cone refers to the mapping cone, Or.G/ to the orbit category of G, andG=H to the groupoid associated to the G-set G=H . This defines a G-homologytheory HG

� on the category of G-CW -complexes. Its coefficients are given byHGm .G=H IE/ D �m.E.H//.An equivariant homology theory H ‹

� in the sense of [23, sec. 1] assigns toevery discrete group G a G-homology theory H G

� on the category of G-CW -complexes. Given a group homomorphism ˛W G ! H , there is a correspond-ing map of abelian groups ind˛W H G

m .X;A/ ! H Hm .H �˛ .X;A//. The ax-

ioms for an equivariant homology theory are satisfied when EW GROUPOIDS!SPECTRA is a homotopy invariant functor and H G

m .X/ is defined as above, see [25,prop. 157, p. 796].

Examples of such GROUPOIDS-spectra are K and Lh�1i defined in [11, sec. 2].Here �m.K.G=H// D Km.ZH/ and �m.L.G=H// D L

h�1im .ZH/.

A family F of subgroups ofG is a collection of subgroups that is nonempty andclosed under conjugation and under taking subgroups. The classifying spaceEFG

for group actions with isotropy in F is characterized up to G-homotopy equiva-lence as a G-CW-complex where .EFG/

H is empty if H 62 F and contractibleif H 2 F . We write EG for the classifying space when F is the family of finitesubgroups and G for the classifying space when the F is the family of virtuallycyclic subgroups. For more information about these spaces we refer to [24].

The Farrell-Jones conjecture for the groupG, which was originally stated in [14,1.6 on p. 257], predicts that for all m 2 Z the projection G ! � induces isomor-phisms

HGm .GIK/! HG

m . � IK/ D Km.ZG/;

HGm .GIL

h�1i/! HGm . � IL

h�1i/ D Lh�1im .ZG/:

We now specialize to the group � D ZnÌZ=p. The first point is that the Farrell-Jones conjecture in K- and L-theory holds for � by [7]. Since the only subgroups


that are virtually cyclic and not finite are infinite cyclic, and since the Farrell-Jones conjecture holds for infinite cyclic groups, the transitivity principle [14, the-orem A.10], or [25, theorem 65, p. 742] shows that

H�m.E�IL

h�1i/Š�! H�

m.�ILh�1i/;

H�m.E�IK/

Š�! H�

m.�IK/;

are bijective for all m 2 Z. Hence we get:

THEOREM 2.1 (Farrell-Jones conjecture for �). The projection E� ! � inducesfor all m 2 Z bijections

H�m.E�IK/! H�

m. � IK/ D Km.Z�/;

H�m.E�IL

h�1i/! H�m. � IL

h�1i/ D Lh�1im .Z�/:

Remark 2.2. Note that � D Zn Ì Z=p acts affinely on Zn where Zn acts bytranslation and Z=p acts by the map �. By extending scalars, � acts properly andcocompactly on Rn. Hence � is a crystallographic group and Rn can be taken as amodel for E� .

3 Algebraic K -theory of the Group RingFor a group G and an integer m, define Whm.G/ to be the homotopy groups of

the homotopy cofiber of the assembly map in algebraic K-theory, i.e.,

Whm.G/ D HGm .EG ! � IK/

where K D KZ is the algebraic K-theory spectrum over the orbit category of [11,sec. 2] with �n.K.G=H// D Kn.ZH/. Hence Wh1.G/ is the classical White-head group Wh.G/ D cok.f˙1g � Gab ! K1.ZG//, Wh0.G/ is zK0.ZG/ Dcok.K0.Z/! K0.ZG//, and Whn.G/ is Kn.ZG/ for n � �1.

The spaceE� can be profitably analyzed using the following cellular �-pushout;see [28, cor. 2.11],

(3.1)

F.P /2P � �P EP //

��

E�

��F.P /2P �=P // E�:

This leads to the following result taken from [13, lemma 7.2(ii)].

LEMMA 3.1. Let H ‹� be an equivariant homology theory in the sense of [23,

sec. 1]. Then there is a long exact sequence

� � � !H �mC1.E�/

ind�!1��!HmC1.B�/!

MP2.P/

�H Pm . � /

'm��!H �

m .E�/ind�!1��!Hm.B�/! � � � ;


where �H Pm . � / is the kernel of the induction map

indP!1W H Pm . � /!Hm. � /;

the map 'm is induced by the various inclusions P ! � , and B� WD �nE� .The map

ind�!1Œ1=p�W H �m .E�/Œ1=p�!Hm.B�/Œ1=p�

is split surjective.

THEOREM 3.2 (Computation of Whm.�/). For every m 2 Z,MP2.P/

Whm.P /Š�!Whm.�/:

Furthermore, for p an odd prime, Wh.Z=p/ Š Z.p�3/=2, zK0.ZŒZ=p�/ is theideal class group C.ZŒexp.2�i=p/�/ and hence is finite, and Km.ZŒZ=p�/ D 0

for m � �1.

PROOF. The isomorphism ˚Whm.P / Š Whm.�/ is a direct consequence ofTheorem 2.1 and the G-pushout (3.1). See also [26, theorem 1.8], [12, theorem5.1(d)], and [27, theorem 0.2].

The computation of Wh.Z=p/ for an odd prime p (and much more informationabout the Whitehead group for finite groups) can be found in [30], a discussion ofzK0.ZŒZ=p�/ in [29], and the vanishing of Km.ZŒZ=p�/ for m � �1 in [8]. �

Theorem 3.2 is consistent with [26, theorem 1.8(i)].

4 Algebraic L-theory of the Group RingIn this section we compute the L-groups of Z� for all decorations.

4.1 Decorated L-groups

We first discuss the so-called decorated versions of L-theory Lh�ii� .ZG/ forany group G for i D 2; 1; 0;�1;�2; : : : and i D �1. We recall briefly afew facts; more information can be found in [31]. For m 2 Z, these are func-tors Lhiim W GROUPS! ABELIAN GROUPS that are 4-periodic in the sense thatLhiim .ZG/ D L

hiimC4.ZG/. There are natural maps

LhiC1i� .ZG/! L

hii� .ZG/;

and one definesLh�1i� .ZG/ D colim

i!�1Lhii� .ZG/:

One sometimes writes

Ls�.ZG/ D Lh2i� .ZG/; Lh�.ZG/ D L

h1i� .ZG/I L

p� .ZG/ D L

h0i� .ZG/:

The Ls-groups are bordism groups of algebraic Poincaré complexes with basedmodules and simple Poincaré duality; they are useful in classifying manifolds. The


Lh-groups are bordism groups of algebraic Poincaré complexes with free mod-ules; they are useful for studying the existence question of when a space has thehomotopy type of a manifold. The Lp-groups are bordism groups of algebraicPoincaré complexes with projective modules. The Lh�1i-groups are useful for theFarrell-Jones conjecture.

We will often use the Ranicki-Rothenberg exact sequence for a groupG; see [31,theorem 7.12, p. 146]:

(4.1) � � � ! LhiC1im .ZG/! Lhiim .ZG/! �Hm.Z=2IWhi .G//

! LhiC1im�1 .ZG/! L

hiim�1.ZG/! � � � :

For any decoration i , there is a homotopy invariant functor

LhiiW GROUPOIDS! SPECTRA

satisfying �m.Lhii.G=H// D Lhiim .ZH/. Farrell and Jones [14] conjecture that

HG� .GIL

h�1i/Š�! HG

� . � ILh�1i/

for all groups G. However, the decorated version of the assembly map

HG�

�GILhii

� Š�! HG

� . � ILhii/

need not be bijective in general; for example, it is not bijective for the group G DZ2 � Z=29 for the decorations p, h, and s; see [15, example 14]. However, theFarrell-Jones conjecture for the group � holds for all i as we show below. Thiswill be important since the Ls-version is the geometrically significant one.

4.2 The h�1i-decoration

In this subsection we compute Lh�1im .Z�/ using the Farrell-Jones conjecture;see Theorem 2.1. The L-theory of � D ZnÌ Z=p is, in some sense, built from theL-theory of Zn and Z=p, so we first review these.

The Farrell-Jones conjecture inK-theory holds for the torsion-free group Zn. Itfollows that Whm.Zn/ D 0 for all m 2 Z and for all n 2 Z�0. Thus the maps

Lhiim .ZŒZn�/Š�! Lh�1im .ZŒZn�/(4.2)

are bijections for i , m, and n, and the map of spectra

Lhii.ZŒZn�/'�! Lh�1i.ZŒZn�/(4.3)

is a weak homotopy equivalence for all i and n. Hence we will omit the decorationand refer to Lm.ZŒZn�/ and L.ZŒZn�/.


When n D 0, the L-groups are well-known, essentially due to Kervaire-Milnor,

Lm.Z/ D

8̂̂̂<̂ˆ̂:

Z; m � 0 .mod 4/;0; m � 1 .mod 4/;Z=2; m � 2 .mod 4/;0; m � 3 .mod 4/;

where the map to Z is given by the signature divided by 8 and the map to Z=2 isgiven by the Arf invariant. Since the Farrell-Jones conjecture in L-theory holds forthe torsion-free group Zn,

(4.4) Lm.ZŒZn�/

Š � Hm.BZnIL.Z// D

nMiD0

Lm�i .Z/.ni /:

For p an odd prime, we get from [4, theorem 1], [5, theorems 1, 2, and 3],and [16, theorem 10.1]

Lh0im .ZŒZ=p�/ D

(Z.p�1/=2 ˚ Lm.Z/; m even,0; m odd.

Since Whi .Z=p/ D 0 for i < 0 by [8], we have

Lh0im .ZŒZ=p�/Š�! Lh�1im .ZŒZ=p�/

Š�! Lh�2im .ZŒZ=p�/

Š�! � � �

Š�! Lh�1im .ZŒZ=p�/:

Thus

(4.5) Lh�1im .ZŒZ=p�/ D

(Z.p�1/=2 ˚ Lm.Z/; m even,0; m odd.

THEOREM 4.1 (Lh�1i-theory). There is a long exact sequence

� � � ! HmC1.B�IL.Z//!M.P /2P

zLh�1im .ZP /! Lh�1im .Z�/

ˇm��! Hm.B�IL.Z//!

M.P /2P

zLh�1i

m�1 .ZP /! � � � ;

where zLh�1im .ZP / is the kernel of the map Lh�1im .ZP / ! Lm.Z/ induced byinduction with P ! 1.

The map ˇmŒ1=p� is a split surjection, and thus there is an isomorphism ofZŒ1=p�-modules

Lh�1im .Z�/Œ1=p� Š

� M.P /2P

zLh�1im .ZP /Œ1=p�

�˚Hm.B�IL.Z//Œ1=p�:

PROOF. This follows directly from Theorem 2.1 and Lemma 3.1. �


Next we want to improve Theorem 4.1 by comparing it with the computation ofKO�.C

�r .G// of [13, theorem 10.1].

The next result is motivated by Sullivan’s thesis [33], but its proof requires muchmore, in particular recent results of Land and Nikolaus.

THEOREM 4.2 (ComparingL-theory and topologicalK-theory). There is a naturaltransformation of equivariant homology theories

T ‹�.�/W H‹�.�IL

h�1i/Œ1=2�! KO‹�.�/Œ1=2�

such that for every group G, every proper G-CW -complex X , and every m 2 Zthe map

TGm .X/W HGm .X IL

h�1i/Œ1=2�! KOGm .X/Œ1=2�

is an isomorphism.

PROOF. Notice that we invert 2 so that the decorations do not matter and wetherefore ignore them. The key ingredient is a rigorous construction of a weakhomotopy equivalence of spectra KO.A/Œ1=2�! L.A/Œ1=2� for a real C �-algebraA from its topological K-theory spectrum to its algebraic L-theory spectrum afterinverting 1=2, which is natural in A, [21, theorem C]. In particular, this applies tothe real group C �-algebra and can be extended from groups to groupoids; see [21,sec. 5.2]. For any group there is a natural map from the integral group ring tothe real group C �-algebra, which yields a natural map between the correspondingL-theory spectra. This construction also extends to groupoids. Combining thesetwo transformations and using the fact that the one in [21, theorem C] is a weakhomotopy equivalence yields the desired transformation T ‹�.�/. Recall that a G-CW -complex is proper if all isotropy groups are finite. In order to show thatTGn .X/ is an isomorphism for every proper G-CW -complex X , it suffices to dothis in the case X D G=H for a finite subgroup H � G; see [6, lemma 1.2].Hence one needs to show that the change-of-coefficients map Lm.ZH/Œ1=2� !Lm.RH/Œ1=2� is bijective for allm, which is done in [32, prop. 22.34, p. 253]. �

If G is a group and M is a left ZG-module, the invariants of M are MG WD

HomZG.Z;M/ D H 0.BGIM/, and the coinvariants of M are MG WD Z ˝ZG

M D H0.BGIM/. If G is finite of order q, the norm map MG ! MG sending1˝ x to

Pg2G gx is an isomorphism after tensoring with ZŒ1=q�.

