PIECEWISE LINEAR STRUCTURES ON
TOPOLOGICAL MANIFOLDS
YULI B. RUDYAK
Abstract. This is a survey paper where we expose the Kirby
Siebenmann results on classification of PL structures on topo
logical manifolds and, in particular, the homotopy equivalence
T OP=P L = K(Z=2.3) and the Hauptvermutung for manifolds.
Prologue
In his paper [42 ] Novikov wrote:
Sullivan's Hauptvermutung theorem was announced first in
early 1967. After the careful analysis made by Bill Brow
der and myself in Princeton, the first version in May 1967
(before publication), his theorem was corrected: a necessary
restriction on the 2torsion of the group H3(M) was miss
ing. This gap was found and restriction was added. Full
proof of this theory has never been written and published.
Indeed, nobody knows whether it has been finished or not.
Who knows whether it is complete or not? This question is
not clarified properly in the literature. Many pieces of this
theory were developed by other topologists later. In particu
lar, the final KirbySiebenmann classification of topological
multidimensional manifolds therefore is not proved yet in the
literature.
I do not want to discuss here whether the situation is so dramatic
as Novikov wrote. However, it is definitely true that there is no de
tailed enough and wellordered exposition of KirbySiebenmann clas
sification, such that can be recommended to advanced students which
are willing to learn the subject. The fundamental book of Kirby
Siebenmann [28 ] was written by pioneers and, in a sense, posthaste.
It contains all the necessary results, but it is really "Essays", and one
must do a lot of work in order to do it readable for general audience.
____________
Date: May 17, 2003.
1991 Mathematics Subject Classification. 57Q25.
1
2 YULI B. RUDYAK
The job on hand is an attempt of (or, probably, an approximation
to) such an expository paper.
Contents
Prologue 1
Introduction 3
Notation and conventions 9
Chapter 1. Architecture 10
1. A result from homotopy theory 10
2. Preliminaries on bundles and classifying spaces 13
3. Structures on manifolds and bundles 19
4. From manifolds to bundles 24
5. Homotopy PL structures on T kx Dn 27
6. The product structure theorem, or from bundles to manifolds 29
7. Noncontractibility of T OP=P L 31
8. Homotopy groups of T OP=P L 32
9. Do it 36
Chapter 2. Tools 37
10. Stable equivalences of spherical bundles 37
11. Proof of Theorem 4.6 39
12. Normal maps and F=P L 44
13. The Sullivan map s : [M, F=P L] ! PdimM 46
14. The homotopy type of F=P L[2] 49
15. Splitting theorems 54
16. Detecting families 58
17. A special case of the theorem on the normal invariant of a
homeomorphism 59
Chapter 3. Applications 61
18. Topological invariance of rational Pontryagin classes 61
19. The space F=T OP 62
20. The map a : T OP=P L ! F=P L 64
21. The theorem on the normal invariant of a homeomorphism 66
22. A counterexample to the Hauptvermutung, and other
examples 67
Epilogue 68
References 69
PL STRUCTURES 3
Introduction
Throughout the paper we use abbreviation PL for "piecewise linear".
Hauptvermutung (main conjecture) is an abbreviation for die Haupt
vermutung der kombinatorischen Topologie (the main conjecture of
combinatorial topology). It seems that the conjecture was first for
mulated in the papers of Steinitz [54 ] and Tietze [59 ] in 1908.
The conjecture claims that the topology of a simplicial complex
determines completely its combinatorial structure. In other words,
two simplicial complexes are simplicially isomorphic whenever they are
homeomorphic. This conjecture was disproved by Milnor [35 ] in 1961.
However, for manifolds one can state a refined version of the Haupt
vermutung. A PL manifold is defined to be a simplicial complex such
that the star of every point (the union of all closed simplices containing
the point) is simplicially isomorphic to the ndimensional ball. (Equiv
alently, a PL manifold is a manifold with a fixed maximal PL atlas.)
There are examples of simplicial complexes which are homeomorphic
to topological manifolds but, nevertheless, are not PL manifolds (the
double suspensions over Poincar'e spheres, see [6]). Moreover, there
exists a topological manifold which is homeomorphic to a simplicial
complex but do not admit a PL structure, see Example 22.5.
Now, the Hauptvermutung for manifolds asks whether any two home
omorphic PL manifolds are PL isomorphic. Furthermore, the related
question asks whether every topological manifold is homeomorphic to
a PL manifold. Both these questions were solved (negatively) by Kirby
and Siebenmann [27 , 28]. In fact, Kirby and Siebenmann classified PL
structures on highdimensional topological manifolds. It turned out
that a topological manifold can have different PL structures, or not to
have any. Below we give a brief description of these results.
Let BT OP and BP L be the classifying spaces for stable topological
and PL bundles, respectively. We regard the forgetful map p : BP L !
BT OP as a fibration and denote its fiber by T OP=P L.
Let f : M ! BT OP classify the stable tangent bundle of a topolog
ical manifold M. It is clear that every PL structure on M gives us a
plifting of f and that every two such liftings are fiberwise homotopic.
(By the definition, a map bf: M ! BP L is a plifting of f if pfb= f.)
It is remarkable that the inverse is also true provided that dim M
5. In greater detail, M admits a PL structure if f admits a plifting
(the Existence Theorem 6.3), and P L structures on M are in a bijective
correspondence with fiberwise homotopy classes of pliftings of f (the
4 YULI B. RUDYAK
Classification Theorem 6.2). Kirby and Siebenmann proved these the
orems and, moreover, they proved that T OP=P L is the EilenbergMac
Lane space K(Z=2, 3). Thus, there is only one obstruction
(M) 2 H4(M; Z=2)
to a plifting of f, and the set of fiberwise homotopic pliftings of f (if
they exist) is in bijective correspondence with H3(M; Z=2). In other
words, for every topological manifold M, dim M 5 there is a class
(M) 2 H4(M; Z=2) with the following property: M admits a PL
structure if and only if (M) = 0. Furthermore, given a homeomor
phism h : V ! M of two PL manifolds, there exists a class
(h) 2 H3(M; Z=2)
with the following property: (h) = 0 if and only if h is concordant to
a PL isomorphism (or, equivalently, to the identity map 1M ). Finally,
every class a 2 H3(M; Z=2) has the form a = (h) for some homeo
morphism h. These results give us the complete classification of PL
structures on a topological manifold of dimension 5.
We must explain the following. It can happen that two different
PL structures on M yield PL isomorphic PL manifolds (like that two
pliftings f : M ! BP L of f can be nonfiberwise homotopic). In
deed, roughly speaking, a PL structure on a topological manifold M
is a concordance class of PL atlases on M (see Section 3 for accurate
definitions). However, a PL automorphism of a PL manifold can turn
the atlas into a nonconcordant to the original one, see Example 22.3.
So, in fact, the set of pairwise nonisomorphic PL manifolds which are
homeomorphic to a given PL manifold is in a bijective correspondence
with the set H3(M; Z=2)=R where R is the following equivalence re
lation: two PL structure are equivalent if the corresponding PL man
ifolds are PL isomorphic. The Hauptvermutung for manifolds claims
that H3(M; Z=2)=R is oneelement. But this is wrong.
Namely, there exists a PL manifold M which is homeomorphic but
not PL isomorphic to RP2n+1, see Example 22.1. So, here we have a
counterexample to the Hauptvermutung.
For completeness of the picture, we mention again that there are
topological manifolds which do not admit any PL structure, see Exam
ple 22.4.
Comparing the classes of smooth, PL and topological manifolds, we
see that there is a big difference between first and second classes,
and not so big difference between second and third ones. From the
homotopytheoretical point of view, one can say that the space P L=O
PL STRUCTURES 5
(which classifies smooth structures on PL manifold, see Remark 6.7)
has many nontrivial homotopy groups, while TOP/PL is an Eilenberg
Mac Lane space. Geometrically, one can mention that there are many
smooth manifolds which are PL isomorphic to Sn but pairwise non
diffeomorphic, while any PL manifold Mn , n 5 is PL isomorphic to
Sn provided that it is homeomorphic to Sn.
It is worthwhile to go one step deeper and explain the following.
Let M4k be a closed connected almost parallelizable manifold (i.e. M
becomes parallelizable after deletion of a point). Let oek denote the
minimal natural number which can be realized as the signature of the
manifold M4k. In fact, for every k we have three numbers oeSk, oePLk and
oeTOPk while M4k is a smooth, PL or topological manifold, respectively.
Milnor and Kervaire [37 ] proved that
oeSk= ck(2k  1)!
where ck 2 N. On the other hand,
oePL1= 16 and oePLk= 8 fork > 1.
Finally,
oeTOPk = 8 for allk.
So, here we can see again the big difference between smooth and PL
cases. On the other hand, oePLk = oeTOPk for k > 1. Moreover, we will
see below that the number
2 = 16=8 = oePL1=oeTOP1
yields the group Z=2 = ß3(T OP=P L).
It makes sense to say here about low dimensional manifolds, because
of the following remarkable contrast. There is no difference between PL
and smooth manifolds in dimension < 7: every PL manifold V n, n < 7
admits a unique smooth structure. However, there are infinitely many
smooth manifolds which are homeomorphic to R4 but pairwise non
diffeomorphic, see [17 , 26].
The paper is organized as follows. The first chapter contains the
architecture of the proof of the Main Theorem: T OP=P L ' K(Z=2, 3).
In fact, we comment the following graph there:
6 YULI B. RUDYAK
_____________________________________________________________________
 Theorem on the normal invariant of a homeomorphism for T kx Sn 
____________________________________________________________________



_________________________________? ________________________________*
*_
 Classification of homotopy ___________ Classification of homotopy*
* 
 PL structures on T kx Dn   k *
* 
________________________________ ___PL_structures_on_T__________*
*_ 
 
 
 
 ________________________ _____________________________*
*?