If H� is a (nonequivariant) generalized homology theory, andG acts onX , thenthe homology quotient map factors as H�.X/ ! H�.X/G ! H�.GnX/. If thequotient map X ! GnX is a regular G-cover and G is finite of order q, thenthere is a transfer map H�.GnX/!H�.X/G so that the composite H�.X/G !H�.GnX/!H�.X/G is an isomorphism after tensoring with ZŒ1=q�.


LEMMA 4.3. Let H� be any generalized homology theory taking values in ZŒ1=p�-modules. For all m 2 Z, the following maps are isomorphisms:

Hm.BZn�/Z=pŠ�!Hm.B�/

Š�!Hm.BZn�/

Z=p;

Hm.B�/Š�!Hm.B�/:

PROOF. Consider the diagram

H�.BZn�/Z=p˛

// H�.B�/ˇ//

��

H�.BZn�/Z=p

H�.B�/

;

where ˛ and are the functorial H�-maps and ˇ is the transfer. We will show that˛, ˇ, and are isomorphisms.

Given a Z=p-CW -complex X , the map

jmW Hm.X/Z=p !Hm

�.Z=p/nX

�is natural in X . Since the functor sending a ZŒ1=p�ŒZ=p�-module M to MZ=pis an exact functor, the assignments sending a Z=p-CW -complex to Hm.X/Z=pand to Hm

�.Z=p/nX

�are Z=p-homology theories and j� is a transformation of

Z=p-homology theories. One easily checks that jm is a bijection ifX is .Z=p/=Hfor any subgroup H � Z=p. It follows that jm is a bijection for any Z=p-CW -complex. Taking X to be ZnnE� , we see ˛ is an isomorphism and taking X to beZnnE� we see that ı ˛ is an isomorphism. We commented above that ˇ ı ˛ isan isomorphism. It follows that all the maps are isomorphisms. �

We also need some numbers defined in our previous paper [13]. For j; k 2 Z�0and for p an odd prime, define

(4.6) rj WD rk�ƒj .ZŒ�p�

k/Z=p�;

where ƒj means the j th exterior power of a Z-module and �p is a primitive pth

root of unity. Thus rj D rkH j .Tn� /

Z=p. When k D 1 we worked out thesenumbers in [13, lemma 1.22]: rj D 0 for j � p, and for 0 � j � .p � 1/,

rj D1

p

p � 1

j

!C .�1/j .p � 1/

!:

THEOREM 4.4 (Computation of Lh�1i.Z�/).(1) There is an isomorphism

Lh�1i

2mC1.Z�/ Š H2mC1.B�IL.Z//:

Transferring to the finite index subgroup Zn of � induces an isomorphism

Lh�1i

2mC1.Z�/Š�! L2mC1.ZŒZ

n��/

Z=p:


(2) There is an exact sequence

0!M.P /2P

zLh�1i

2m .ZP /! Lh�1i

2m .Z�/! H2m.B�IL.Z//! 0;

which splits after inverting p.(3) We have

Lh�1im .Z�/ Š

(Zp

k.p�1/=2 ˚�Ln

iD0Lm�i .Z/ri�; m even;Ln

iD0Lm�i .Z/ri ; m odd:

PROOF. (1) and (2). The computation of Lh�1i� .ZŒZ=p�/ in (4.5) and Theo-rem 4.1 implies that for every m 2 Z we obtain an isomorphism

Lh�1i

2mC1.Z�/Š�! H2mC1.B�IL.Z//

and a short exact sequence

0!M.P /2P

zLh�1i

2m .ZP /! Lh�1i

2m .Z�/! H2m.B�IL.Z//! 0;

which splits after inverting p.It remains to show that transferring to subgroup Zn of � induces an isomorphism

Lh�1i

2mC1.Z�/Š�! L2mC1.ZŒZ

n�/Z=p:

It suffices to do this after localizing at p and after inverting p.We first invert 2. Because of Theorem 4.2 and the fact that the assembly maps

H�m.E�IL

h�1i/Œ1=2�Š�! H�

m. � ILh�1i/Œ1=2� D Lh�1im .Z�/Œ1=2�;

KO�m.E�/Œ1=2�Š�! KOm.C

�r .�IR//Œ1=2�;

are bijections (see Theorem 2.1 and [17]), we obtain a commutative diagram ofZŒ1=2�-modules

Lh�1i

2mC1.Z�/Œ1=2��Œ1=2�

//

Š

��

L2mC1.ZŒZn�/Z=pŒ1=2�

Š

��

KO2mC1.C�r .�IR//Œ1=2�

��Œ1=2�// KO2mC1.C

�r .Z

nIR//Z=pŒ1=2�

with bijective vertical arrows. The lower horizontal arrow is bijective by theorem[13, sec. 11.2]. Hence

��.p/W Lh�1i

2mC1.Z�/.p/Š�! L2mC1.ZŒZ

n�/Z=p.p/

is bijective for every m 2 Z.


Next consider the commutative diagram

(4.7) H2mC1.B�IL.Z//��//

A��

H2mC1.BZnIL.Z//Z=p

AZ=pZn

��

Lh�1i

2mC1.Z�/��

// L2mC1.ZŒZn�/Z=p:

We will show that all maps are isomorphisms after inverting p. Lemma 4.3 showsthis is true for the top horizontal map. The Farrell-Jones conjecture in L-theoryfor Zn (see (4.4)) shows this holds for the right vertical map. To show this holdsfor A� , first rewrite it as A� W H2mC1.B�IL.Z// D H�

2mC1.E�ILh�1i/ !

H�2mC1.E�IL

h�1i/, using the Farrell-Jones conjecture in L-theory for � . Themap ind�!1 W H�

2mC1.E�ILh�1i/ ! H2mC1.B�IL.Z// is an isomorphism

after inverting p by Lemma 3.1 and the vanishing of Lh�1i2mC1.ZŒZ=p�/. The com-posite ind�!1 ıA� is an isomorphism after inverting p by Lemma 4.3. Thus wecan conclude that A� is an isomorphism after inverting p.

Thus the bottom row ��W Lh�1i

2mC1.Z�/! L2mC1.ZŒZn�/Z=p of (4.7) is also anisomorphism after inverting p. We have shown that ��

.p/and ��Œ1=p� are isomor-

phisms, so we conclude that �� is an isomorphism.

(3) It suffices to show that the abelian groups in question are isomorphic af-ter inverting 2 and after inverting p. We conclude from [13, theorem 10.1] andTheorem 4.2 that this is the case after inverting 2.

It remains to treat the case where we invert p. Because of assertion (2), (4.5),and Lemma 4.3, it remains to prove

Lm.ZŒZn�/Z=pŒ1=p� Š

nMiD0

.Lm�i .Z//ri Œ1=p�:(4.8)

Since Lm.ZŒZn�/ Š Hm.BZnIL.Z//, it remains to show

Hm.BZnIL.Z//Z=pŒ1=p� ŠnMiD0

.Lm�i .Z//ri Œ1=p�:(4.9)

The Atiyah-Hirzebruch spectral sequence converging to Hm.BZnIL.Z// collap-ses, since BZn is Tn and therefore one can compute Hm.BZnIL.Z// directly.Hence we obtain a filtration of ZŒZ=p�-modules

0 D FnC1;m�n�1 � Fn;m�n � � � � � F1;m�1 � F0;m D Hm.BZnIL.Z//

together with exact sequences of ZŒZ=p�-modules

0! FiC1;m�i�1 ! Fi;m�iq�! Hi

�BZn�

�˝Z L

h�1i

m�i .Z/! 0;


which splits as short exact sequence of Z-modules. Let sW Hi .BZn�/˝ZLm�i .Z/!Fi;m�i be a Z-map with q ı s D id. Then

zsW Hi .BZn�/˝Z Lm�i .Z/! Fi;m�i ; x 7!Xg2Z=p

g � s.g�1 � x/;

is a ZŒZ=p�-map with q ı zs D p � id.We obtain an exact sequence of ZŒ1=p�-modules,

0! .FiC1;n�iC1/Z=pŒ1=p�! .Fi;n�i /

Z=pŒ1=p�

qZ=pŒ1=p��! Hi .BZn�/

Z=pŒ1=p�˝Z Lm�i .Z/! 0:

Since qZ=pŒ1=p�ızsZ=pŒ1=p� is the automorphism p � id ofHi .BZn�/Z=pŒ1=p�˝Z

Lm�i .Z/, this short exact sequence of ZŒ1=p�-modules splits and we obtain anZŒ1=p�-isomorphism

.Fi;m�i /Z=pŒ1=p�

Š .FiC1;m�i�1/Z=pŒ1=p�˚

�Hi .BZn�/

Z=pŒ1=p�˝Z Lm�i .Z/�:

This implies by induction over i that there is an isomorphism of ZŒ1=p�-modules

Hm.BZnIL.Z//Z=pŒ1=p� ŠnMiD0

Hi .BZn�/Z=pŒ1=p�˝Z Lm�i .Z/:

Since Hi .BZn�/Z=p Š Zri , the claim follows. This finishes the proof of Theo-

rem 4.4. �

4.3 Arbitrary DecorationsFinally, we extend Theorem 4.4 to all decorations. Recall we abbreviate Lhii.Z/

by L.Z/, which is justified by (4.3).

THEOREM 4.5 (Computation ofLhii.Z�/). Let i 2 f2; 1; 0;�1;�2; : : : gtf�1g.Then:

(1) The assembly map

Ahiim W H�m.E�IL

hii/Š�! H�

m. � ILhii/ D Lhiim .Z�/

is an isomorphism for m 2 Z.(2) For m 2 Z there is an exact sequence

0!M.P /2P

zLhiim .ZP /! Lhiim .Z�/ˇhiim��! Hm.B�IL.Z//! 0

that splits after inverting p. The first map is given by the various inclusionsP ! � , and ˇhiim is defined by the composite

ˇhiim W Lhiim .Z�/

.Ahiim /�1

��! H�m.E�IL

hii/ind�!f1g��! Hm.B�IL.Z//:


(3) For m 2 Z the maps

Ls2mC1.Z�/Š�! L

h�1i

2mC1.Z�/;

ˇhii2mC1W L

s2mC1.Z�/

Š�! H2mC1.B�IL.Z//;

are bijective and transferring to the finite index subgroup Zn of � inducesan isomorphism

Ls2mC1.Z�/Š�! L2mC1.ZŒZ

n�/Z=p:

(4) We have

Lsm.Z�/ Š

(Zp

k.p�1/=2 ˚�Ln

iD0Lm�i .Z/ri�; m even;Ln

iD0Lm�i .Z/ri ; m odd:

PROOF. (1) and (2). Since Whi .P / D 0 because of [8] and Whi .�/ D 0 byTheorem 3.2 for i � �2, we obtain from (4.1) for i � �1, m 2 Z, isomorphisms

Lh�1im .ZP /Š�! Lhi�1im .ZP /

Š�! Lhiim .ZP /;

Lh�1im .Z�/Š�! Lhi�1im .Z�/

Š�! Lhiim .Z�/;

H�m.E�IL

h�1i/Š�! H�

m.E�ILhi�1i/

Š�! H�

m.E�ILhii/:

This together with Theorem 4.4 implies that assertions (1) and (2) are true fori 2 f�1;�2; : : : g t f�1g.