  Classification Theorem__________________ProductoStructureeTheorem *
* 
 _______________________  ____________________________*
*
  
  
  
  _________________________________ 
 _____________________?  Local contractibility of the  
 _Reduction_Theorem_______oehomeomorphism group  
  ________________________________  
  
  
  
_______________________________?? _____________________?
 T OP=P L = K(ß, 3), ß Z=2   Existence Theorem 
______________________________  ____________________
H H
H H
H H
HH___________________________j
 
  T OP=P L = K(Z=2, 3) 
_________________________
6


_________________________________________________________________________
 Existence a highdimensional topological manifold without PL structure 
________________________________________________________________________ 
6 6
 
 
 
_____________________________ ______________________
 Rokhlin Signature Theorem   Freedman's Example 
____________________________  _____________________ 
PL STRUCTURES 7
Namely, we formulate without proofs the boxed claims (and provide
the necessary definitions), while we prove all the implications (arrows),
i.e., we explain how a claim can be deduced from another one.
The second chapter contains a proof of the Sullivan Theorem on a
normal invariant of a homeomorphism for T kx Sn, and also a proof
of the BrowderNovikov Theorem 4.6 about homotopy properties of
normal bundles. We need this theorem in order to define the concept
of normal invariant.
The third chapter contains several application of the Main theorem
and, in particular, the counterexample to the Hauptvermutung.
Let me tell something more about the graph. As we have already
seen, the classification theory of PL structures on topological manifolds
splits into two parts. The first part reduces the original geometric
problem to a homotopy one (classification of pliftings of a map M !
BT OP ), the second part solves this homotopy problem by proving that
T OP=P L = K(Z=2, 3).
The key result for the first part is the Product Structure Theorem
6.1. Roughly speaking, this theorem establishes a bijection between PL
structures on M and M x R. The Classification Theorem 6.2 and the
Existence Theorem 6.3 are the consequences of the Product Structure
Theorem.
Passing to the second part, the description of the homotopy type of
T OP=P L, we have the following. Because of the Classification Theo
rem, for n 5 there is a bijection between the set ßn(T OP=P L) and
the set of PL structures on Sn. By the Smale Theorem, every PL man
ifold Mn , n 5, is PL isomorphic to Sn whenever it is homeomorphic
to Sn. So, ßn(T OP=P L) = 0 for n 5.
What about n < 5? Again, because of the Classification Theorem,
the group ßn(T OP=P L) is in a bijective correspondence with the set
of PL structures on Rk x Sn provided that k + n 5. However, this
set of PL structures is uncontrollable. In order to make the situation
more manageable, one can consider the PL structures on the compact
manifold T kx Sn and then pass to the universal covering. We can't
do it directly, but there is a trick (the Reduction Theorem 8.7) which
allows us to estimate PL structure on Rk x Sn in terms of socalled ho
motopy PL structures on T kx Sn (more precisely, we should consider
the homotopy PL structures on T kx Dn modulo the boundary), see
Section 3 for the definitions. Now, using results of Hsiang and Shane
son [23 ] and Wall [62 , 63] about homotopy PL structures on T kx Dn,
one can prove that ßi(T OP=P L) = 0 for i 6= 3 and that ß3(T OP=P L)
8 YULI B. RUDYAK
has at most 2 elements. Finally, there exists a highdimensional topo
logical manifold which does not admit any PL structure. Thus, by the
Existence Theorem, the space T OP=P L is not contractible. Therefore
T OP=P L = K(Z=2, 3)
It is worthwhile to mention that the proof of the Product Structure
Theorem uses the classification of homotopy PL structures on T k.
Now I say some words about the top box of the above graph. Let
Fn be the monoid of pointed homotopy equivalences Sn ! Sn, let BFn
be the classifying space for Fn, and let BF = limn!1 BFn. There is
an obvious forgetful map BP L ! BF , and we denote by F=P L the
homotopy fiber of this map. For every homotopy equivalence of closed
PL manifolds h : V ! M Sullivan [56 , 57] defined the normal invariant
of h to be a certain homotopy class jF (h) 2 [M, F=P L], see Section 4.
Sullivan proved that, for every homeomorphism h : V ! M, jF (h) = 0
whenever H3(M) is 2torsion free. Moreover, this theorem implies that
if, in addition, M is simplyconnected then h is homotopic to a PL
isomorphism. Thus the Hauptvermutung holds for simplyconnected
manifolds with H3(M) 2torsion free.
Definitely, the above formulated Sullivan Theorem on the Normal
Invariant of a Homeomorphism is interesting by itself. However, in the
paper on hand this theorem plays also an additional important role.
Namely, the Sullivan Theorem for T kx Sn is a lemma in classifying of
homotopy structures on T kx Dn. For this reason, we first prove the
Sullivan Theorem for T kx Sn, then use it in the proof of the Main
Theorem, and then (in Chapter 3) prove the Sullivan Theorem in full
generality.
You can also see that the proof of the Main Theorem uses the diffi
cult Freedman's example of a 4dimensional almost parallelizable topo
logical manifold of signature 8. This example provides the equality
oeTOP1 = 8. Actually, the original proof of the Main Theorem appeared
before Freedman's Theorem and therefore did not use the last one.
However, as we noticed above, the Freedman results clarify the rela
tions between PL and topological manifolds, and thus they should be
incorporated in the exposition of the global picture.
Acknowledgments. I express my best thanks to Andrew Ranicki
who read the whole manuscript and did many useful remarks and com
ments. I am also grateful to HansJoachim Baues for useful discussions.
PL STRUCTURES 9
Notation and conventions
We work mainly with CW spaces and topological manifolds. How
ever, when we quit these clases by taking products or functional spaces,
we equip the last ones with the compactly generated topology, (follow
ing Steenrod [53 ] and McCord [33 ], see e.g.[48 ] for the exposition). All
maps are supposed to be continuous. All neighbourhoods are supposed
to be open.
Given two topological spaces X, Y , we denote by [X, Y ] the set of
homotopy classes of maps X ! Y . We also use the notation [X, Y ]o
for the set of pointed homotopy classes of pointed maps X ! Y of
pointed spaces.
It is quite standard to denote by [f] the homotopy class of a map f.
However, here we usually do not distinguish a map and its homotopy
class and use the same symbol, say f for a map as well as for the
homotopy class. In this paper this does not lead to any confusion.
We use the term inessential map for nullhomotopic maps; otherwise
we say that a map is called essential.
We use the sign ' for homotopy of maps or homotopy equivalence
of spaces.
We reserve the term bundle for locally trivial bundles and the term
fibration for Hurewicz fibrations. Given a space F , an F bundle is a
bundle whose fiber is F , and an F fibration is a fibration whose fibers
are homotopy equivalent to F .
Given a bundle or fibration , = {p : E ! B}, we say that B is the
base of , and that E is the total space of ,. Furthermore, given a space
X, we set
, x X = {p x 1 : E x X ! B x X}.
Given two bundles , = {p : E ! B} and j = {q : Y ! X}, a bundle
morphism ' : , ! j is a commutative diagram
g
E  ! Y
? ?
p?y ?yq
f
B  ! X.
We say that f is the base of the morphism ' or that ' is a morphism
over f. We also say that g is a map over f. If X = B and f = 1B we
say that g is a map over B (and ' is a morphism over B).
10 YULI B. RUDYAK
Given a map f : Z ! B and a bundle (or fibration) , over B, we
use the notation f*, for the induced bundle over Z. Recall that there
is a canonical bundle morphism If,,: , ! j over f, see [48 ] (or [16 ]
where it is denoted by ad(f)). Following [16 ], we call If,,the adjoint
morphism of f, or just the fadjoint morphism. Furthermore, given
a bundle morphism ' : , ! j with the base f, there exists a unique
bundle morphism c(') : , ! f*j over the base of , such that the
composition
c(') * If,j
,  ! f j   ! j
coincides with '. Following [16 ], we call c(') the correcting morphism.
Given a subspace A of a space X and a bundle , over X, we denote
by ,A the bundle i*, where i : A X is the inclusion.
Chapter 1. Architecture
1. A result from homotopy theory
Recall that an Hspace is a space F with a base point f0 and a
multiplication map ~ : F x F ! F such that f0 is a homotopy unit,
i.e. the maps f 7! ~(f, f0) and f 7! ~(f0, f) are homotopic to the
identity rel {f0}. For details, see [4].
1.1. Definition. (a) Let (F, f0) be an Hspace with the multiplication
~ : F x F ! F . A principal F fibration is an F fibration p : E ! B
equipped with a map m : E x F ! E such that the following holds:
(i) the diagrams
E x F x F mx1!E x F E x F  m! E
? ? ? ?
1x~?y ?ym p1?y ?yp
p
E x F m! E E   ! B
commute;
(ii) the map
E ! E, e 7! m(e, f0)
is a homotopy equivalence;
(iii) for every e0 2 E, the map
F  ! p1(p(e0)), f 7! m(e0, f)
is a homotopy equivalence.
(b) A trivial principal F fibration is the fibration p2 : X x F ! F with
the action m : E x F ! E of the form
m : X x F x F ! X x F, m(x, f1, f2) = (x, ~(f1, f2)).
PL STRUCTURES 11
It is easy to see that if the fibration j is induced from a principal
fibration , then j turns into a principal fibration in a canonical way.
1.2. Definition. Let ß1 : E1 ! B and ß2 : E2 ! B be two principal
F fibrations over the same base B. We say that a map h : E1 ! E2 is
an F equivariant map over B if h is a map over B and the diagram
E1 x F hx1!E2 x F
? ?
m1?y ?ym2
E1 h! E2
commutes up to homotopy over B.
Notice that, for every b 2 B, the map
hb : ß11(b) ! ß12(b), hb(x) = h(x)
is a homotopy equivalence.
Now, let p : E ! B be a principal F fibration, and let f : X ! B
be an arbitrary map. Given a plifting g : X ! E of f and a map
u : X ! F , consider the map
gxu m
gu : X  ! X x X  ! E x F   ! E.