It remains to prove assertions (1) and (2) for i D 0; 1; 2, which we will do byinduction on i . So we want to show for i � 2 that assertions (1) and (2) are truefor i if they are true for i�1. Consider the commutative diagram in Figure 4.1. Theleft vertical column is the direct sum over P of the Rothenberg sequences (4.1) as-sociated to P . The middle column is the Rothenberg sequences (4.1) associated to� . The isomorphism in the right column come from (4.3). Notice that all columnsare exact. In each row the left arrow comes from the various inclusions P ! � .The rows involving the Tate cohomology are exact sequences

0!M.P /2P

�Hm.Z=2IWhi�1.P //! �Hm.Z=2IWhi�1.�//! 0! 0:

The sequences for the decorations hi � 1i are short exact sequences by inductionhypothesis

0!M.P /2P

zLhi�1im .ZP /! Lhi�1im .Z�/ˇhi�1im��! Hm.B�IL.Z//! 0:


:::

��

:::

��

:::

Š

��L.P /2P

zLhi�1im .ZP / //

��

Lhi�1im .Z�/

ˇhi�1im

//

��

Hm.B�ILhi�1i.Z//

��L.P /2P

�Hm.Z=2IWhi�1.P //Š//

��

�Hm.Z=2IWhi�1.�// //

��

0

��L.P /2P

zLhiim�1.ZP /

//

��

Lhiim�1.Z�/

x̌hiim

//

��

Hm�1.B�ILhii.Z//

Š

��L.P /2P

zLhi�1im�1 .ZP /

//

��

Lhi�1im�1 .Z�/

ˇhi�1im

//

��

Hm�1.B�ILhi�1i.Z//

��L.P /2P

�Hm�1.Z=2IWhi�1.P //Š//

��

�Hm�1.Z=2IWhi�1.�// //

��

0

��

::::::

:::

FIGURE 4.1. Rotherberg exact sequences.

The map x̌hiim W Lhiim .Z�/ ! Hm.B�IL.Z// is defined so that the diagram com-

mutes. The remaining columns yield short exact sequences

0!M.P /2P

zLhiim .ZP /! Lhiim .Z�/x̌hiim��! Hm.B�IL.Z//! 0:(4.10)

Lemma 3.1 shows that there is a long exact sequence

� � � ! HmC1.B�IL.Z//!M.P /2P

zLhiim .ZP /! H�m.E�IL

hii/

! Hm.B�IL.Z//!M.P /2P

zLhiim�1.ZP /! � � �

and that the ZŒ1=p�-map

H�m.E�IL

hii/Œ1=p�! Hm.B�IL.Z//Œ1=p�

is split surjective. We have already shown (see (4.10)) that the compositeM.P /2P


hii/! Lhiim .Z�/


is injective. Hence the long exact sequence reduces to short exact sequences

0!M.P /2P


hii/! Hm.B�IL.Z//! 0

which split after inverting p. We obtain the following commutative diagram:

0 //L.P /2P

zLhiim .ZP / //

id��

H�m.E�ILhii/ //

Ahiim��

Hm.B�IL.Z// //

id

��

0

0 //L.P /2P

zLhiim .ZP / // L

hiim .Z�/ // Hm.B�IL.Z// // 0:

Since the rows are exact and the first and third vertical arrows are bijective, themiddle arrow is bijective by the Five Lemma. This finishes the proof of asser-tions (1).

A direct computation shows that the map x̌hiim agrees with the map ˇhiim . Nowassertion (2) follows from (4.10).

(3) The following isomorphism is due independently to Bak [3] and Wall [35,cor. 2.4.3]:

zLsm.ZP / Š

(Z.p�1/=2; m even;0; m odd:

(4.11)

Hence assertion (2) implies that we obtain an isomorphism

Ls2mC1.Z�/Š�! H2mC1.B�IL.Z//:

The following diagram commutes:

Ls2mC1.Z�/

��

// H2mC1.B�IL.Z//

id��

Lh�1i

2mC1.Z�/// H2mC1.B�IL.Z//:

The lower horizontal arrow is an isomorphism by Theorem 4.4 (1), and the rightvertical arrow is an isomorphism by (4.3). Hence the map

Ls2mC1.Z�/Š�! L

h�1i

2mC1.Z�/

is an isomorphism. The following diagram commutes

Ls2mC1.Z�/

��

// Ls2mC1.ZŒZn�/Z=p

��

Lh�1i

2mC1.Z�/// Lh�1i

2mC1.ZŒZn�/Z=p:


The lower horizontal arrow is an isomorphism by Theorem 4.4 (1). Since for i � 1we have Whi .Zn/ D 0, the right vertical arrow is an isomorphism by the Rothen-berg sequence (4.1). Hence the upper horizontal arrow is an isomorphism. Thisfinishes the proof of assertion (3).

(4) It suffices to prove the claim after inverting 2 and after inverting p. If weinvert 2, the natural comparison maps between the various decorated L-groupsare isomorphisms by the Rothenberg sequence (4.1), and the claim follows fromTheorem 4.4 (3). If we invert p, the claim follows from assertion (2), Lemma 4.3,and isomorphisms (4.2), (4.9), and (4.11). �

5 Structure SetsGiven a closed oriented m-dimensional manifold N , we denote its geometric

simple structure set by S geo;s.N /. An element is represented by a simple ho-motopy equivalence N 0 ! N with a closed manifold N 0 as source and N astarget. Two such maps g0W N 0 ! N and g00W N 00 ! N define the same element inS geo;s.N / if and only if there is a homeomorphism uW N 0 ! N 00 such that g00 ı uand g0 are homotopic. This structure set appears in the geometric surgery exactsequence due to Browder, Novikov, Sullivan, and Wall; see [36, theorem 10.3]and [22, chap. 5], valid when m D dimN � 5:

(5.1) � � � ! N .N � .D1; S0//! LsmC1.ZŒ�1.N /�/! S geo;s.N /

! N .N /! Lsm.ZŒ�1.N /�/:

In the following we abbreviate

L WD Ls D Lh2iW GROUPOIDS! SPECTRA

so that we have �m.L.G=H// D Lsm.ZH/ for a group G and subgroup H � G.Given any CW -complex X , there is an exact algebraic surgery sequence of

abelian groups

(5.2) � � ��mC2.X/��! HmC1.X IL.Z//

AmC1.X/��! LsmC1.ZŒ�1.X/�/

�mC1.X/��! S

per;smC1.X/

�mC1.X/��! Hm.X IL.Z//

Am.X/��! Lsm.ZŒ�1.X/�/

�m.X/��! � � � ;

natural in X . There is also a 1-connective version of the sequence (5.2)

(5.3) � � ��h1imC2

.X/

��! HmC1.X ILh1i/Ah1imC1

.X/

��! LsmC1.ZŒ�1.X/�/

�h1imC1

.X/

��! S h1i;smC1 .X/�h1imC1

.X/

��! Hm.X ILh1i/

Ah1im .X/��! Lsm.ZŒ�1.X/�/

�h1im .X/��! � � � ;


where pW Lh1i ! L is the 1-connective cover of the L-theory spectrum L WDL.Z/. Here �i .p/W �i .Lh1i/ ! �i .L.Z// is an isomorphism for i � 1 and�i .Lh1i/ D 0 for i � 0.

The algebraic surgery exact sequences can be constructed in two ways. It canbe constructed by defining the structure groups to be the homotopy groups of thecofiber of a spectrum-level assembly map (defined, for example, in [11, example5.5]) or at the level of representatives (see the quadratic structure group Sm.Z; X/appearing in [32, def. 14.6, p. 148]). Using the second definition, Ranicki identi-fied the geometric surgery sequence (5.1) with the 1-connective algebraic surgerysequence (5.3) truncated at Ls5.ZŒ�1.X/�, see [32, theorem 18.5, p. 198] and [20].In particular, we get an identification

(5.4) S geo;s.N / Š S h1i;snClC1

.N /

and a map

j.N /W S geo;s.N / Š S h1i;snClC1

.N /! Sper;snClC1

.N /(5.5)

from the canonical map pW Lh1i ! L.

6 Periodic Simple Structure Set of BP for a Finite p-group

In this section we compute the periodic simple structure groups Sper;s� .BP / for

a finite p-group P and p an odd prime.Let k be a nonzero integer. A map of abelian groups f W A ! B is a 1=k-

equivalence if f ˝ idW A˝ZŒ1=k�! B˝ZŒ1=k� is an isomorphism. An abeliangroup A is 1=k-local if the map A˝Z! A˝ZŒ1=k� is an isomorphism. A mapA ! B is a 1=k-localization if it is a 1=k-equivalence and B is 1=k-local. If kand l are relatively prime integers, then A is 1=k-local if and only if A˝ ZŒ1=l�is 1=k-local.

THEOREM 6.1 (Periodic structure set of BP for a finite p-group P ). Let p be anodd prime and P be a finite p-group. Let ˛.P IC/ be the number of irreduciblereal representations of P of complex type.

(1) zLsm.ZP / Š

(Z˛.P IC/; m even;0; m odd;

where zLsm.ZP / is the cokernel of the map Lsm.Z/! Lsm.ZP / induced bythe inclusion 1! P .

(2) The homomorphism ��.BP /W Ls�.ZP / ! S

per;s� .BP / induces a 1=p-lo-

calization z��.BP /W zLs�.ZP /! Sper;s� .BP /.

PROOF.(1) This is a result due independently to Bak and Wall; see [3] and [35, cor. 2.4.3].


(2) The assembly map A�. � /W H�. � IL.Z// ! Ls�.Z/ is an isomorphism,essentially by definition. Thus there is a reduced algebraic surgery exact sequence

(6.1) � � �z�mC1.X/��! zHm.X IL.Z//

zAm.X/��! zLsm.ZŒ�1.X/�/

z�m.X/��! S

per;sm .X/

z�m.X/��! zHm�1.X IL.Z//

zAm�1.X/��! zLsm�1.ZŒ�1.X/�/

z�m�1.X/��! � � �

and

(6.2) � � �z�h1imC1

.X/

��! zHm.X ILh1i/zAh1im .X/��! zLsm.ZŒ�1.X/�/

z�h1im .X/��! S geo;s.X/

z�h1im .X/��! zHm�1.X ILh1i/

zAh1im�1.X/��! zLsm�1.ZŒ�1.X/�/

z�m�1.X/��! � � � :

Using the Atiyah-Hirzebruch spectral sequence and the Z-flatness of ZŒ1=p�,one sees that zH�.BP IL.Z//˝ ZŒ1=p� D 0. Using the flatness of ZŒ1=p� again itfollows that z��.BP / is a 1=p-equivalence. Thus it suffices to show that S per;s

� .BP /

is 1=p-local.We will express the structure set in terms of an equivariant homology theory.

We get from [9, theorem B.1] an isomorphism

Sper;sm .X/ Š HG

m .zX ! � IL.Z//

for a connected CW-complex X with fundamental group G. Thus it suffices toshow that HP

� .EP ! � IL.Z// is 1=p-local. Since p is odd, it remains to showthat HP

� .EP ! � IL.Z//Œ1=2� is 1=p-local since p is odd. Because of Theo-rem 4.2 it is enough to show that KO�.EP ! � IL.Z//Œ1=2� is 1=p-local.

The composite of complexification c with the forgetful map r

KOc�! K

r�! KO

is multiplication by 2, so

KOPm .EP ! � /Œ1=2� is a summand of KPm .EP ! � /Œ1=2�:

Since a summand of a 1=p-local abelian group is 1=p-local, it suffices to show thatKP� .EP ! � / is 1=p-local. Hence the proof of Theorem 6.1 is finished after wehave proved the next result. �

LEMMA 6.2. Let p be a prime and P be a p-group. Let ˛.P IC/ be the numberof irreducible complex representations of P .

(1) zKPm . � / Š

(Z˛.P IC/�1; m even;0; m odd:

(2) The map zKP� . � / �! KP� .EP ! � / is a 1=p-localization.


PROOF.(1) Let R.P / be the complex representation ring. Note that equivariant K-

cohomology and equivariant K-homology are 2-periodic,

K0P . � / D R.P /; KP0 . � / D HomZ.R.P /;Z/; K1P . � / D 0 D KP1 . � /:

(2) Our method for computing the relative equivariant K-homology will be toapply a universal coefficient theorem. This universal coefficient theorem appliesonly to finite complexes. Hence we take models of BP and EP whose n-skeletonsBP n andEP n have a finite number of cells. For each nwe have the exact sequence

(6.3) 0! K1P .EPn/! K0P .EP

n! � /! K0P . � /

! K0P .EPn/! K1P .EP

n! � /! 0:

It induces an exact sequence of pro-Z-modules indexed by n D 0; 1; 2; 3 : : :

(6.4) 0! fK1P .EPn/g ! fK0P .EP

n! � /g ! fK0P . � /g

! fK0P .EPn/g ! fK1P .EP

n! � /g ! 0;

where fK0P . � /g is a constant pro-Z-module. For an introduction to the abeliancategory of pro-Z-modules, see [1]. By the Atiyah-Segal completion theorem [1],there is an isomorphism of pro-Z-modules fK1P .EP

n/g Š f0g and a commutativediagram of pro-Z-modules,

fR.P /g //

Š

��

fR.P /=Ing

Š

��

fK0P . � /g// fK0P .EP

n/g;

where I D ker.R.P / ! Z/ is the augmentation ideal of the representation ring.It follows that the sequence (6.4) can be identified with

0! f0g ! fIng ! fR.P /g ! fR.P /=Ing ! f0g ! 0;

and in particular,˚K1P .EP

n! � /

D f0g;

˚K0P .EP

n! � /

D fIng:

A universal coefficient theorem for KP� proven in [18] states that for an equi-variant map X ! Y of finite P -CW-complexes, there is a short exact sequence

0! ExtZ.K�C1P .X ! Y /;Z/! KP� .X ! Y /

! HomZ.K�P .X ! Y /;Z/! 0:


It is an algebraic fact that fIng D fpnI g and, setting I� D HomZ.I;Z/ � I�˝Q,

that .pnI /� D p�nI�. Thus

KP0 .EP ! � / Š colimn!1

KP0 .EPn! � /

Š colimn!1

HomZ.K0P .EP

n! � /;Z/

Š colimn!1

.pnI /�

Š I� ˝Z ZŒ1=p�:

The map zKP0 . � / ! KP0 .EP ! � / can be identified with the inclusion I� DI� ˝ Z ! I� ˝ ZŒ1=p�. Similar reasoning shows that KP1 .EP ! � / D 0.Hence Lemma 6.2 follows. �

7 Periodic Simple Structure Set of B�

In this section we compute the periodic simple structure set of B� . Recall thatP is the set of conjugacy classes of subgroups of � of order p and that the orderof P is pk .

THEOREM 7.1 (Computation of S per;s.B�/).

(1) The map M.P /2P

Sper;sm .BP /

Š�! S

per;sm .B�/

induced by the various inclusions P ! � is for all m 2 Z an isomorphism.We get

Sper;sm .B�/ Š

(.ZŒ1=p�/p

k �.p�1/=2 m odd;0 m even;

(2) Let prW � ! �ab D �=Œ�; �� be the projection onto the abelianization of � .Let S

per;sm .Bpr/W S per;s

m .B�/ ! Sper;sm .�ab/ be the induced map. Recall

�ab Š .Z=p/kC1. For every .P / 2 P , let P 0 � �ab be the image of Punder pr. Let

resP0

�abW S

per;sm .B�ab/! S

per;sm .BP 0/

be the map induced by transfer to the subgroup P 0 � �ab.Then the mapY.P /2P

resP0

�abıS

per;sm .Bpr/W S per;s

m .B�/Š�!

Y.P /2P

Sper;sm .BP 0/

is an isomorphism.


PROOF.(1) By [9, theorem B.1], we have the identifications

Sper;sm .B�/ D H�

m.E� ! � ILs/;

Sper;sm .BP / D HP

m .EP ! � ILs/:

Theorem 4.5(1) implies

H�� .E� ! E�ILs/ D H�

� .E� ! � ILs/:

We get for m 2 Z from the induction structure (see [23, sec. 1]) isomorphisms

(7.1) H f1gm .BP ILs/Š � HP

m .EP ILs/Š�! H�

m.� �P EP ILs/:

We conclude from the �-pushout (3.1) and the identification (7.1),M.P /2P

HP� .EP ! � IL

s/ D H�� .E� ! E�ILs/:

Hence M.P /2P

Sper;sm .BP /

Š�! S

per;sm .B�/

is an isomorphism. Assertion (1) follows from Theorem 6.1 and Lemma 1.1(4).(2) Because of assertion (1) it suffices to show that the compositeM.P /2P

Sper;sm .BP /

Š�! S

per;sm .B�/Q

.P/2P resP0

�abı S

per;sm .Bpr/

��!

Y.P /2P

Sper;sm .BP 0/

is an isomorphism. We conclude from Theorem 6.1(2) that it suffices to show thatthe compositeM.P /2P

zLsm.ZP /! Lsm.Z�/pr��! zLsm.ZŒ�ab�/

Q.P/2P resP

0

�ab��!

Y.P /2P

zLsm.ZP0/

is an isomorphism after inverting p, where the first map is given by inductionwith the various inclusions P ! � , the second by induction with prW � ! �ab,and the third is the product over the various transfer homomorphisms resP

0

�ab. This

composite agrees with the composite

M.P /2P

zLsm.ZP0/

�Q.Q/2P resQ

0

�ab

�ı

�L.P/2P ind�ab

P 0

��!

Y.Q/2P

zLsm.ZQ0/:

Hence it suffices to show that for .P /; .Q/ 2P

resQ0

�abı ind�ab

P 0 WzLsm.ZP

0/! zLsm.ZQ0/


is trivial for .P / 6D .Q/ and pm � id for .P / D .Q/. Notice that P 0 D Q0 ,

.P / D .Q/ holds for .P /; .Q/ 2 P by Lemma 1.1, items (3) and (7). By thedouble coset formula the composite

res�abQ0 ı ind�ab

P 0 W Lsm.ZP

0/! Lsm.ZQ0/

factorizes through Lsm.Z/ if P 0 6D Q0, and isP�ab=P

id if P 0 D Q0. Since �ab=P

contains pk elements by Lemma 1.1(3) and (7), Theorem 7.1 follows. �

The splitting of the periodic structure set appearing in Theorem 7.1(1) has al-ready been established in [26, theorem 2.12].

8 Periodic Simple Structure Set of M

In this section we compute the periodic simple structure set Sper;snClC1

.M/ ofM .Recall that M D Tn

� �Z=p Sl and �1.M/ D � D Zn �� Z=p. Let f W M ! B�

be a classifying map for the universal covering of M .

THEOREM 8.1 (Periodic simple structure set of M ). There is a homomorphism

� W Sper;snClC1

.M/! Hn.TnIL.Z//Z=p

such that the following holds:(1) The map

� �Sper;snClC1

.f /W Sper;snClC1

.M/! Hn.TnIL.Z//Z=p �S

per;snClC1

.B�/

is injective.(2) The cokernel of � is a finite abelian p-group.(3) Consider the composite

�WM.P /2P

zLsnClC1.ZP /!zLsnClC1.Z�/

z�nClC1.M/��! S

per;snClC1

.M/

where the first map is given by induction with the various inclusions P ! �

and z�nClC1.M/ comes from (6.1).Then � is injective, the image of � is contained in the kernel of � , and

ker.�/=im.�/ is a finite abelian p-group.(4) After inverting p we obtain an isomorphism�� S

per;snClC1

.f /�Œ1=p�W S

per;snClC1

.M/Œ1=p�

! Hn.TnIL.Z//Z=pŒ1=p� �S

per;snClC1

.B�/Œ1=p�:

(5) As an abelian group we have

Sper;snClC1

.M/ Š Zpk.p�1/=2

˚

nMiD0

Ln�i .Z/ri ;

where the numbers ri are defined in (4.6).


Remark 8.2. There are several different points of views on the codomain of � .Indeed, there are isomorphisms

Ln.ZŒZn�/Z=p Š Hn.T

nIL.Z//Z=p Š

nMiD0

.Hi .Tn/Z=p ˝ Ln�i .Z//

Š

nMiD0

Ln�i .Z/ri :

The first isomorphism is due to the Shaneson splitting/Farrell-Jones conjecture,the second isomorphism is due to the collapse of the Atiyah-Hirzebruch spectralsequence, and the last isomorphism comes from (4.6).

In the proof of Theorem 8.1 it will be convenient to define �W S per;snClC1

.M/ !

Ln.ZŒZn��/Z=p and then define the composite

� W Sper;snClC1

.M/��! Ln.ZŒZ

n��/

Z=pŠ Hn.T

nIL.Z//Z=p;

noting that ker� D ker � and cok� Š cok � .

PROOF OF ASSERTION (1) OF THEOREM 8.1. We will have proved assertion (1)once we accomplish the following two goals:

Goal (1) Construct the commutative diagram in Figure 8.1.Goal (2) Show that the induced maps

ker�S

per;snClC1

.f /�! ker

�HnCl.f IL.Z//

�! ker.E1l;n.f //

are isomorphisms.Once we accomplish these two goals, a diagram chase gives the proof of Theo-

rem 8.1(1).We now construct the commutative diagram of abelian groups (Figure 8.1).

Some explanations are in order. Here and elsewhere in the paper if A is a nontrivialZG-module andX is a freeG-space, we writeHG

i .X IA/ forHi .C�.X/˝ZGA/,and we also write Hi .GIA/ for HG

i .EGIA/.The symbol inc always stands for an obvious inclusion.Given a free Z=p-CW -complex X and any homology theory H�, there is a

Leray-Serre spectral sequence converging to HiCj .X �Z=p Tn� / whose E2-term

is HZ=pi .X IHj .Tn

� //. In particular, we have a spectral sequence

E2i;j D HZ=pi

�X IHj .T

n� IL.Z//

�H) HiCj .X �Z=p Tn

� IL.Z//:

The symbols Fl;n.M/, Fl;n.B�/,Erl;n.M/, andErl;n.B�/ denote the correspond-

ing filtration terms andEr -terms of the spectral sequences applied to the free Z=p-CW -complex X D Sl and X D EZ=p. This explains the third, fourth, and fifthrow in the diagram of Figure (8.1) except for the map gnCl , which we describebelow. In order to show the equality of the fourth, fifth, and the sixth row, we needto show the following:


Sper;snClC1

.M/S

per;snClC1

.f ///

�nClC1.M/

��

�

��

Sper;snClC1

.B�/

�nClC1.B�/

��

HnCl.M IL.Z//HnCl .f IL.Z//

// HnCl.B�IL.Z//

Fl;n.M/

incŠ

OO

Fl;n.f ///

pr

��

Fl;n.B�/ D .Hl.M IL.Z//! Hl.B�IL.Z///

pr

��

inc

OO

E1l;n.M/

E1l;n.f /

//

idŠ

��

E1l;n.B�/

idŠ

��

E2l;n.M/

E2l;n.f /

//

idŠ

��

E2l;n.B�/

idŠ

��

HZ=pl

�Sl IHn.Tn

� IL.Z//� gnCl

//

idŠ

��

HZ=pl

�EZ=pIHn.Tn

� IL.Z//�

idŠ

��

Hn.Tn� IL.Z//Z=p

gnCl//

An.Tn� /Z=pŠ

��

Hl�Z=pIHn.Tn

� IL.Z//�

idŠ

��

Lsn.ZŒZn��/

Z=pgnClı.An.Tn� /

Z=p/�1// Hl

�Z=pIHn.Tn

� IL.Z//�

FIGURE 8.1. Commutative surgery diagram.

LEMMA 8.3. All differentials in the following two spectral sequences vanish:

(1) E2i;j D Hi�Z=pIHj .Tn

� IL.Z//�H) HiCj .B�IL.Z//.

(2) E2i;j D HZ=pi

�Sl IHj .Tn

� IL.Z//�H) HiCj .M IL.Z//.

PROOF.(1) It suffices to show that all differentials vanish after inverting p and after

localizing at p. Since for i 6D 0

E2i;j Œ1=p� D Hi�Z=pIHj .T

n� IL.Z//

�Œ1=p� D 0;

this is obvious after inverting p.If we localize at p, we get a natural isomorphism of homology theories

KO�.�/.p/Š�! H�.�IL.Z//.p/;


since p is odd, by Sullivan’s KOŒ1=2�-orientation; see Theorem 4.2. Hence itsuffices to show that all the differentials of the Leray-Serre spectral sequence con-verging to KOiCj .B�/ with E2-term Hi

�Z=p;KOj .Tn

� /�

are trivial. The edgehomomorphism

H0�Z=pIKOm.T

n� /�D KOm.T

n� /˝ZŒZ=p� Z

Š�! KOm.B�/

is bijective for even m by [13, theorem 6.1(ii)]. Thus all differentials involvingEr0;m are trivial for m even. Hence it suffices to show

Hi�Z=pIKOj .T

n� /�.p/D 0 if i > 0; i C j � 0 mod 2:

Since there are natural transformation of homology theories i W KO� ! K� andr W K� ! KO� with r ı i D 2 � id, it suffices to show

Hi�Z=pIKj .T

n� /�.p/D 0 if i > 0; i C j � 0 mod 2:

This vanishing is explicitly given in the proof of [13, theorem 4.1 (ii)].(2) Let gW Sl ! EZ=p be the classifying map of the free Z=p-CW-complex

Sl . It induces a map of the spectral sequence of assertion (2) to the one of asser-tion (1). We know already that all the differentials of the latter one vanish. Theinduced maps on the E2-terms

E2i;j .M/ D HZ=pi

�Sl IHj .T

n� IL.Z//

�! E2i;j .B�/

D HZ=pi

�EZ=pIHj .T

n� IL.Z//

�(8.1)

are bijective for i � l � 1 and surjective for i D l since Sl ! EZ=p is l-connected. Since Sl is l-dimensional, we have

E2i;j .M/ D 0 if i � l C 1:

This finishes the proof of Lemma 8.3. �

Let

gnCl W HZ=pl

�Sl IHn.T

n� IL.Z//

�! H

Z=pl

�EZ=pIHn.T

n� IL.Z//

�be the map induced by the classifying map gW Sl ! EZ=p.