It is easy to see that the correspondence (g, u) 7! gu yields a well
defined map (action)
(1.1) [Liftpf] x [X, F ] ! [Liftpf].
In particular, for every plifting g of f the correspondence u 7! gu
induces a map
Tg : [X, F ] ! [Liftpf].
1.3. Theorem. Let , = {p : E ! B} be a principal F fibration, and
let f : X ! B be a map where X is assumed to be paracompact and
locally contractible. If F is an Hspace with homotopy inversion, then
the above action (1.1) is free and transitive provided [Liftpf] 6= ;. In
particular, for every plifting g : X ! E of f the map Tg is a bijection.
Proof. We start with the following lemma.
1.4. Lemma. The theorem holds if X = B, f = 1X and , is the trivial
principal F fibration.
Proof. In this case every plifting g : X ! X x F of f = 1X determines
and is completely determined by the map
__g: X :g! X x F p2! F.
12 YULI B. RUDYAK
In other words, we have the bijection [Liftpf] ~=[X, F ], and under this
bijection the action (1.1)turns into the multiplication
[X, F ] x [X, F ] ! [X, F ].
Now the result follows since [X, F ] is a group.
We finish the proof of the theorem. Consider the induced fibration
f*, = {q : Y ! X} and notice that there is an [X, F ]equivariant
bijection
(1.2) [Liftpf] ~=[Liftq1X ].
Now, suppose that [Liftpf] 6= ; and take a plifting g of f. Regarding
Y as the subset of X x E, define the F equivariant map
h : X x F ! Y, h(x, a) = (x, g(x)a), x 2 X, a 2 F.
It is easy to see that the diagram
X x F h! Y
? ?
p1?y ?yq
X _______X
commutes, i.e. h is a map over X. Since X is a locally contractible
paracompact space, and by a theorem of Dold [9], there exists a map
k : Y ! X x F over X which is homotopy inverse over X to h. It
is easy to see that k is an equivariant map over X. Indeed, if m1 :
X x F x F ! X x F and m2 : Y x F ! Y are the corresponding
actions then
m1(k x 1) ' khm1(k x 1) ' km2(h x 1)(k x 1) =' km2(hk x 1) ' km2,
where ' denotes the homotopy over X.
In particular, there is an [X, F ]equivariant bijection
[Liftq1X ] ~=[Liftp11X ]
where p1 : X xF ! X is the projection. Now we compose this bijection
with (1.2)and get [X, F ]equivariant bijections
[Liftpf]] ~=[Liftq1X ] ~=[Liftp11X ],
__
and the result follows from Lemma 1.4. __
1.5. Example. If p : E ! B is an F fibration then p : E ! B
is a principal F fibration. Here denotes the loop functor.
PL STRUCTURES 13
2. Preliminaries on bundles and classifying spaces
Here we give a brief recollection on Rn bundles, spherical fibrations
and their classifying spaces. For details, see [48 ].
We define a topological Rnbundle over a space B to be an Rnbundle
p : E ! B equipped with a fixed section s : B ! E. Given two
topological Rnbundles , = {p : E ! B} and j = {q : Y ! X}, we
define a topological morphism ' : , ! j to be a commutative diagram
g
E  ! Y
? ?
(2.1) p?y ?yq
f
B  ! X
where g preserves the sections and yields homeomorphism of fibers.
The last one means that, for every b 2 B, the map
gb : Rn = p1(b) ! q1 (f(b) = Rn, gb(a) = g(a) for everya 2 p1(b)
is a homeomorphism. As usual, we say that f is the base of the mor
phism '.
Topological Rnbundles can also be regarded as (T OPn, Rn)bundles,
i.e. Rnbundles with the structure group T OPn. Here T OPn is the
topological group of selfhomeomorphism f : Rn ! Rn, f(0) = 0. The
classifying space BT OPn of the group T OPn turns out to be a clas
sifying space for topological Rnbundles over CW spaces. This means
there exists a universal topological Rnbundle flnTOP over BT OPn with
the following
2.1. Universal Property. For every topological Rnbundle , over a
CW space B, every CW subspace A of B and every morphism
_ : ,A ! flnTOP
of topological Rnbundles, there exists a morphism ' : , ! flnTOP which
is an extension of _.
In particular, for every topological Rnbundle , over B there exists
a morphism ' : , ! flnTOP of topological Rnbundles. We call such ' a
classifying morphism for ,. The base f : B ! BT OPn of ' is called a
classifying map for ,. It is clear that , is isomorphic over B to f*flnTOP.
2.2. Proposition. Let '0, '1 : , ! flnTOP be two classifying mor
phisms for ,. Then there exists a classifying morphism : , x I !
flnTOP such that , x {i} = 'i, i = 0, 1. In particular, a classifying
map f for , is determined by , uniquely up to homotopy.
14 YULI B. RUDYAK
Proof. This follows from the universal property 2.1 applied to , x_I,_if
we put A = X x {0, 1} where X denotes the base of ,. __.
A piecewise linear (in future PL) Rnbundle is a topological Rn
bundle , = {p : E ! B} such that E and B are polyhedra and
p : E ! B and s : B ! E are PL maps. Furthermore, we re
quire that, for every simplex B, there is a PL homeomorphism
h : p1( ) ~= x Rn with h(s( )) = x {0}. (For definitions of PL
maps, see [24 , 47].)
A morphism of PL Rnbundles is a morphism of topological Rn
bundles where the maps g and f in (2.1)are PL maps.
There exists a universal PL Rnbundle flnPL over a certain space
BP Ln. This means that the universal property 2.1 remains valid if we
replace flnTOP by flnPLand öt pological Rn bundle" by "PL Rnbundle"
there. So, BP Ln is a classifying space for PL Rnbundles.
Notice that BP Ln can also be regarded as the classifying space of a
certain group P Ln (which is constructed as the geometric realization
of a certain simplicial group), [29 , 30].
A sectioned Snfibration is defined to be an Snfibration p : E ! B
equipped with a section s : B ! E. Morphisms of sectioned Sn
fibrations are defined to be diagrams like (2.1) where each map gb is
assumed to be a pointed homotopy equivalence.
There exists a universal sectioned Snfibration flnFover a certain space
BFn. This means that the universal property 2.1 remains valid if we
replace flnTOP by flnF and öt pological Rn bundle" by "sectioned Sn
fibration" there. So, BFn is a classifying space sectioned Snfibrations.
The space BFn can also be regarded as the classifying space for the
monoid Fn of pointed homotopy equivalences (Sn, *) ! (Sn, *). Be
cause of this, we shall use the brief term "(Sn, *)fibration" for sectioned
Snfibrations. Furthermore, we will also use the term "Fnmorphism"
for morphism of sectioned Snfibrations. Finally, an Fnmorphism over
a space is called an Fnequivalence.
We need also to recall the space BOn which classifies ndimensional
vector bundles. The universal vector bundle over BOn is denoted by
flnO. This space is wellknown and described in many sources, e.g. [38 ].
Since flnPLcan be regarded as the (underlying) topological Rnbundle,
there is a classifying morphism
(2.2) ! = !PLTOP(n) : flnPL! flnTOP.
PL STRUCTURES 15
We denote by ff = ffPLTOP(n) : BP Ln ! BT OPn the base of this mor
phism.
Similarly, given a topological Rnbundle , = {p : E ! B}, let ,o
denotes the Snbundle ,o = {po : Eo ! B} where Eo is the fiber
wise onepoint compactification of E. Notice that the added points
("infinities") give us a certain section of ,o.
In other words, the T OPnaction on Rn extends uniquely to a T OPnaction on
the onepoint compactification Sn of Rn, and ,o is the (T OPn, Sn)bundle assoc*
*iated
with ,. Furthermore, the fixed point 1 of the T OPnaction on Sn yields a secti*
*on of
,o.
So, ,o can be regarded as an (Sn, *)fibration over B. In particular,
(flnTOP)o can be regarded as an (Sn, *)fibration over BT OPn. So, there
is a classifying morphism
!TOPF(n) : flnTOP ! flnF.
We denote by ffTOPF(n) : BT OPn ! BFn the base of !TOPF(n).
Finally, we notice that an ndimensional vector bundle over a poly
hedron X has a canonical structure of PL Rnbundle over X. Similarly
to above, this gives us a (forgetful) map
ffOPL(n) : BOn ! BP Ln.
So, we have a sequence of forgetful maps
0 ff00 ff000
(2.3) BOn ff!BP Ln ! BT OPn ! BFn
where ff0= ffOPL(n), etc.
2.3. Constructions. 1. Given an F bundle , = {p : E ! B} and an
F 0bundle ,0 = {p0 : E0 ! B0}, we define the product , x ,0 to be the
F x F 0bundle
p x p0: E x E0 ! B x B0.
2. Given an F bundle , = {p : E ! B} with a section s : B ! E
and an F 0bundle ,0 = {p0 : E0 ! B0} with a section s0 : B0 ! E0,
we define the smash product , ^ ,0 to be the F ^ F 0bundle as follows.
The map p x p0 : E x E0 ! B x B0 passes through the quotient map
q : E x E0 ! E x E0=(E x s(B0) [ E0x s(B), and we set
, ^ j = {ß : E x E0=(E x s(B0) [ E0x s(B) ! B x B0,
where ß is the unique map with p x p0= ßq. Finally, the section s and
s0 yield an obvious section of ß.
16 YULI B. RUDYAK
3. Given an Rm bundle , and an Rnbundle j over the same space
X, the Whitney sum of , and j is the Rm+n bundle , j = d*(, x j)
where d : X ! X x X is the diagonal.
Notice that if , and j are a PL Rm and PL Rnbundle, respectively,
then , x j is a PL Rm+n bundle.
4. Given a sectioned Sm bundle , and sectioned Snbundle j over
the same space X, we set , y j = d*(, ^ j).