For an appropriate choice of generator t 2 Z=p the chain ZŒZ=p�-chain com-plex of Sl is ZŒZ=p�-chain homotopy equivalent to the l-dimensional ZŒZ=p�-chain complex

� � � ! 0! ZŒZ=p�t�1��! ZŒZ=p�

N�! � � �

N�! ZŒZ=p�

t�1��! ZŒZ=p�! 0! � � � :

Thus we obtain an identification

HZ=pl

�Sl IHn.T

n� IL.Z//

�D Hn.T

n� IL.Z//

Z=p:

The assembly map

An.Tn� /W Hn.T

n� IL.Z//

Š�! Ln.ZŒZ

n��/


is an isomorphism because of the Shaneson splitting (or the Farrell-Jones conjec-ture in L-theory for the group Zn) and the Rothenberg sequences since Wh.Zn/ D0 and zKm.ZŒZn�/ D 0 for m � 0, and is a map of ZŒZ=p�-modules because ofnaturality of the assembly map.

Define the map

�W Sper;snClC1

.M/! Ln.ZŒZn��/

Z=p(8.2)

to be the composite of the vertical arrows in the diagram (8.1) from Sper;snClC1

.M/

to Lsn.ZŒZn��/

Z=p, namely, to be the composite An.T n� /Z=p ı pr ı�nClC1.M/.

We have explained all modules and maps in the diagram (8.1). One easily checksthat it commutes. Hence we have accomplished Goal (1).

LEMMA 8.4. Consider the following commutative square:

Sper;snClC1

.M/S

per;snClC1

.f ///

�nClC1.M/

��

Sper;snClC1

.B�/

�nClC1.B�/

��


// HnCl.B�IL.Z//

where the vertical maps come from (5.2).Then the vertical maps induce an isomorphism

ker�S

per;snClC1

.f /�Š�! ker

�HnCl.f IL.Z//

�:

PROOF. Consider the following commutative diagram of abelian groups, whichcomes from (5.2):

HnClC1.M IL.Z//HnClC1.f IL.Z//

//

AnClC1.M/

��

HnClC1.B�IL.Z//

AnClC1.B�/

��

LsnClC1

.Z�/id

//

�nClC1.M/

��

LsnClC1

.Z�/

�nClC1.B�/

��

Sper;snClC1

.M/S

per;snClC1

.f ///

�nClC1.M/

��

Sper;snClC1

.B�/

�nClC1.B�/

��


//

AnCl .M/

��

HnCl.B�IL.Z//

AnCl .B�/

��

LsnCl

.Z�/id

// LsnCl

.Z�/:

An easy diagram chase shows that it suffices to show that the map

HnClC1.f IL.Z//W HnClC1.M IL.Z//! HnClC1.B�IL.Z//


is surjective. We prove surjectivity after localizing at p and after inverting p.We begin with localizing at p. Since p is odd, for every CW -complex X and

m 2 Z there is a natural isomorphism (see Theorem 4.2),

Hm.X IL.Z//.p/Š�! KOm.X/.p/:

Hence it suffices to show that

KOnClC1.f /.p/W KOnClC1.M/.p/ ! KOnClC1.B�/.p/

is surjective. This follows from the following commutative diagram:

KOnClC1.Sl � BZn/ //

KOnClC1.pr/��

KOnClC1.M/

KOnClC1.f /

��

KOnClC1.BZn/ // KOnClC1.B�/

since the left vertical arrow is induced by the projection and is hence surjective andthe lower horizontal arrow is surjective, asKOk.BZn/! KOk.B�/ is surjectivefor k even by [13, theorem 6.1(ii)].

Next we invert p. Then a standard transfer argument (see, for example, propo-sition A.4 of [13]) shows that

Hm.BZn� IL.Z//˝ZŒZ=p� ZŒ1=p�Š�! Hm.B�IL.Z//Œ1=p�

is an isomorphism for all m 2 Z. It follows, as above, that HnClC1.M IL.Z//!HnClC1.B�IL.Z// is surjective, as desired. �

LEMMA 8.5. The composite

HnCl.M IL.Z//id�! Fl;n.M/

pr�! E1l;n.M/

induces a bijection

ker�HnCl.f IL.Z//W HnCl.M IL.Z//! HnCl.B�IL.Z//

�Š�! ker

�E1l;n.f /W E

1l;n.M/! E1l;n.B�/

�:

PROOF. We have shown in Lemma 8.3, items (1) and 2, that all the differentialsof the spectral sequence converging to HiCj .B�IL.Z// with

E2i;j D Hi�Z=pIHj .T

n� IL.Z//

�and all the differentials of the spectral sequence converging to HiCj .M IL.Z//with E2i;j D H

Z=pi

�Sl IHj .Tn

� IL.Z//�

vanish. The map

E2i;j .f /W E2i;j .M/! E2i;j .B�/

is bijective for i � l � 1 and all j since the map Sl ! EZ=p is l-connected.Hence the map

Fi;j .f /W Fi;j .M/! Fi;j .B�/


is bijective for i � l � 1 and all j . This implies

Fl�1;nC1.M/ \ ker�HnCl.f IL.Z//W

HnCl.M IL.Z//! HnCl.B�IL.Z//�D 0:

Since Sl is l-dimensional and hence HnCl.M IL.Z// D Fl;n.M/, Lemma 8.5follows. �

Lemma 8.4 and Lemma 8.5 give Goal (2) and hence complete the proof of as-sertion (1) of Theorem 8.1 �

PROOF OF ASSERTION (2) OF THEOREM 8.1. Consider the commutative dia-gram

(8.3)

Fl�1;nCl.M/Š

//

inc��

Fl�1;nCl.B�/

Š1=p

��

HnCl.M IL.Z//

AnCl .M/

��

HnCl .f IL.Z//// HnCl.B�IL.Z//

Š1=p

��AnCl .B�/uu

LsnCl

.Z�/Š

// HnCl.B�IL.Z//

where the two maps ending at LsnCl

.Z�/ are the assembly maps AnCl.M/ andAnCl.B�/ and the bottom horizontal isomorphism is discussed in Theorem 4.5(3).We have already explained that the top horizontal arrow is an isomorphism; see(8.1). The spectral sequence appearing in Lemma 8.3(1) implies that the verticalinclusion at the top right induces an isomorphism after tensoring with ZŒ1=p�. Weconclude from Lemma 4.3 thatHnCl.B�IL.Z//Œ1=p�! HnCl.B�IL.Z//Œ1=p�is an isomorphism.

Now the diagram (8.3) shows that

.AnCl.M/ ı inc/Œ1=p�W Fl�1;nC1.M/Œ1=p�Š�! LsnCl.Z�/Œ1=p�

is bijective.


We have the following commutative diagram:

(8.4)

0

��

Fl�1;nC1.M/

inc��

AnCl .M/ıinc

**

Sper;snClC1

.M/�nClC1.M/

//

�

**

HnCl.M IL.Z//AnCl .M/

//

AnCl .Tn� /Z=pıpr

��

LsnCl

.Z�/

Ln.ZŒZn��/Z=p

��

0

with exact row and exact column. An easy diagram chase proves that

�Œ1=p�W Sper;snClC1

.M/Œ1=p�! Ln.ZŒZn��/

Z=pŒ1=p�

is surjective, and we get

(8.5) ker.�Œ1=p�/ D ker.�nClC1.M/Œ1=p�/:

Since Lsn.ZŒZn��/ is a finitely generated abelian group (see Theorem 6.1(1), the

cokernel of� is a finite abelian p-group. Recalling that the cokernel of � is isomor-phic to the cokernel of�, this finishes the proof of assertion (2) of Theorem 8.1 �

PROOF OF ASSERTION 3 OF THEOREM 8.1. The composite of

�WM.P /2P

zLsnClC1.ZP /! Sper;snClC1

.M/

withS

per;snClC1

.f /W Sper;snClC1

.M/! Sper;snClC1

.B�/

becomes an isomorphism after inverting p since the composite is also the compos-ite of M

.P /2P

zLsnClC1.ZP /!M.P /2P

Sper;snClC1

.BP /! Sper;snClC1

.B�/;

where the first map is an isomorphism after inverting p by Theorem 6.1(2) andthe second map is an isomorphism by Theorem 7.1(1). Hence � is injective afterinverting p. Since zLs

nClC1.ZP / is a finitely generated free abelian group, � is

injective.Note

im.�/ � im.z�nClC1.M// D im.�nClC1.M// D ker.�nClC1.M// � ker�;

where the first equality holds since the simply connected surgery exact sequence isshort exact and the second equality holds because of the periodic surgery exact se-quence (5.2). We have shown in (8.5) that ker.�Œ1=p�/ D ker.�nClC1.M/Œ1=p�/


holds. Therefore ker.�/= ker �nClC1.M/ D ker.�/= im.�nClC1.M// is a p-torsion abelian group. Since S

per;snClC1

.M/ is a finitely generated abelian group be-cause of the surgery exact sequence (5.2), we conclude that ker.�/= im.�nClC1.M//

is a finite abelian p-group.Next we consider the following commutative diagram

0

��L.P /2P

zLsnClC1

.ZP /

��

�

**

HnClC1.B�IL.Z//AnClC1.B�/

//

HnClC1.i IL.Z// **

LsnClC1

.Z�/�nClC1.B�/

//

��

Sper;snClC1

.M/

HnClC1.B�IL.Z//

��

0

Here the exact column is taken from Theorem 4.5(2), the row is exact because ofthe surgery exact sequence (5.2) for M and the surjectivity of

HnClC1.M IL.Z//! HnClC1.B�IL.Z//;

which we have shown in the proof of Lemma 8.4. The map HnClC1.i IL.Z//induced by the obvious map i W B� ! B� is bijective after inverting p; seethe proof of part 8.1(2). Since there is a finite CW -model for B� and henceHnClC1.B�IL.Z// is finitely generated, the cokernel ofHnClC1.i IL.Z// is a fi-nite abelian p-group. Since this cokernel is isomorphic to im.�nClC1.B�//= im.�/by the commutative diagram above, we have shown that both

ker.�/= im.�nClC1.M// and im.�nClC1.B�//= im.�/

are finite abelian p-groups. Hence ker.�/=im.�/ is a finite abelian p-group. Re-calling that ker.�/ D ker.�/, this finishes the proof of assertion (3) of Theo-rem 8.1. �

PROOF OF ASSERTION (4) OF THEOREM 8.1. Because of assertion (1) it suf-fices to show that�

� �Sper;snClC1

.f /�Œ1=p�W S

per;snClC1

.M/Œ1=p�

! Ln.ZŒZn��/

Z=pŒ1=p� �Sper;snClC1

.B�/Œ1=p�


is surjective. We have already shown that �Œ1=p� is surjective and that its kernelagrees with the image of �Œ1=p�. We have already explained that S per;s

nClC1.f /Œ1=p�ı

�Œ1=p� is an isomorphism. Assertion (4) follows. �

PROOF OF ASSERTION (5). If we invert p, we conclude from assertion (4) us-ing Theorem 7.1(1), Theorem 6.1, and Lemma 1.1(4),

Sper;snClC1

.M/Œ1=p� Š Zpk.p�1/=2Œ1=p�˚ Ln.ZŒZ

n��/

Z=pŒ1=p�:

The abelian groups Sper;snClC1

.M/ and Zpk.p�1/=2 ˚ Ln.ZŒZn��/

Z=p are finitely

generated. The abelian group Zpk.p�1/=2˚Ln.ZŒZn��/

Z=p contains no p-torsion.We conclude from assertion (1) that the abelian group S

per;snClC1

.M/ contains nop-torsion. Hence

Sper;snClC1

.M/ Š Zpk.p�1/=2

˚ Ln.ZŒZn��/

Z=p:

It remains to prove

LnCl.ZŒZn��/

Z=pŠ

nMiD0

Ln�i .Z/ri :

This follows from Remark 8.2. This finishes the proof of Theorem 8.1. �

9 Geometric Simple Structure Set of M

In this section we compute the geometric simple structure set S geo;s.M/ ofM .For the rest of this paper, we simplify our notation and write L for the spectrumL.Z/ and Lh1i for its 1-connective cover.