We denote by rn = rTOPn : BT OPn ! BT OPn+1 the map which
classifies flnTOP `1BTOPn. The maps rPLn : BP Ln ! BP Ln+1 and
rOn: BOn ! BOn+1 are defined in a similar way.
We can also regard the above map rn : BT OPn ! BT OPn+1 as a
map induced by the standard inclusion T OPn T OPn+1. Using this
approach, we define rFn : BFn ! BFn+1 as the map induced by the
standard inclusion Fn Fn+1, see [31 , p. 45].
2.4. Remarks. 1. Regarding Rm as the bundle over the point, we see
that (Rm )o = (Sm ) and, moreover,
(Rm x Rn)o = Sm ^ Sn, i.e.(Rm Rn)o = Sm y Sn.
Therefore (, j)o = ,o y jo for every Rm bundle , and Rnbundle j.
2. Generally, the smash product of (sectioned) fibrations is not a
fibrations. But we apply it to bundles only and so do not have any
troubles. On the other hand, there is an operation ^h, the homotopy
smash product, such that , ^h j is the (F ^ G)fibration over X x Y if
, is an F fibration over X and j is an Gfibration over Y , see [48 ]. In
particular, one can use it in order to define an analog of Whitney sum
for spherical fibrations and then use this one in order to construct the
map BFn ! BFn+1.
The spaces BOn, BP Ln, BT OPn and BFn are defined uniquely up
to weak homotopy equivalence. However, it is useful for us to work
with more or less concrete models of classifying spaces BOn, etc. In
greater detail, we do the following.
Choose classifying spaces B0Fn for (Sn, *)fibrations (i.e., in the weak
homotopy type BFn) and consider the maps rFn : B0Fn ! B0Fn+1 as
above. We can assume that every B0Fn is a CW complex and every rn
is a cellular map. We define BF to be the telescope (homotopy direct
limit) of the sequence
. . .! B0Fn rn! B0Fn+1  ! . .,.
PL STRUCTURES 17
see e.g. [48 ]. Furthermore, we define BFn to be the telescope of the
finite sequence
rn1 0
. . .! B0Fn1  ! B Fn.
(Notice that BFn ' B0Fn.) So, we have the sequence (filtration)
. . .BFn BFn+1 . ...
S
Here BF = BFn, BFn is closed in BF and BF has the direct limit
topology with respect to the filtration {BFn}. Moreover, if f : K !
BF is a map of a compact space K then there exists n such that
f(K) BFn.
Now, for every n consider a CW space B0T OPn in the weak ho
motopy type BT OPn and define B00T OP to be the telescope of the
sequence
. .. ! B0T OPn rn! B0T OPn+1   ! . . ..
Furthermore, we define B00T OPn to be the telescope of the finite se
quence
rn1 0
. . . ! B0T OPn1  ! B T OPn.
So, we have the diagram
. . . B00T OPn B00T OPn+1 . . .B00T OP
? ? ?
? ? ?
y y y p
. . . BFn BFn+1 . . . BF
where the map p is induced by maps ffTOPF(n). Now we convert every
vertical map to a fibration (using Serre construction). Namely, we set
fi
BT OP = {(x, !) fix 2 B00T OP, ! 2 (BF )I, p(x) = !(0)}
and define ffTOPF : BT OP ! BF by setting ffTOPF(x, !) = !(1). Fi
nally, we set
fi
BT OPn = {(x, !) 2 BT OP fix 2 B00T OPn, ! 2 (BFn)I (B00T OP )I}
So, we have the commutative diagram
. . .BT OPn BT OPn+1 . . .BT OP
? ? ?
? ? ?
y y y p
. . . BFn BFn+1 . . . BF
where all the vertical maps are fibrations.
18 YULI B. RUDYAK
Now it is clear how to proceed and get the diagram
. . . BOn BOn+1 . . . BO
? ? ?
? ? ? O
y y y ffPL
. . . BP Ln BP Ln+1 . . . BP L
? ? ?
? ? ? PL
(2.4) y y y ffTOP
. . .BT OPn BT OPn+1 . . .BT OP
? ? ?
ffTOPF(n)?y ?y ?yffTOPF
. . . BFn BFn+1 . . . BF
where all the vertical maps are fibrations and all the filtrations have
nice properties. Moreover, each of limit spaces has the direct limit
topology with respect to the corresponding filtration, and every com
pact subspace of, say, BO is contained in some BOn.
Furthermore, the fiber of ffOPLis denoted by P L=O, the fiber of ffPLTOP
is denoted by T OP=P L, etc. Similarly, the fiber of the composition,
say,
ffPLF:= ffTOPFOffPLTOP: BP L ! BF
is denoted by F=P L. In particular, we have a fibration
(2.5) T OP=P L a! F=P L  b! F=T OP.
S
Finally, F=T OP = Fn=T OPn where Fn=T OPn denotes the fiber of
the fibration BT OPn ! BFn, and F=T OP has the direct limit topol
ogy with respect to the filtration {Fn=T OPn}. The same holds for
other öh mogeneous spaces" F=P L, T OP=P L, etc.
Notice that, because of wellknown results of Milnor [34 ], all these
öh mogeneous spaces" have the homotopy type of CW spaces. Fur
thermore, all the spaces BO, BP L, BT OP, BF, F=P L, T OP=P L, etc.
are infinite loop spaces and, in particular, Hspaces, see [4].
We mention also the following useful fact.
2.5. Theorem. Let Z denote on of the symbols O, P L, F . The above
described map BZn ! BZn+1 induces an isomorphism of homotopy
groups in dimensions n  1 and an epimorphism in dimension n.
Proof. For Z = O and Z = F it is well known, see e.g [5], for Z = P L __
it can be found in [20 ]. __
PL STRUCTURES 19
2.6. Remark. Let Gn denote the topological monoid of homotopy
equivalences Sn1 ! Sn1 . Then the classifying space BGn of Gn
classifies Sn1 fibrations. Every h 2 T OPn induces a map Rn \ {0} !
Rn \ {0} which, in turn, yields a selfmap
ßh : Sn1 ! Sn1 , ßh(x) = h(x)=h(x).
So, we have a map T OPn ! Gn which, in turn, induces a map
BT OPn ! BGn
of classifying spaces. In the language of bundles, this map converts
a topological Rnbundle into a spherical fibration via deletion of the
section.
We can also consider the space BG by tending n to 1. In particular,
we have the spaces G=P L and G=T OP .
There is an obvious forgetful map Fn ! Gn, and it turns out that
the induced map BF ! BG (as n ! 1) is a homotopy equivalence.
see e.g. [31 , Chapter 3]. In particular, F=P L ' G=P L and F=T OP '
G=T OP .
3. Structures on manifolds and bundles
A PL atlas on a topological manifold is an atlas such that all the
transition maps are PL ones. We define a PL manifold as a topological
manifold with a maximal PL atlas. Furthermore, given two PL man
ifolds M and N, we say that a homeomorphism H : M ! N a PL
isomorphism if h is a PL map. (One can prove that in this case h1 is
a PL map as well, [24 ].)
3.1. Definition. (a) We define a @PL manifold to be a topological
manifold whose boundary @M is a PL manifold. In particular, every
closed topological manifold is a @PL manifold. Furthermore, every PL
manifold can be canonically regarded as a @PL manifold.
(b) Let M be a @PL manifold. A PL structuralization of M is a
homeomorphism h : V ! M such that V is a PL manifold and
h : @V ! @M induces a PL isomorphism @V ! @M of boundaries (or,
equivalently, PL isomorphism of corresponding collars). Two PL struc
turalizations hi : Vi ! X, i = 0, 1 are concordant if there exist a PL
isomorphism ' : V0 ! V1 and a homeomorphism H : V0 x I ! M x I
such that h0 = HV x {0} and HV0 x {1} = h1' and, moreover,
H : @V0 x I ! @M x I coincides with h0 x 1I. Any concordance class
of PL structuralizations is called a PL structure on M. We denote by
TPL (M) the set of all PL structures on X.
20 YULI B. RUDYAK
(c) If M itself is a PL manifold then TPL (M) contains the distin
guished element: the concordance class of 1M . We call it the trivial
element of TPL (M).
3.2. Remarks. 1. Clearly, every PL structuralization of M equips M
with a certain PL atlas. Conversely, if we equip M with a certain PL
atlas then the identity map can be regarded as a PL structuralization
of M.
2. Clearly, if M itself is a PL manifold then the concordance class
of any PL isomorphism is the trivial element of TPL (M).
3. Recall that two homeomorphism h0, h1 : X ! Y are isotopic if
there exists a homeomorphism H : X x I ! Y x I (isotopy) such that
p2H : X x I ! Y x I ! I coincides with p2 : X x I ! I. Given
A X, we say that h0 and H1 are isotopic relA if there exists an
isotopy H such that H(a, t) = h0(a) for every a 2 A and every t 2 I.
In particular, if two PL structuralization h0, h1 : V ! M are isotopic
rel@V then they are concordant.
4. Given two PL structuralizations hi : Vi ! M, i = 0, 1, they are
not necessarily concordant if V0 and V1 are PL isomorphic. We are not
able to give examples here, but we do it later, see Remark 3.10(2) and
Example 22.3.
3.3. Definition (cf. [5, 48]).Given a topological Rnbundle ,, a PL
structuralization of , is a morphism ' : , ! flnPL of topological Rn
bundles. We say that two PL structuralizations '0, '1 : , ! flnPL are
concordant if there exists a morphism : , x 1I ! flnPL of topological
Rnbundles such that , x 1{i}= 'i, i = 0, 1.
Let f : X ! BT OPn classify a topological Rnbundle ,, and let
g : X ! BP Ln
be an ffPLTOP(n)lifting of f. Then we get a morphism (defined uniquely
up to concordance in view of 2.2)
, ~=f*flnTOP = g*ff(n)*flnTOP = g*flnPLclassif!flNPL, ff(n) := ffPLTOP(n).
Clearly, this morphism , ! flnPL is a PL structuralization of ,. It is
easy to see that in this way we have a correspondence
(3.1) [Liftff(n)f] ! {PL structures on ,}.
3.4. Theorem. The correspondence (3.1) is a bijection.
PL STRUCTURES 21
Proof. This can be proved similarly to [48 , Theorem IV.2.3], cf. also_
[5, Chapter II]. __
Consider now the map
~f: X  f! BT OPn BT OP
and the map ff : BP L ! BT OP as in (2.4). Then every ff(n)lifting
of f is the fflifting of ~f. So, we have a correspondence
(3.2) u, : {PL structures on ,} ! [Liftff(n)f] ! [Liftff~f]
where the first map is the inverse to (3.1). Furthermore, there is a
canonical map
v, : {PL structures on ,} ! {PL structures on , `1},
and these maps respect the maps u,, i.e. u, `1 = v,u,. So, we have
the map
(3.3) lim {PL structures on , `n} ! [Liftff~f]
n!1
where lim means the direct limit of the sequence of sets.
3.5. Proposition. If X is a finite CW space then the map (3.3) is a
bijection.
Proof. The surjectivity follows since every compact subset of BT OP is
contained in some BT OPn. Similarly, every map X x I ! BP L passes __
through some BP Ln, and therefore the injectivity holds. __
Furthermore, if , itself is a PL bundle then, by Theorem 1.3, there
is a bijection
[Liftff~f] ~=[X, T OP=P L].
Thus, in this case the bijection (3.3)turns into the bijection
(3.4) lim { PL structures on , `n} ! [X, T OP=P L].
n!1
3.6. Definition. Let M be a @PL manifold. A homotopy PL struc
turalization of M is a homotopy equivalence h : V ! X such that V is
a PL manifold and h : @V ! @M is a PL isomorphism. Two homotopy
PL structuralizations hi : Vi ! X, i = 0, 1 are equivalent if there exists
a PL isomorphism ' : V0 ! V1 and a homotopy H : V0 x I ! M
such that h0 = HV x {0} and HV0 x {1} = h1' and, moreover,
HV x {t} : @V0 ! @M coincides with h0. Any equivalence class of
homotopy PL structuralizations is called a homotopy PL structure on
X. We denote by SPL (X) the set of all homotopy PL structures on X.
If M itself is a PL manifold, we define the trivial element of SPL (M)
as the equivalence class of 1M : M ! M.
22 YULI B. RUDYAK
3.7. Definition. Given an (Sn, *)fibration , over X, a homotopy PL
structuralization of , is an (Sn, *)morphism ' : , ! (flnPL)o. We say
that two PL structuralizations '0, '1 : , ! (flnPL)o are equivalent if
there exists a morphism : , x 1I ! (flnPL)o of (Sn, *)fibrations such
that , x 1{i}= 'i, i = 0, 1. Every such an equivalence class is called
a homotopy PL structure on ,.
Now, similarly to (3.4), for a finite CW space X we have a bijection
(3.5) lim {homotopy PL structures on , `n} ! [X, F=P L].
n!1
However, here we can say more.
3.8. Proposition. The sequence
{{homotopy PL structures on , `n}}1n=1
stabilizes. In particular, the map
{homotopy PL structures on , `n} ! [F=P L]
is a bijection if dim , >> dim X
__
Proof. This follows from 2.5. __
Summarizing, for every PL RN bundle , we have a commutative
diagram
{PL structure on , }   ! [X, T OP=P L]
? ?
? ?
y a*y
{homotopy PL structure on ,o }   ! [X, F=P L]
Here the map a in (2.5) induces the map a* : [X, T OP=P L] !
[X, F=P L]. The left vertical arrow converts a morphism of RN bundles
into a morphism of (SN , *)bundles and regards the last one as a mor
phism of (SN , *)fibrations.
For a finite CW space X, the horizontal arrows turn into bijections if
we stabilize the picture. i.e. pass to the limit as in (3.4). Furthermore,
the bottom arrow is an isomorphism if N >> dim X.
3.9. Remark. Actually, following the proof of the Main Theorem, one
can prove that T OPm =P Lm = K(Z=2, 3) for m 5, see [28 , Essay IV,
x9]. So, an obvious analog of 2.5 holds for T OP also, and therefore the
top map of the above diagram is a bijection for N large enough. But,
of course, we are not allowed to use these arguments here.
PL STRUCTURES 23
3.10. Remark. 1. We can also consider smooth (=differentiable C1 )
structures on topological manifolds. To do this, we must replace the
words "PL" in Definition 3.1 by the word "smooth". The related set
of smooth concordance classes is denoted by TD (M).
The set SD (M) is defined in a similar way, we leave it to the reader.
Moreover, recall that every smooth manifold can be canonically con
verted into a PL manifold (the CairnsHirschWhitehead Theorem, see
e.g [22 ]). So, we can define the set PD (M) of smooth structures on a
PL manifold M. To do this, we must modify definition 3.1 as follows:
M is a PL manifold with a compatible smooth boundary, Viare smooth
manifolds, hi and H are PL isomorphisms.
2. We can now construct an example of two smooth structural
izations hi : V ! Sn, i = 1, 2 which are not concordant. First,
notice that there is a bijective correspondence between SD (Sn) and
the KervaireMilnor group n of homotopy spheres, [25 ]. Indeed, n
consists of equivalence classes of oriented homotopy spheres: two ori
ented homotopy spheres are equivalent if they are orientably diffeomor
phic (=hcobordant). Now, given a homotopy smooth structuralization
h : n ! Sn, we orient n so that h has degree 1. In this way we get
a welldefined map u : SD (Sn) ! n. Conversely, given a homotopy
sphere n, consider a homotopy equivalence h : n ! Sn of degree 1.
In this way we get a welldefined map n ! SD (Sn) which is inverse
to u.
Notice that, because of the Smale Theorem, every smooth homotopy
sphere n, n 5, possesses a smooth function with just two critical
points. Thus, SD (Sn) = TD (Sn) = PD (Sn) for n 5. Kervaire and
Milnor [25 ] proved that 7 = Z=28, i.e., because of what we said above,
SD (S7) = TD (S7) = PD (S7) consists of 28 elements.
On the other hand, there are only 15 smooth manifolds which are
homeomorphic (and PL isomorphic, and homotopy equivalent) to S7
but mutually nondiffeomorphic. Indeed, if an oriented smooth 7
dimensional manifold is homeomorphic to S7 then bounds a par
allelizable manifold W , [25 ]. We equip W an orientation which is
compatible with and set
oe(W )
a( ) = _______ mod 28
8
where oe(W ) is the signature of W . Kervaire and Milnor [25 ] proved
that the correspondence
7 ! Z=28, 7! a(W )
is a welldefined bijection.
24 YULI B. RUDYAK
However, if a( 1) = a( 2) then 1 and 2 are diffeomorphic:
namely, 2 is just the 1 with the opposite orientation. So, there are
only 15 smooth manifolds which are homeomorphic (and homotopy
equivalent, and PL isomorphic) to S7 but mutually nondiffeomorphic.
In terms of structures, it can be expressed as follows. Let æ : Sn !
Sn be a diffeomorphism of degree 1. Then the smooth structural
izations h : 7 ! S7 and æh : 7 ! S7 are not concordant, if
a( 7) 6= 0, 14.
For convenience of references, we fix here the following theorem of
Smale [50 ]. Actually, Smale proved it for smooth manifolds, a good
proof can also be found in Milnor [36 ]. However, the proof can be
transmitted to the PL case, see Stallings [52 , 8.3, Theorem A].
3.11. Theorem. Let M be a closed PL manifold which is homotopy
equivalent to the sphere Sn, n 5. Then M is PL isomorphic to Sn. __
__
4. From manifolds to bundles
Recall that, for every topological manifold Mn , its tangent bundle øM
and (stable) normal bundle M are defined. Here øM is a topological
Rnbundle, and we can regard N as a topological RN bundle with
N >> n. Furthermore, if M is a PL manifold then øM and M turns
into PL bundles in a canonical way, see [28 , 48].
4.1. Construction. Consider a PL manifold M and a PL structural
ization h : V ! M. Let g = h1 : M ! V . Since g is a homeo
morphism, it yields a topological morphism ~ : M ! V , and so we
have the correcting topological morphism c(~) : M ! g* V . Now, the
morphism
c(~) * classif N
M  ! g V   ! flPL
is a PL structuralization of M . It is easy to see that in this way we
have the correspondence
jTOP : TPL (M) ! {PL structures on M } ! [M, T OP=P L]
where the last map comes from 3.4. Moreover, it is clear that, in fact,
jTOP passes through the map
[(M, @M), (T OP=P L, *)] ! [M, T OP=P L].
So, we can and shall regard jTOP as the map
jTOP : TPL (M) ! [(M, @M), (T OP=P L, *)].
PL STRUCTURES 25
4.2. Remark. We constructed the map jTOP using PL structuraliza
tions of M . However, we can also construct the map jTOP by consid
ering other bundles related to M. For example, consider the tangent
bundle øM . The topological morphism
øM  ! g*øV classif!flnPL
can be regarded as a PL structuralization of øM , and so we have an
other way of constructing of jTOP . Moreover, we can also consider the
topological morphism
`N+n = øM NM   ! g*øV NM classif!flN+nPL
and regard it as a PL structuralization of `N+n , etc. One can prove
that all these constructions are equivalent.
Now we construct a map jF : SPL (M) ! [M, F=P L], a öh motopy
analogueö f jTOP . This construction is more delicate, and we treat
only the case of closed manifolds here. So, let M be a connected closed
PL manifold.
4.3. Definition. Given an (Sn, *)fibration , = {E ! B} with a
section s : B ! E, we define its Thom space T , as the quotient space
E=s(B), Given a topological RN bundle j, we define the Thom space
T j as T j := T (jo).