In general, we have the following relationship between the periodic and thegeometric simple structure set.

LEMMA 9.1. Let N be a connected, oriented, closed m-dimensional manifold.Then we obtain an exact sequence

0! S geo;s.N /j.N/��! S

per;smC1.N /

@.N/��! Hm.N IL=Lh1i/

where the map j.N / is taken from (5.5), the map @.N / factors as

Sper;smC1.N /

�mC1.N/��! Hm.N IL/! Hm.N IL=Lh1i/;

and we have Hm.N IL=Lh1i/ Š Hm.N IL0.Z// Š Z.


PROOF. We have the following commutative diagram with exact columns:

HmC1.N ILh1i/ // //

��

HmC1.N IL/

��

LsmC1.Z�/id

//

��

LsmC1.Z�/

��

S geo;s.N /j.N/

//

��

Sper;smC1.N /

��

Hm.N ILh1i/ // //

��

Hm.N IL/

��

Lsm.Z�/id

// Lsm.Z�/;

where the columns are the exact sequences (5.2) and (5.3) using the identifica-tion (5.4) and the horizontal maps are given by the passage Lh1i ! L. LetL=Lh1i be the homotopy cofiber of the canonical map iW Lh1i ! L and denoteby prW L! L=Lh1i the canonical map of spectra. We get an exact sequence

HmC1.N ILh1i/! HmC1.N IL/! HmC1.N IL=Lh1i/! Hm.N ILh1i/! Hm.N IL/! Hm.N IL=Lh1i/:

Since �q.L=Lh1i/ vanishes for q � 1 and is L0.Z/ for q D 0, an easy spectralsequence argument shows that HmC1.N IL=Lh1i/ D 0. Thus the top horizontalmap is surjective. The fourth horizontal map is injective and its cokernel mapsinjectively to Hm.N IL=Lh1i/ Š Hm.N IL0.Z// Š Z. A diagram chase yieldsthe desired exact sequence. �

Recall thatM D Tn� �Z=pSl and � D Zn��Z=p is �1.M/. Let f W M ! B�

be a classifying map for the universal covering of M .

THEOREM 9.2 (Geometric simple structure set of M ). There is a homomorphism

�geoW S geo;s.M/! Hn.T

nILh1i/Z=p;

such that the following holds:(1) The map

�geo� .S

per;snClC1

.f / ı j.M//W

S geo;s.M/! Hn.TnILh1i/Z=p �S

per;snClC1

.B�/

is injective.(2) The cokernel of �geo is a finite abelian p-group.


(3) Consider the composite

�geoW

M.P /2P


z�h1i

nClC1.M/

��! S geo;s.M/

where the first map is given by induction with the various inclusions P ! �

and z�h1inClC1

.M/ comes from (6.2).Then �geo is injective, the image of �geo is contained in the kernel of �geo

and ker.�geo/= im.�geo/ is a finite abelian p-group.(4) After inverting p we obtain an isomorphism��geo�

�S

per;snClC1

.f / ı j.M/��Œ1=p�W S geo;s.M/Œ1=p�

! Hn.TnILh1i/Z=pŒ1=p� �S

per;snClC1

.B�/Œ1=p�:

(5) As an abelian group we have

S geo;s.M/ Š Zpk.p�1/=2

˚

n�1MiD0

Ln�i .Z/ri ;

where the numbers ri are defined in (4.6).

(6) The cokernel of the map @.M/W Sper;sdClC1

.M/�dClC1.M/��! HdCl.M IL=Lh1i/

appearing in Lemma 9.1 is a finite cyclic p-group.

Remark 9.3. There are several different points of views on the codomain of �geo

(see Remark 8.2). Indeed there are isomorphisms

Hn.TnILh1i/Z=p Š

n�1MiD0

.Hi .Tn/Z=p ˝ Ln�i .Z// Š

n�1MiD0

Ln�i .Z/ri :

PROOF. We first prove (4). We construct a commutative diagram whose columnsare exact after inverting p.

(9.1)

0

��

0

��

S geo;s.M/

��

�geo�.Sper;snClC1

.f /ıj.M//// Hn.Tn

� ILh1i/Z=p �Sper;snClC1

.B�/

��

Sper;snClC1

.M/

��

��Sper;snClC1

.f /

Š1=p

// Hn.Tn� IL/Z=p �S

per;snClC1

.B�/

��

HnCl.M IL=Lh1i/ ˛

Š1=p

// Hn.Tn� IL=Lh1i/Z=p:

The exact left column is due to Lemma 9.1. In the right column, the first nontrivialmap is induced by the product of the change of coefficients map Lh1i ! L with


the identity on the structure group and the second nontrivial map is induced by thecomposite of projection on the torus factor and the change of coefficients map L!L=Lh1i. The right column is exact after inverting p, since HnC1.Tn

� IL=Lh1i/ D0; thus we have the exact sequence of ZŒZ=p�-modules

0! Hn.Tn� ILh1i/! Hn.T

n� IL/! Hn.T

n� IL=Lh1i/:

Recall that all differentials of the spectral sequence

E2i;j D HZ=pi

�Sl IHj .T

n� IL/

�H) HiCj .M IL/

vanish by Lemma 8.3(2). This implies that for the spectral sequence

E2i;j D HZ=pi

�Sl IHj .T

n� ILh1i/

�H) HiCj .M ILh1i/

all differentials which end or start at place .s; t/ vanish, provided that sCt D nCl .Hence we can define analogously to � a map

(9.2) �geoW S geo;s.M/! Hn.T

n� ILh1i/

Z=p

such that the following diagram commutes

S geo;s.M/�geo

//

j.M/

��

Hn.Tn� ILh1i/Z=p

Hn.Tn� Ii/Z=p��

Sper;snClC1

.M/�

// Hn.Tn� IL/Z=p:

The homomorphism ˛ in diagram (9.1) is given by the edge isomorphism

HZ=pl

�Sl IHn.T

n� IL=Lh1i/

� Š�! HnCl.M IL=Lh1i/

at i D l and j D n of the spectral sequence

E2i;j D HZ=pi

�Sl IHj .T

n� IL=Lh1i/

�H) HiCj .M IL=Lh1i/;

the canonical map

HZ=pl

�Sl IHn.T

n� IL=Lh1i/

�! Hn.T

n� IL=Lh1i/Z=p;

which is an isomorphism after inverting p since l is odd, the isomorphism ofZŒZ=p�-modules

Hn.Tn� IL=Lh1i/

Š�! Hn.T

n� IL0.Z//;

which is the obvious edge homomorphism in the spectral sequence

E2i;j D Hi .TnI�j .L=Lh1i// H) HiCj .T

nIL=Lh1i/;

and the bijectivity of the inclusion

Hn.Tn� IL0.Z//

Z=p Š�! Hn.T

n� IL0.Z//:


The second horizontal arrow � �Sper;snClC1

.f / in diagram (9.1) is an isomorphismafter inverting p by Theorem 8.1(4). We leave it to the reader to check that thediagram (9.1) commutes. By the Five Lemma the upper horizontal arrow

�geo� .S

per;snClC1

.f / ı j.M//W S geo;s.M/! Hn.Tn� ILh1i/

Z=p�S

per;snClC1

.M/

is an isomorphism after inverting p. This finishes the proof of assertion (4).

(1) This follows from assertion (4) since S geo;s.M/ is a subgroup of Sper;snClC1

.M/

and Sper;snClC1

.M/ contains no p-torsion by Theorem 8.1(5).(2) This follows from assertion (4) since Hn.Tn

� ILh1i/Z=p is a finitely gener-ated abelian group.

(3) We get from the diagram (9.1) the following commutative diagram withexact columns:

0

��

// 0

��

// 0

��L.P /2P

zLsnClC1

.ZP /�geo

//

id��

S geo;s.M/�geo

//

j.M/

��

Hn.Tn� ILh1i/Z=p

Hn.Tn� Ii/Z=p��L

.P /2PzLsnClC1

.ZP /�

//

��

Sper;snClC1

.M/�

//

��

Hn.Tn� IL/Z=p

��

0 // HnCl.M IL=Lh1i/ ˛// Hn.Tn

� IL=Lh1i/Z=p:

We have already shown that the map ˛ is an isomorphism after inverting p, and itssource is an infinite cyclic group. Hence ˛ is injective. Theorem 8.1(3) shows that� is injective, the kernel of � contains im.�/, and ker.�/= im.�/ is a finite abelianp-group. An easy diagram chase shows that �geo is injective, the image of �geo iscontained in the kernel of �geo, and there is obvious isomorphism

ker.�geo/= im.�geo/Š�! ker.�/= im.�/:

(5) This follows from Theorem 8.1(5) and the diagram (9.1) by inspecting thedefinition of the various maps and the identification HnCl.M IL0.Z// Š L0.Z/.

(6) This follows from the diagram (9.1) after we have shown that the the mapHd .T

nIL/ ! Hd .TnIL=Lh1i/ is surjective. The latter claim follows from the

fact Tn is stably homotopy equivalent to a wedge of spheres. This finishes theproof of Theorem 9.2. �

10 Invariants for Detecting the Structure Set of M

Let G be a finite group; let R.G/ be its complex representation ring. LetzR.G/ D R.G/=hregi be the reduced regular representation ring. (AdditivelyR.G/ and zR.G/ are K0.CG/ and zK0.CG/, respectively.) For " D ˙1, let


R.G/" and zR.G/" be the subgroups invariant under V 7! "V . For example,zR.Z=p/ Š ZŒe2�i=p� and both zR.Z=p/C1 and zR.Z=p/�1 are free abelian ofrank .p � 1/=2 for p an odd prime.

Let V be a finitely generated RG-module and let

BW V � V ! R

be a G-equivariant "-symmetric form. Atiyah-Singer [2, sec. 6] define the G-signature signG.V; B/ 2 R.G/". If W is a compact 2d -dimensional orientedmanifold with a G-action, define its G-signature signG.W / to be the G-signatureof its intersection form.

This defines the multisignature maps

(10.1)signG W L

s2d .ZG/! R.G/.�1/

d

;

esignG W zLs2d .ZG/! zR.G/.�1/d :

LEMMA 10.1. The multisignature homomorphisms signG and esignG are ZŒ1=2�-isomorphisms.

PROOF. See Wall [36, theorems 13A.3 and 13A.4] or Ranicki [32, props. 22.14and 22.34]. �

When G is cyclic and odd order, according to [36, theorem 13A.4],esignG W zLs2d .ZG/! 4 zR.G/.�1/

d

is an isomorphism.

THEOREM 10.2 (Geometric structure set of M ). Let d D .n C l C 1/=2. Thereare injective maps

� � �W Sper;snClC1

.M/! Hn.TnIL/Z=p ˚

� M.P /2P

zR.P 0/.�1/d

Œ1=p��;

and

�geo� �geo

W S geo;s.M/! Hn.TnILh1i/Z=p ˚

� M.P /2P

zR.P 0/.�1/d

Œ1=p��:

The cokernels of these maps are trivial after tensoring with ZŒ1=2p�.

PROOF. We have already defined � and �geo in Theorems 8.1 and 9.2 respec-tively.

Let y� be the composite

(10.2) y�W S per;snClC1

.B�/

�Q.P/2P resP

0

�ab

�ıS

per;snClC1

.Bpr/��!

Y.P /2P

Sper;snClC1

.BP 0/

Š �

Y.P /2P

zLsnClC1.ZP0/Œ1=p�

Q.P/2P fsignP 0��!

Y.P /2P

zR.P 0/.�1/d

Œ1=p�;


where for .P / 2P the subgroup P 0 � �ab is the image of P under the projectionprW � ! �ab, the first map is the isomorphism taken from Theorem 7.1(2), thesecond map is given by the product of the inverse of the isomorphism appearing inTheorem 6.1(2), and the third map is the product of the homomorphisms definedin (10.1). Define � to be the composite

�W Sper;snClC1

.M/S

per;snClC1

.f /

��! Sper;snClC1

.B�/y��!

Y.P /2P

zR.P 0/.�1/d

Œ1=p�:

Note that all these maps are isomorphisms after tensoring with ZŒ1=2p�.We thus see that y�Œ1=2� is an isomorphism and, thus, since the domain of y� is

torsionfree by Theorem 7.1(1), y� is injective.The map � �S

per;snClC1

.f / is injective and an isomorphism after tensoring withZŒ1=p� by Theorem 8.1, so it follows that � � � D .id�y�/ ı .� �S

per;snClC1

.f // isinjective and is an isomorphism after tensoring with ZŒ1=2p�.

Define �geo D � ı j.M/ and note that j.M/ is injective by Lemma 9.1:

0

��

0

��

S geo;s.M/

j.M/

��

�geo��geo//

�Ln�1iD0 Ln�i .Z/

.ni /�Z=p

˚

�L.P /2P

zR.P 0/.�1/d

Œ1=p��

inc� id��

Sper;snClC1

.M/

��

��//

�LniD0Ln�i .Z/

.ni /�Z=p

˚

�L.P /2P

zR.P 0/.�1/d

Œ1=p��

��

cok.j.M//��

//

��

cok.inc� id/

��

0 0

The map � � � is the map induced on cokernels by the commutative square aboveit. The columns are short exact sequences so the snake lemma applies and yieldsthe following exact sequence:

0! ker.�geo� �geo/! ker.� � �/! ker.� � �/!

cok.�geo� �geo/! cok.� � �/! cok.� � �/! 0:

Since we have already shown that ker.� � �/ D 0 it follows that �geo � �geo isinjective. Since we have already shown that .� ��/Œ1=p� is an isomorphism, it fol-lows that both cok.� ��/Œ1=p� and cok.� � �/Œ1=p� vanish. Note that cok.j.M//

is an infinite cyclic group by Lemma 9.1 and Theorem 9.2 (6). Since the domainand codomain of � � � are infinite cyclic and the cokernel is p-torsion, its kernel


is trivial. Hence there is a short exact sequence

0! cok.�geo� �geo/! cok.� � �/! cok.� � �/! 0

where the middle term is p-torsion. Hence all the groups are p-torsion and .�geo�

�geo/Œ1=p� is an isomorphism as desired. �

We next interpret the detecting maps from Theorem 10.2 in terms of geometricinvariants. The first map �geo will be given by splitting invariants, coming fromsurgery obstructions along submanifolds and the second map �geo will be given by�-invariants that arise both in index theory for manifolds with boundary as wellas in the homeomorphism classification of homotopy lens spaces with odd orderfundamental group.

10.1 Splitting InvariantsLet X be a closed oriented manifold with dimension d � 5. Recall that in Sec-

tion 5 we mentioned an identification of the geometric surgery exact sequence withthe 1-connective algebraic surgery exact sequence. To discuss splitting invariantswe say precisely what this identification is. In particular, we discuss the bijectionN .X/! Hd .X ILh1i/.

Recall that a degree one normal map .f; zf / is a degree one map f W N d ! Xd

and a trivialization zf W TN ˚ f �� Š Rl for stable some bundle � over X . ThenN .X/ is the set of normal bordism classes of degree one normal maps to X .The map �W S geo;s.X/ ! N .X/ is given by considering a simple homotopyequivalence hW X 0 ! X as a degree one map and taking � D h�1��X 0 , where h�1

is a homotopy inverse of h and �X 0 is the normal bundle ofX 0 with respect to someembedding in Euclidean space. The map � W N .X/! Ls

d.ZŒ�1.X/�/ is given by

the surgery obstruction. In particular, a normal bordism class maps to zero if andonly if it has a representative given by a simple homotopy equivalence.

A Pontryagin-Thom construction gives a bijection

PT W N .X/Š�! ŒX;G=TOP �:

Sullivan/Quinn/Ranicki give a 4-fold periodicity,�4.Z�G=TOP / ' Z�G=TOP .The corresponding spectrum is homotopy equivalent to L.Z/ (which we have beenabbreviating L). It follows formally that the 0-space of the 1-connective cover Lh1iis homotopy equivalent to G=TOP and that ŒX;G=TOP � D H 0.X ILh1i/. Fur-thermore, Ranicki shows that closed, oriented, topological manifolds are orientedwith respect to these two spectra; hence there are Poincaré duality isomorphisms

PDW H i .X IL/Š�! Hd�i .X IL/ and PDW H i .X ILh1i/

Š�! Hd�i .X ILh1i/:

The following foundational theorem is due to Quinn and Ranicki; references are [32,theorem 18.5] and [20, prop. 14.8].


THEOREM 10.3. There is a commutative diagram, with vertical bijections,

� � � // LsdC1

.ZŒ�1.X/�/@// S geo;s.X/

s Š��

�// N .X/

t Š

��

�// Lsd.ZŒ�1.X/�/

� � � // LsdC1

.ZŒ�1.X/�/�h1i

dC1// S h1i;sdC1

.X/ // Hd .X ILh1i/Ah1i

d.X/// Lsd.ZŒ�1.X/�/;

where t is the composite

N .X/PT��! ŒX;G=TOP � D H 0.X ILh1i/

PD��! Hd .X ILh1i/:

The philosophy of splitting invariants is to detect N .X/ by submanifolds. LetY j be a closed, oriented submanifold of Xd . (We assume our submanifolds havea normal bundle so we can apply transversality.) The restriction map

resYX W N .Xd /! N .Y j /

is defined by representing a normal bordism class by a normal map .f; zf / wheref W N ! X is transverse to Y . Then resŒf; zf � is represented by the restricted mapf jW f �1Y ! Y with the trivialization Tf �1Y ˚ f j�.�.Y ,! X/ ˚ �jY / D

TN jf �1Y ˚ f j�� Š Rl .

It follows from the Pontragin-Thom construction that the map resYX is the com-posite

N .Xd /PT��! ŒX;G=TOP �

Œ�;G=TOP��! ŒY;G=TOP �

PT�1

��! N .Y /;

but we state this result slightly differently:

LEMMA 10.4. resYX is the composite

N .Xd /PT��! ŒX;G=TOP � D H 0.X ILh1i/

inc��! H 0.Y ILh1i/ D ŒY;G=TOP �

PT�1

��! N .Y j /:

DEFINITION 10.5 (Sullivan [33, 34]). A characteristic variety of Xd is a collec-tion of closed, oriented, connected, submanifolds fY ji g of Xd so that any simplehomotopy equivalence h W X 0 ! X that vanishes under the composite

N .Xd /res�!

Yi;j

N .Yji /

��!

Yi;j

Lj .ZŒ�1Yji �/

"�!

Yi;j

Lj .Z/

is homotopic to a homeomorphism.

Sullivan’s original example is that a characteristic variety for CP n is given byfCP j g with 0 < j < n. If a manifold satisfies topological rigidity [19] (forexample a sphere or a torus), then the empty set is a characteristic variety.

THEOREM 10.6. Choose a point � 2 Sl in the sphere.


(1) The following map is an isomorphism

inc� �.prTn� ıPD/W

H 0.Tn� Sl ILh1i/! H 0.Tn

� � ILh1i/˚HnCl.TnILh1i/:

(2) For a subset J � f1; 2; : : : ; ng, let T J � � � Tn � Sl be the obviousjJ j-dimensional submanifold. Then fT J � � j ¿ 6D J � f1; 2; : : : ; ngg isa characteristic variety for Tn � Sl .

(3) Let �geoW S geo;s.M/ ! Hn.TnILh1i/Z=p be the map defined in Theo-rem 9.2. Then �geo.h W N ! M/ D 0 if and only if the p-fold coverxh W N ! Tn � Sl is homotopic to a homeomorphism.

PROOF.(1) It is an exercise to show the analogue in ordinary homology

inc� �.prTn� ıPD/W H�.Tn

� Sl/Š�! H�.Tn

� � /˚HnCl��.Tn/:

The general case follows since the Atiyah-Hirzebruch spectral sequence collapsesand the L-spectrum Poincaré duality is compatible with classical Poincaré dualitysince the fundamental class ŒTn � Sl �L� 2 HnCl.T

n � Sl IL � / maps to the fun-damental class ŒTn�Sl � 2 HnCl.T

n�Sl/ �LHnCl��.T

n�Sl IL�.Z//. (Seesection 16, “The L-theory orientation of topology” in [32].)

(2) Let hW X 0 ! Tn � Sl be a simple homotopy equivalence. We first claimthat h is homotopic to a homeomorphism if and only if

PT .�.h// D 0 2 H 0.Tn� Sl ILh1i/:

Since HnClC1.Tn � Sl ILh1i/ ! HnClC1.TnILh1i/ is a (split) surjection, and

the Farrell-Jones conjecture for Zn, which is due to Shaneson in this case, impliesthatAh1i

nClC1.Tn/W HnClC1.T

nILh1i/! LnClC1.ZŒZn�/ is an isomorphism, the

composite

Ah1i

nClC1.Tn/ ıHnClC1.prTn ILh1i/ D A

h1i

nClC1.Tn� Sl/

is surjective. Hence the boundary map @ in the surgery exact sequence (see Theo-rem 10.3) is the trivial map and thus � is injective. Our first claim follows.

Our second claim is that the mapYJ

" ı � ı resTJ

Tn ı.PT /�1W H 0.Tn

ILh1i/!MJ

LjJ j.Z/

is an isomorphism where J runs over all nonempty subsets of f1; 2; : : : ; ng. Tosee that this is an isomorphism, we exhibit its inverse map, which sends the gen-erator E jJ j ! S jJ j of LjJ j.Z/ to PT ..T J #E jJ j/ � T J

0

! Tn/ where J 0 is thecomplement of J .

Our third claim is that for a simple homotopy equivalence h W X 0 ! Tn � Sl ,

(10.3) prTn�

�PD

�PT .�.h//

��D 0 2 HnCl.T

nILh1i/:


Indeed,

0 D �.�.h// by exactness of the surgeryexact sequence

D Ah1i

nCl.Tn� Sl/.PD.PT .�.h//// by Theorem 10.3

D Ah1i

nCl.Tn/ prTn�.PD.PT .�.h//// by naturality of the assembly map.

Since Ah1inCl

.Tn/ is an isomorphism, (10.3) follows.Now back to the proof of (2). Let hW X 0 ! Tn � Sl be a simple homotopy

equivalence so that all splitting invariants along T J � � vanish. Our goal is toshow that h is homotopic to a homeomorphism. By our first claim, part (1), andour third claim, it suffices to show that

inc�.PT .�.h/// D 0 2 H 0.Tn� � ILh1i/:

By Lemma 10.4, this is equivalent to showing

PT�resTn��

Tn�Sl .�.h//�D 0 2 H 0.Tn

� � ILh1i/By our second claim, this is equivalent to showing that for any nonempty subset Jof f1; 2; : : : ; ; ng,

".�.resTJ��

Tn�Sl .�.h//// D 0 2 LjJ j.Z/:

It follows that fT J � � g is a characteristic variety for Tn � Sl .(3) The composite �geoW S geo;s.M/ ! Hn.TnILh1i/Z=p with the inclusion

Hn.TnILh1i/Z=p ,! Hn.TnILh1i/ can be identified with the composite

S geo;s.M/tr�! S geo;s.Tn

� Sl/��! N .Tn

� Sl/

res�! N .Tn

� � /PT��! H 0.Tn

ILh1i/PD��! Hn.T

nILh1i/;

where the transfer map tr is passing to the p-fold cover. By Theorem 10.6(1) andLemma 10.4, the result follows. �

DEFINITION 10.7. A map f W N ! X splits along a submanifold Y if f is homo-topic to a map g, transverse to Y so that g�1Y ! Y is a homotopy equivalence.The map f s-splits along Y , if, in addition, g�1Y ! Y is a simple homotopyequivalence.

Thus if fY ji g is a characteristic variety for X , then a simple homotopy equiva-lence is homotopic to a homeomorphism if and only if it splits along fY ji g.

We now mention the relationship between restriction and splitting invariants.The key result is due to Browder (see [37, sec. 4.3]).

THEOREM 10.8. Suppose hW X 0 ! X is a simple homotopy equivalence and thatY j is a submanifold of Xd with codimension d � j � 3 and dimension j � 5.Then h is splitable along Y if and only if �.resYX .h// D 0 2 L

sj .ZŒ�1Y �/.