Given a morphism ' : , ! j of (Sn, *)fibrations, we define T ' :
T , ! T j to be the unique map such that the diagram
E  ! E0
? ?
? ?
y y
T'
T ,  ! T j
commutes. Here E0 is the total space of j.
4.4. Definition. A pointed space X is called reducible if there is a
pointed map f : Sm ! X such that f* : Hei(Sm ) ! Hei(X) is an
isomorphism for i m. Every such map f (as well as its homotopy
class or its stable homotopy class) is called a reduction for X.
We embed M in RN+n , N >> n, and let M , dim M = N be a normal
bundle of this embedding. Recall that M is a PL bundle E ! M whose
total space E is PL isomorphic to a (tubular) neighbourhood U of M
in RN+n . We choose such isomorphism and denote it by ' : U ! E.
One can prove that, for N large enough, the normal bundle always
exists, [20 , 29, 30].
26 YULI B. RUDYAK
4.5. ConstructionDefinition. Let T M be the Thom space of M .
Then there is a unique map
_ : RN+n =(RN+n \ U) ! T
such that _U = '. We define the collapse map ' : SN+n ! T M (the
BrowderNovikov map) to be the composition
quotientN+n N+n N+n N+n _
' : SN+n   ! S =(S \ U) = R =(R \ U)  ! T .
See [5, II.2.11] or [48 , 7.15] for details.
It is well known and easy to see that ' is a reduction for T , see
Corollary 11.7 below.
It turns out that, for N large enough, the normal bundle of a given
embedding M ! RN+n is unique. For detailed definitions and proofs,
see [20 , 29, 30]. The uniqueness gives us the following important fact.
Let 0 = {E0 ! M} be another normal bundle and '0 : U0 ! E0 be
another PL isomorphism. Let ' : SN+n ! T and '0: SN+n ! T 0 be
the corresponding BrowderNovikov maps. Then there is a morphism
! 0 of PL bundles which carries ' to a map homotopic to '0.
4.6. Theorem. Consider a PL RN bundle j over M such that T j is
reducible. Let ff 2 ßN+n (T j) be an arbitrary reduction for T j. Then
there exist an FN equivalence ~ : oM ! jo such that (T ~)*(') = ff, and
such a ~ is unique up to fiberwise homotopy over M.
__
Proof. We postpone it to the next Chapter, see 11.11. __
4.7. ConstructionDefinition. Given a homotopy equivalence h :
V ! M of closed connected PL manifolds, let V be a normal bundle
of a certain embedding V RN+n , and let u 2 ßN+n (T V ) be the
homotopy class of a collapsing map SN+n ! T V . Let g : M ! V be
homotopy inverse to h and set j = g* V . The adjoint to g morphism
' := Ig, V: j ! V
yields the map T ' : T j ! T V . It is easy to see that T ' is a homo
topy equivalence, and so there exists a unique ff 2 ßN+n (T j) with
(T ')*(ff) = u. Since u is a reduction for T V , we conclude that
ff is a reduction for T j. By Theorem 4.6, we get an F equivalence
~ : oM ! jo with (T ~)*(') = ff. Now, the morphism
~ o classif N
( M )o  ! j   !flF
PL STRUCTURES 27
is a homotopy PL structuralization of M . Because of the uniqueness
of the normal bundle, the concordance class of this structuralization is
well defined. So, in this way we have the correspondence
jF : SPL (M) ! {homotopy PL structures on M } ~=[M, F=P L]
where the last bijection comes from 3.8.
The map jF is called the normal invariant, and its value on a homo
topy PL structure is called the normal invariant of this structure.
Notice that there is a commutative diagram
jTOP
TPL (M)  ! [M, T OP=P L]
? ?
(4.1) fi?y ?ya*
jF
SPL (M)  ! [M, F=P L]
where fi regards a PL structuralization as the homotopy PL structural
ization.
5. Homotopy PL structures on T kx Dn
Below T kdenotes the kdimensional torus.
5.1. Theorem. Assume that k + n 5. If x 2 SPL (T kx Sn) can_be_
represented by a homeomorphism M ! T kx Sn then jF (x) = 0. __
This is a special case of the Sullivan Normal Invariant Homeomor
phism Theorem. We prove 5.1 (in fact, a little bit general result) in
the next chapter.
We also prove the Sullivan Theorem in full generality in Chapter 3, section 2*
*1. We
must do this repetition since the proof in Chapter 3 uses 5.1.
5.2. ConstructionDefinition. Let x 2 TPL (M) be represented by
a map h : V ! M, and let p : fM ! M be a covering. Then we have a
commutative diagram
Ve  eh!Mf
? ?
q?y ?yp
V  h! M
where q is the induced covering. Since ehis defined uniquely up to deck
transformations, the concordance class of ehis well defined. So, we have
a welldefined map
p* : TPL (M) ! TPL (fM )
28 YULI B. RUDYAK
where p*(x) is the concordance class of eh. Similarly, one can construct
a map
p* : SPL (M) ! SPL (fM ).
If p* is a finite covering, we say that a class p*(x) 2 SPL (fM ) finitely
covers the class x.
5.3. Theorem. Let k + n 5 Then the following holds:
(i) if n > 3 then the set SPL (T kx Dn) consists of precisely one
(trivial) element;
(ii) if n < 3 then every element from SPL (T kx Dn) can be finitely
covered by the trivial element;
(iii) the set SPL (T kxD3) contains at most one element which cannot
be finitely covered by the trivial element.
Some words about the proof. First, we mention the proof given
by Wall, [62 ] and [63 , Section 15 A]. Wall proved the bijection w :
SPL (T kx Dn) ! H3n (T k). Moreover, he also proved that finite cov
erings respect this bijection, i.e. if p : T kx Dn ! T kx Dn is a finite
covering then there is the commutative diagram
SPL (T kx Dn) w! H3n (T k; Z=2)
x x
p*?? ??p*
SPL (T kx Dn) w! H3n (T k; Z=2) .
Certainly, this result implies all the claims(i)(iii). Walls proof uses
difficult algebraic calculations.
Another proof of the theorem can be found in [23 , Theorem C].
Minding the complaint of Novikov concerning Sullivan's results (see
Prologue), we must mention that Hsiang and Shaneson [23 ] use a Sulli
van's result. Namely, they consider the socalled surgery exact sequence
 @! S k n jF k n
PL (S x T )  ! [S x T , F=P L]  ! . . .
and write (page 42, Section 10):
By [44], every homomorphism h : M ! Sk x T n, k = n 5,
represents an element in the image of @.
Here the item [44] of the citation is our bibliographical item [56 ]. So,
in fact, Hsiang and Shaneson use Theorem 5.1. As I already said, we __
prove 5.1 in next Chapter and thus fix the proof. __
PL STRUCTURES 29
6. The product structure theorem, or from bundles to
manifolds
Let M be an ndimensional @PL manifold. Then every PL struc
turalization h : V ! M yields a PL structuralization
h x 1 : V x Rk ! M x Rk.
Thus, we have a welldefined map
e : TPL (M) ! TPL (M x Rk).
6.1. Theorem (The Product Structure Theorem). For every n 5
and every k 0, the map e : TPL (M) ! TPL (M x Rk) is a bijection.
In particular, if TPL (M x Rk) 6= ; then TPL (M) 6= ;.
Concerning the proof. I did not find a proof which is essentially better
then the original one. So, I refer the reader to the original source [28 ].
I want also to mention here that the proof of Theorem 6.1 uses the
Theorem 5.3 for n = 0. For another approach to the proof of Theorem __
6.1, see [13 , Remark 5.3]. __
6.2. Corollary (The Classification Theorem). If dim M 5 and M
admits a PL structure, then the map
jTOP : TPL (M) ! [(M, @M), (T OP=P L, *)]
is a bijection.
Proof. We construct a map
(6.1) oe : [(M, @M), (T OP=P L, *)] ! TPL (M)
which is inverse to jTOP . For simplicity of notations, we consider the
case of M closed. Take an element a 2 [M, T OP=P L] and, using 3.4,
interpret it as a concordance class of a morphism ' : `NM ! flNPL of
topological RN bundles (cf. Remark 4.2). The morphism ' yields a
correcting isomorphism `NM ! b*flNPL of topological RN bundles over
M, where b : M ! BP L is the base of the morphism '. So, we have
the commutative diagram
M x RN h! W
? ?
? ?
y y
M _______M
where h is a fiberwise homeomorphism and W ! M is a PL RN bundle
b*flNPL. In particular, W is PL manifold. Regarding h1W ! M x RN
as a PL structuralization of M x RN , we conclude that, by the Product
30 YULI B. RUDYAK
Structure Theorem 6.1, h1 is concordant to a map g x 1 for some PL
structuralization g : V ! M. We define oe(a) 2 TPL (M) to be the
concordance class of g. One can check that oe is a welldefined map __
which is inverse to jTOP . Cf. [28 , Essay IV]. __
6.3. Corollary (The Existence Theorem). A topological manifold M
with dim M 5 admits a PL structure if and only if the tangent bundle
to M admits a PL structure.
Proof. Only claim "if" needs a proof. Let ø = {ß : D ! M} be the
tangent bundle of M, and let = {r : E ! M} be a stable normal
bundle of M, dim = N. Then E is homeomorphic to an open subset
of RN+n , and therefore we can (and shall) regard E as a PL manifold.
Since ø is a PL bundle, we conclude that r*ø is a PL bundle over E.
In particular, the total space M x RN+n of r*ø turns out to be a PL
manifold. Now, because of the Product Structure Theorem 6.1, M __
admits a PL structure. __
Let f : M ! BT OP classify the stable tangent bundle of a closed
topological manifold M, dim M 5.
6.4. Corollary. The following conditions are equivalent:
(i) M admits a PL structure;
(ii) ø admits a PL structure;
(iii) there exists k such that ø `k admits a PL structure;
(iv) the map f admits an ffPLTOPlifting to BP L.
Proof. It suffices to prove that (iv) =) (iii) =) (i). The implication
(iii) =) (i) can be proved similarly to 6.3. Furthermore, since M is
compact, we conclude that f(M) BT OPm for some m. So, if (iv) __
holds then f lifts to BP Lm , i.e. ø `mk admits a PL structure. __
6.5. Remark. It follows from 1.3, 6.3 and 6.2 that the set TPL (M)
of PL structures on M is in a bijective correspondence with the set of
fiber homotopy classes of ffPLTOPliftings of f. (We leave it to the reader
to extend the result on @PL manifolds.)
6.6. Remark. It is well known that jF is not a bijection in general.
The "kernel"' and öc kernelö f jF can be described in terms of so
called Wall groups, [63 ]. (For M simplyconnected, see also Theorem
13.2.) On the other hand, the bijectivity of jTOP (the Classification
Theorem) follows from the Product Structure Theorem. So, informally
speaking, kernel and cokernel of jF play the role of obstructions to
splitting of structures. It seems interesting to develop and clarify these
naive arguments.
PL STRUCTURES 31
6.7. Remark. Since tangent and normal bundles of smooth manifolds
turn out to be vector bundles, one can construct a map
k : PD (M) ! [M, P L=O]
which is an obvious analog of jTOP (here we assume M to be closed).
Moreover, the obvious analog of the Product Structure Theorem (as
well as of the Classification and Existence Theorems) holds without
any dimensional restriction. In particular, k is a bijection for every
smooth manifold, [22 ].
It is well known (although difficult to prove) that ßi(P L=O) = 0
for i 6. (See [48 , IV.4.27(iv)] for the references.) Thus, every PL
manifold M of dimension 7 admits a smooth structure, and this
structure is unique if dim M 6.
7. Noncontractibility of T OP=P L
7.1. Theorem (Freedman's Example). There exists a closed topolog
ical 4dimensional manifold F with w1(F ) = 0 = w2(F ) and such that
the signature of F is equal to 8.
Here wi denotes the ith StiefelWhitney class.
__
Proof. See [14 ] or the original work [15 ]. __
7.2. Theorem (Rokhlin Signature Theorem). Let M be a closed 4
dimensional PL manifold with w1(M) = 0 = w2(M). Then the signa
ture of M is divisible by 16.
Proof. See [26 , 37] or the original work [44 ]. In fact, Rokhlin proved the
result for smooth manifolds, but the proof works for PL manifolds as
well. On the other hand, in view of 6.7, there is no difference between_
smooth and PL manifolds in dimension 4. __
7.3. Corollary. The topological manifolds F and F x T k, k 1 do
not admit any PL structure.
Proof. The claim about F follows from 7.2. Suppose that F x T k
has a PL structure. Then F x Rk has a PL structure. So, because
of the Product Structure Theorem 6.1, F x R has a PL structure.
Hence, by 6.7, it possesses a smooth structure. Then the projection
p2 : F x R ! R can be C0approximated by a map f : F x R ! R
which coincides with p2 on F x (1, 0) and is smooth on F x (1, 1).
Take a regular value a 2 (0, 1) of f (which exists because of the Sard
Theorem) and set W = f1 (a). Then W is a smooth manifold (by the
Implicit Function Theorem), and it is easy to see that w1(W ) = 0 =
32 YULI B. RUDYAK
w2(W ) (because it holds for both manifolds R and F x R). On the
other hand, both manifolds F and W cut the üt be" F x R. So, they
are (topologically) bordant, and therefore W has signature 8. But this
contradicts the Rokhlin Theorem 7.2. __
__
7.4. Corollary. The space T OP=P L is not contractible.
Proof. Indeed, suppose that TOP/PL is contractible. Then every map
X ! BT OP lifts to BP L, and so, by 6.3, every topological manifold of
dimension greater than 4 admits a PL structure. But this contradicts __
7.3. __
7.5. Remark. Kirby and Siebenmann [27 , 28] constructed the original
example of a topological manifold which does not admit a PL structure.
Again, the Rokhlin Theorem 7.2 is one of the main ingredients of the
proof.
8. Homotopy groups of T OP=P L
Let M be a compact topological manifold equipped with a metric æ.
Then the spacefHiof homeomorphisms gets a metric d with d(f, g) =
sup{x 2 M fiæ(f(x), g(x))}.
8.1. Theorem. The space H is locally contractible.
__
Proof. See [7, 11]. __
8.2. Corollary. There exists " > 0 such that every homeomorphism __
h 2 H with d(H, 1M ) < " is isotopic to 1M . __
8.3. Construction. We regard the torus T k as a commutative Lie
group (multiplicative) and equip it with the invariant metric æ. Con
sider the map p~ : T k! T k, p~(a) = a~, ~ 2 N. Then p~ is a ~ksheeted
covering. It is also clear that all the deck transformations of the cov
ering torus are isometries.
8.4. Lemma. Let h : T kx Dn ! T kx Dn is a selfhomeomorphism
which is homotopic rel@(T kx Dn) to the identity. Then there exists a
commutative diagram
eh
T kx Dn  ! T kx Dn
? ?
p~?y ?yp~
T kx Dn h! T kx Dn
where the lifting ehof h is isotopic rel@(T kx Dn) to the identity.
PL STRUCTURES 33
Proof. (Cf. [28 , Essay V].) First, consider the case n = 0. Without
loss of generality we can assume that h(e) = e where e is the neutral
element of T k. Consider a covering p~ : T k! T kas in 8.3 and take a
covering eh: T k! T k, p~eh= ehp~ of h such that eh(e) = e. In order to
distinguish the domain and the range of p~, we denote the domain of
p~ by eTand the range of p~ by T . Since all the deck transformations of
eTare isometries, we conclude that the diameter of each of (isometric)
fundamental domain tends to zero as ~ ! 1. Furthermore, since h is
homotopic to 1T , we conclude that every point of the lattice L := p1~(e)
is fixed under eh.
Given " > 0, choose ffi such that æ(eh(x), eh(y)) < "=2 whenever
æ(x, y) < ffi. Furthermore, choose ~ so large that the diameter of any
closed fundamental domain is less then min {"=2, ffi}. Now, given x 2 eT,
choose a 2 L such that a and x belong to the same closed fundamental
domain. Now,
" "
æ(x, eh(x) æ(x, a) + æ(a, eh(x)) = æ(x, a) + æ(eh(a), eh(x)) < __+ __= ".
2 2
So, for every " > 0 there exists ~ such that d(eh, 1Te) < ". Thus, by 8.2,
ehis isotopic to 1Tefor ~ large enough.
The proof for n > 0 is similar but a bit more technical. Let D'' Dn
be the disk centered at 0 and having the radius j. We can always as
sume that h coincides with identity outside of T kx D''. Now, asserting
as for n = 0, take a covering p~ as above and choose ~ and j so small
that the diameter of every fundamental domain in Te x D''is small
enough. Then
eh: eTx D''! eTx D''
is isotopic to the identity (and ehcoincides with identity outside eTxD'').
This isotopy is not an isotopy rel eTx @D''. Nevertheless, we can easily
extend it to the whole Te x Dn so that this extended isotopy is an
isotopy rel@(Tex Dn).
If you want formulae, do the following. Given a = (b, c) 2 eTx D'',
set a = c. Consider an isotopy
' : eTxD''xI ! eTxD''xI, '(a, 0) = a, '(a, 1) = eh(a), a 2 eTxD''.
Define _ : eTx D''x I ! eTx D''= xI by setting
æ
'(a, t) ifa j,
_(a, t) = a1
'(a, ____''1t)ifa j.
__
Then _ is the desired isotopy rel@(Tex Dn). __
34 YULI B. RUDYAK
8.5. Corollary. Let fi : TPL (T kxDn) ! SPL (T kxDn) be the forgetful
map as in (4.1). If fi(x) = fi(y) then there exists a finite covering_
p : T kx Dn ! T kx Dn such that p*(x) = p*(y). __
Consider the map
p*2
_ : ßn(T OP=P L) = [(Dn, @Dn), (T OP=P L, *)] !
jTOP k n
[(T kx Dn, @(T kx Dn)), T OP=P L]  ! TPL (T x D )
where oe is the map from (6.1).
8.6. Lemma. The map _ is injective. Moreover, if p*_(x) = p*_(y)
for some finite covering p : T kx Dn ! T kx Dn then x = y.
In particular, if p*_(x) is the trivial element of TPL (T kx Dn) then
x = 0.
Proof. The injectivity of _ follows from the injectivity of p*2and oe.
Furthermore, for every finite covering p : T kx Dn ! T kx Dn we have
the commutative diagram
_ k n
ßn(T OP L=P L)   ! TPL (T x D )
fl x
fl ? *
fl ? p
_ k n
ßn(T OP L=P L)   ! TPL (T x D )
Therefore x = y whenever p*_(x) = p*_(y). Finally, if p*_(x) is trivial_
element then p*_(x) = p*_(0), and thus x = 0. __
Consider the map
_ k n fi k n
' : ßn(T OP=P L)   ! TPL ((T x D )  ! SPL ((T x D )
where fi is the forgetful map described in (4.1).
8.7. Theorem (The Reduction Theorem). The map ' is injective.
Moreover, if p*'(x) = p*'(y) for some finite covering
p : T kx Dn ! T kx Dn
then x = y.
In particular, if p*'(x) is the trivial element of TPL (T kx Dn) then
x = 0.
We call it the Reduction Theorem because it reduces the calculation
of the group ßi(T OP=P L) to the calculation of the sets SPL (T kx Dn).
Proof. If '(x) = '(y) then fi_(x) = fi_(y). Hence, by Corollary
8.5, there exists a finite covering ß : T kx Dn ! T kx Dn such that
ß*_(x) = ß*_(y). So, by Lemma 8.6, x = y, i.e. ' is injective.