10.2 Rho InvariantsDEFINITION 10.9. Let N be a closed, oriented, .2d � 1/-dimensional manifoldmapping to BG for a finite G. The �-invariant

�.N ! BG/ 2 zR.G/.�1/d

Œ1=jGj�

is defined by

�.N ! BG/ D1

k� signG.W / 2 zR.G/

.�1/d Œ1=jGj�;

where k is a power of jGj andW is a compact, oriented, 2d -dimensional manifoldwith orientation-preserving free G-action, whose boundary is k disjoint copies ofthe xN , the induced G-cover of N . This was given an analytic interpretation byAtiyah-Patodi-Singer. The �0-function

�0.N ! BG/W S geo;s.N /! R.G/.�1/d

Œ1=jGj�

is defined by

�0.N ! BG/.N 0 ! N/ D �.N 0 ! N ! BG/ � �.N ! BG/:

The �-invariant and the �0-function only depend on the induced homomorphism�1N ! G, or, equivalently, on the induced G-cover xN ! N . When �1N D G,we will write �.N / and �0.

The passage from R.G/.�1/d

Œ1=jGj� to zR.G/.�1/d

Œ1=jGj� ensures that the �-invariant is independent of the choices of k and W . For the definition of the �-invariant, see [2, remark after cor. 7.5] or [36, sec. 13B].

Here are two fundamental properties of the �0-function.

THEOREM 10.10. Let N ! BG be a map from a closed, oriented, .2d � 1/-dimensional manifold to the classifying space of a finite group. Let �W �1N ! G

be the induced map on fundamental groups. Recall that there is an identificationS geo;s.N / D S h1i;s

2d.N / (see (5.4)). In particular, the geometric structure set is

an abelian group.

(1) The map �0.N ! BG/W S geo;s.N / ! zR.G/.�1/d

Œ1=jGj� is a homomor-phism of abelian groups.

(2) Recall that Ls2d.ZŒ�1N�/ acts on S geo;s.N / (see [36, theorem 10.4]). For

x 2 Ls2d.Z�1N/ and y 2 S geo;s.N /,

�0.N ! BG/.x C y/

D esignG.zLs2d .�/.x//C �0.N ! BG/.y/ 2 zR.G/.�1/

d

Œ1=jGj�:

In particular, by taking y D idN , we have an equality of maps

�0.N ! BG/ ı @ D esignG ı zLs2d .�/W zLs2d .Z�1N/!

zR.G/.�1/d

Œ1=jGj�:


PROOF.(1) This is the main result of the paper [10] by Crowley and Macko.(2) The following is an easy consequence of the definition of the �-invariant:

If W 2d is a compact oriented manifold with a map to BG and if .@W ! BG/ D

.N 0 t �N 00 ! BG/, then �.N 0 ! BG/ D signG.W /C �.N00 ! BG/.

The action is implemented by such a W and the result follows. �

Let Ll be a homotopy lens space with fundamental group P Š Z=p for p anodd prime. Let d D .l C 1/=2. A homotopy lens space is the orbit space of a freeaction of Z=p on Sl ; equivalently, it is a closed manifold having the homotopytype of a lens space.

THEOREM 10.11. �0 W S geo;s.Ll/ ! zR.P /.�1/d

Œ1=p� Š ZŒ1=p�.p�1/=2 is aninjection, and is an isomorphism after tensoring with ZŒ1=2p�.

PROOF. We first show that S geo;s.Ll/ is isomorphic to Z.p�1/=2 and is in par-ticular torsion-free. The Atiyah-Hirzebruch spectral sequence in equivariant ho-mology (see [11, theorem 4.7]) shows that HP

lC1.Sl ! S1ILh1i/ is zero. The

long exact sequence of the triple Sl ! S1 ! � then shows that S h1i;slC1

.Ll/ !

S h1i;slC1

.BP / is injective. But the domain is S geo;s.Ll/, which is finitely gener-ated and has rank .p � 1/=2 by the surgery exact sequence, and the codomain isisomorphic to ZŒ1=p�.p�1/=2 by Theorem 6.1.

The multisignature map zsignG W zLslC1

.ZP /! zR.P /.�1/d

Œ1=p� is a ZŒ1=2p�-isomorphism by Lemma 10.1. We conclude from Theorem 10.10(2) that the com-

posite zLslC1

.ZP /@�! S geo;s.Ll/

�0

�! zR.P /.�1/d

Œ1=p� is a ZŒ1=2p�-isomorphismSince S geo;s.Ll/ is torsion-free of rank .p � 1/=2, the result follows. �

Remark 10.12. This gives a new proof of Wall’s result [36, chap. 14E] that thestructure set of a homotopy lens space is detected by the �0 invariant. He showedthat if Ll is a homotopy lens space with fundamental group Z=k for k odd, that�0W S geo;s.Ll/ ! zR.Z=k/.�1/

d

Œ1=k� is injective. This follows for k a primepower by our argument above, and, we believe, for all k odd by the techniques inour paper. In fact, Wall proved much more and identified the image of the map �0,which takes much more work.

Wall’s result that a structure set of lens spaces is detected by the �0-invariant andthe fact that splitting invariants detect the structure set of Tn�Sl provided a majorimpetus for our paper.

Recall that P is the set of conjugacy classes of subgroups of order p of � DZnÌ Z=p. Recall jPj D pk where n D k.p�1/. Let prW � ! �ab Š .Z=p/kC1

be the quotient map. For each .P / 2 P , let P 0 D pr.P /, let �P D pr�1.pr.P //,and note that ker pr ¨ �P ¨ � . Let MP ! M be the cover corresponding to the


subgroup �P . Consider the transfer map

res�P� W Sgeo;s.M/! S geo;s.MP /;

obtained by sending a simple homotopy equivalence N ! M to the coveringsimple homotopy equivalence NP !MP .

THEOREM 10.13 (Detection Theorem). An element h 2 S geo;s.M/ is the trivialelement if and only if �geo.h/ D 0 and �0.MP ! BP 0/.res�P� .h// D 0 holds forall .P / 2P .

PROOF. Obviously it suffices to show that the restriction of the homomorphismof abelian groups

�geoW S geo;s.M/!

Y.P /2P

zR.P 0/.�1/d

Œ1=p�

appearing in Theorem 10.2 to the kernel of homomorphism �geo appearing in The-orem 10.2 sends h to .�0.MP ! BP 0/.res�P� .h///.P /2P . The image of the map

�geoW

M.P /2P


z�h1i

nClC1.M/

��! S geo;s.M/

defined in Theorem 9.2(3) is contained in the kernel of �geo and has finite p-powerindex. The map �geo is a homomorphism of abelian groups by Theorem 10.10 (1).Since the image of �geo is a ZŒ1=p�-module, we conclude that it suffices to showthat the restriction of the homomorphism �geo to the image of the map �geo sends hto .�0.res�P� .h///.P /2P . Since im.�geo/ is contained in the image of �h1i

nClC1.M/ D

@, it suffices to show that restriction of the homomorphism �geo to the image of�h1i

nClC1.M/ D @ sends h to .�0.MP ! BP 0/.res�P� .h///.P /2P . This follows

from Theorem 10.10 (2). �

Now Theorem 0.2 follows directly from Theorems 10.2, 10.6, and 10.13.

EXAMPLE 10.14. Take p D 3, k D 1, n D 2, and the Z=3-action in Z2 given bythe .n1; n2/ 7! .�n2; n1 � n2/. Then Z2� is ZŒexp.2�=3/�, and we have

S geo;s.M/ Š Z3 ˚ Z=2:

There is one nontrivial splitting obstruction taking values in L2.Z/ Š Z=2, de-fined by making the corresponding map transversal to T 2 � � � T 2 � S3.

There are three conjugacy classes of subgroups of order p in � , and each yieldsa �-invariant type obstruction.


11 Appendix: Open QuestionsAs mentioned earlier, our inspiration, our muse, for this paper is Wall’s classi-

fication [36, chap. 14E] of homotopy lens spaces Ll with odd order fundamentalgroup. This classification is complete. It consists of the classification of homo-topy types of homotopy lens spaces, the classification of the simple homotopytypes within a homotopy type, the computation of the geometric structure set, andthe computation of the moduli space M .LL/ of homeomorphism classes of theclosed manifolds within a simple homotopy type. In the case of homotopy lensspaces, it is a bit easier to state if one considers the polarized homotopy type, fix-ing an orientation and an identification of the fundamental group with Z=k. Thenthe polarized homotopy type is given by the first k-invariant, lying in the group.Z=k/� � Z=k D H lC1.B.Z=k//. The simple homotopy types of polarizedlens spaces are given by the Reidemeister torsion �.Ll/ and Wall determines theset of possible values which occur. Fixing a simple homotopy type he showed, asmentioned earlier, that

�0W S geo;s.Ll/! zR.Z=k/.�1/.lC1/=2

is injective and he computed the image.From this, it is not difficult to compute the moduli space. For any closed man-

ifold X , let sAut.X/ be the group of homotopy classes of simple self-homotopyequivalences. Then sAut.X/ acts on S geo;s.X/ with orbit space the moduli space.In the case of a homotopy lens space, then sAut.X/ can be computed directlysince Ll is a skeleton of K.Z=k; 1/, or better yet, one can compute the action ofsAut.X/ on the set of k-invariants, Reidemeister torsions, and �0-invariants. Weomit the details.

The discussion above leads to several questions:(1) Can one describe the image of the injective map of Theorem 10.2

�geo� �geo

W S geo;s.M/! Hn.TnILh1i/Z=p ˚

� M.P /2P

zR.P 0/.�1/d

Œ1=p��‹

A first step is to compute the �-invariant of MP ! BP 0.(2) In the proof of the detection theorem 10.13 we show that �geo restricted to

the kernel of �geo is given by differences of �-invariants. Is this also true for�geo itself? In other words, does �geo D

QP2P �0.MP ! BP 0/ ı res�P� ?

(3) What can we say about the moduli space of homeomorphism classes ofmanifolds (simple) homotopy equivalent toM . Is it infinite? Can we com-pute action of sAut.M/ on the structure set? How does this group act onthe splitting and �-invariants? When is a self-homotopy equivalence ofM homotopic to a homeomorphism? Can we classify all self-homotopyequivalences of M , perhaps up to finite ambiguity? Does the subgroup ofself-homotopy equivalences ofM that are homotopic to a homeomorphismhave finite index in the group of self-homotopy equivalences of M ? Is a


self-homotopy equivalence of M determined by the induced map on thefundamental group up to finite ambiguity? Can we show that within thehomotopy type of M there are infinitely many mutually different homeo-morphism types?

(4) Is a homotopy equivalence hW N !M splitable along

pkf � g � Ll D .Tn/Z=p � Ll � Tn�Z=p Sl DM

if and only if h is a simple homotopy equivalence? (The if direction followsfrom Theorem 10.8 and equation (4.11).)

(5) Is a simple homotopy equivalence hW N ! M D Tn �Z=p Sl homo-topic to a homeomorphism if and only if xChW xCN ! Tn � Sl is splitablealong T J � � for all nonempty J � f1; 2; : : : ; ng and if h ' k where kj Wk�1..Tn/Z=p�Ll/! .Tn/Z=k�Ll is a homeomorphism? (An alternateconjecture is that a simple homotopy equivalence hW N ! M is homo-topic to a homeomorphism if and only if h ' k where kjW k�1..Tn/Z=p �

Ll/! .Tn/Z=p � Ll and kjW k�1.T J � � /! T J � � are homeomor-

phisms for all J .)

Acknowledgment. The first author was supported by National Science Foun-dation Grant DMS 1615056. The paper has been supported financially by the ERCAdvanced Grant “KL2MG-interactions” (no. 662400) of the second author grantedby the European Research Council and by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ2047/1, Projekt-ID 390685813. We wish to thank the referee for careful readingand helpful suggestions.

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JAMES F. DAVISDepartment of MathematicsIndiana UniversityRawles Hall831 East 3rd StBloomington, IN 47405USAE-mail: [email protected]

WOLFGANG LÜCKMathematicians Institut

der Universität BonnEndenicher Allee 6053115 BonnGERMANYE-mail: wolfgang.lueck@

him.uni-bonn.de

Received July 2019.




http://dx.doi.org/doi:10.1090/surv/069

mailto:[email protected]

mailto:wolfgang.lueck@ him.uni-bonn.de

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