PL STRUCTURES 35
Now, suppose that p*'(x) = p*'(y) for some finite covering p :
T kx Dn ! T kx Dn. Then fi*p*_(x) = fi*p*_(y). Now, by Corollary
8.5, there exists a finite covering
q : T kx Dn ! T kx Dn
such that q*p*_(x) = q*p*_(y), i.e. (pq)*_(x) = (pq)*_(y). Thus, by_
Lemma 8.6, x = y. __
8.8. Corollary (The Main Theorem). ßi(T OP=P L) = 0 for i 6= 3.
Furthermore, ß3(T OP=P L) = Z=2. Thus, T OP=P L = K(Z=2.3).
Proof. The equality ßi(T OP=P L) = 0 for i 6= 3 follows from The
orem 5.3(i,ii) and Theorem 8.7. Furthermore, again because of 5.3
and 8.7, we conclude that ß3(T OP=P L) has at most two elements.
In other words, T OP=P L = K(ß, 3) where ß = Z=2 or ß = 0. Fi
nally, by Corollary 7.4, the space T OP=P L is not contractible. Thus,_
T OP=P L = K(Z=2, 3). __
8.9. Remark. Notice that, for i > 5, the set TPL (Si) consists of just
one element by the Smale Theorem 3.11. Because of this, the equality
ßi(T OP=P L) = 0, i > 5 follows from Theorem 6.2. However, for i
small we need Theorem 5.3. (Moreover, the proof of Theorem 6.1 uses
5.3 for n = 0.)
From now on and till the end of the section we fix a closed topological
manifold M and let f : M ! BT OP denote the classifying map for
the stable tangent bundle of M.
8.10. ConstructionDefinition. Let
 2 H4(BT OP ; ß3(T OP=P L)) = H4(BT OP ; Z=2)
be the characteristic class of the fibration
i := {ffPLTOP: BP L ! BT OP },
(see e.g. [39 , 48 ] for the definition of the characteristic class of the
fibration). We define the KirbySiebenmann class (M) 2 H4(M; Z=2)
of M by setting
(M) = f*.
Clearly, the class (M) can also be described as the characteristic
class of the T OP=P Lfibration f*i over M.
8.11. Corollary. The manifold M admits a PL structure if and only
if (M) = 0. In particular, if H4(M; Z=2) = 0 then M admits a PL
structure. Furthermore, if M admits a PL structure then the set of all
PL structure on M is in a bijective correspondence with H3(M; Z=2).
36 YULI B. RUDYAK
Proof. By 6.4, M admits a PL structure if and only if f admits an
ffPLTOPlifting.
BP L
?
?
y
f
M   ! BT OP
By the Main Theorem 8.8, the fiber of the fibration
ffPLTOP: BP L ! BT OP
is the EilenbergMac Lane space K(Z=2, 3). Thus, because of the
obstruction theory, f lifts to BP L if and only if f* = 0.
Finally, by 6.2 and 8.8, we have the bijections
TPL (M) ~=[M, T OP=P L] ~=[M, K(Z=2, 3] = H3(M; Z=2)
__
provided TPL (M) 6= ;. __
9. Do it
We recommend that the reader return to the introduction and look
again the graph of our proof of the Main Theorem.
PL STRUCTURES 37
Chapter 2. Tools
Here we prove two important results which we used without proofs
in Chapter 1. First, we prove Theorem 4.6. Notice that Browder
[5] proved the Theorem for M simplyconnected. In fact, his proof
works for every orientable M. Here we follow Browder's proof, the
only essential modification is that we use the duality theorem 11.6 for
arbitrary M while Browder [5] uses it for M simplyconnected.
Another goal of this chapter is to prove Theorem 5.1. In fact, we
prove here a little bit more general result, Theorem 17.1. The proof
uses the Sullivan's result on the homotopy type of F=P L. Notice that
Madsen and Milgram [31 ] gave a detailed proof of those Sullivan result.
10. Stable equivalences of spherical bundles
We denote by oek = oekX the trivial Skbundle over X with a fixed
section. In another words, oek = (`k)o.
Given a sectioned spherical bundle , over a finite CW space X, let
aut, denote the group of fiberwise homotopy classes of selfequivalences
, ! , over X, where we assume that all homotopies preserve the
section.
10.1. Proposition. There is a natural bijection
aut oek = [X, Fk].
Proof. Because of the exponential law, every map X ! Fk yields a
sectionpreserving map X x Sk ! X x Sk over X, and vice versa. Cf. __
[5, Prop. I.4.7]. __
Consider the map
~ : Fk x Fk ! F2k, ~(a, b) = a ^ b : S2k = Sk ^ Sk ! Sk ^ Sk = S2k
where we regard a, b 2 Fk as pointed maps Sk ! Sk. Let T : FkxFk !
Fk x Fk be the transpose map, T (a, b) = (b, a).
10.2. Lemma. The maps ~ : Fk x Fk ! F2k and ~T : Fk x Fk !
F2k, k > 0 are homotopic.
Proof. Consider the map
ø : S2k = Sk ^ Sk ! Sk ^ Sk = S2k, ø (u, v) = (v, u)
and notice that, for every a, b 2 Fk, we have
(~OT )(a, b) = ø O~(a, b)Oø.
38 YULI B. RUDYAK
First, consider the case of k odd. Then there is a pointed homotopy
ht between ø and 1S2k. Now, the pointed homotopy htO~(a, b)Oht is
a pointed homotopy between (~OT )(a, b) and ~(a, b) which yields a
homotopy between ~T and ~.
Now consider the case of k even. We regard S2k as R2k [ 1 with
R2k = {(x1, . .,.x2k)} and define ø 0, ø 00: S2k ! S2k by setting
ø 0(x1, x2, x3, . .,.x2k)=(x2, x1, x3, . .,.x2k),
ø 00(x1, . .,.x2k2x2k1x2k)= (x1, . .,.x2k1, x2k2, x2k),
(i.e. ø 0permutes the first two coordinates and ø 00permutes the last
two coordinates). Since k is even, we conclude that ø 0' ø ' ø 00.
Furthermore, ø 00ø 0' 1S2k. If we fix such pointed homotopies then we
get the pointed homotopies
(~OT )(a, b) = ø O~(a, b)ø ' ø 00O~(a, b)ø 0= ø 00O(a ^ b)ø 0
= ø 00O(a ^ 1)O(1 ^ b)ø 0= (a ^ ø 00)O(ø 0^ b)
= (a ^ 1)O(ø 00ø 0)O(1 ^ b) ' a ^ b = ~(a, b)
__
which yield the homotopy ~OT ' ~. __
10.3. Corollary. Let ', _ : oek ! oek be two automorphisms of oek.
Then the automorphisms 'y_ and _ y' of oe2k are fiberwise homotopic. __
__
Given two spherical bundles , and j over X, consider the bundle
, ^ j over X x X. We denote by : X ! X x X the diagonal and
consider the adjoint bundle morphism
J := I ,,^'': , y j ! , ^ j.
10.4. Proposition. For every automorphism ' : j ! j the diagram
, y j J! , ^ j
? ?
1y'?y ?y1^'
, y j J! , ^ j
__
commutes __
10.5. Corollary. The diagram
(1y1)^'
, y j y jJ! (, y j) ^ j  ! (, y j) ^ j
fl fl
fl fl
fl fl
(1y')^1
, y j y jJ! (, y j) ^ j  ! (, y j) ^ j
__
commutes up to homotopy. __
PL STRUCTURES 39
11. Proof of Theorem 4.6
We need some preliminaries on stable duality [51 ]. Given a pointed
map f : X ! Y , let Sf : SX ! SY denote the (reduced) suspension
over f. So, we have a welldefined map S : [X, Y ]o ! [SX, SY ]o.
11.1. Proposition. Suppose that ßi(Y ) = 0 for i < n and that X
is a CW space with dim X < 2n  1. Then the map S : [X, Y ]o !
[SX, SY ]o is a bijection.
Proof. This is the wellknown Freudenthal Suspension theorem, see e.g __
[58 ] __
Given two pointed spaces X, Y , we define {X, Y } to be the direct
limit of the sequence
[X, Y ]oS! [SX, SY ]o S! . .. ! [SnX, SnY ]o  S! . . ..
In particular, we have the obvious maps
(Y, *)(X,*)! [X, Y ]o ! {X, Y }.
The image of a pointed map f : X ! Y in {X, Y } is called the stable
homotopy class of f. The standard notation for this one is {f}, but,
as usual, in most cases we will not distinguish f, [f] and {f}.
It is well known that, for n 2, the set [SnX, SnY ]o has a natural
structure of the abelian group, and the corresponding maps S are ho
momorphisms, [58 ]. So, {X, Y } turns out to be a group. Furthermore,
by Theorem 11.1, if X is a finite CW space then the map
[SN X, SN Y ]o ! {SN X, SN Y }
is a bijection for N large enough.
11.2. Definition. A map f : Sd ! A^A? is called a (stable) dduality
if the maps
uE : {A, E} ! {S, E ^ A? }, uE (') = (' ^ 1A? )u
and
uE : {A? , E} ! {S, A ^ E}, uE (') = (1A ^ ')u
are isomorphisms.
11.3. Proposition. Let u : Sd ! A ^ A? be a dduality between two
finite CW spaces. Then, for every i and ß, u yields an isomorphism
Hi(u; ß) : eHi(A? ; ß) ! eHdi(A, ß).
40 YULI B. RUDYAK
Proof. Recall that
Hn (A? ; ß) = [A, k(ß, n)]o = [SN A, K(ß, N + n)]o
where K(ß, i) is the EilenbergMac Lane space. Because of Theorem
11.1, the last group coincides with {SN A, K(ß, N + n)} for N large
enough, and therefore
Hn (A? ; ß) = {SN A, K(ß, N + n)} for N large enough .
Furthermore, let "n : SK(ß, n) ! K(ß, n + 1) be the map which is
adjoint to the standard homotopy equivalence K(ß, n) ! K(ß, n+1),
see e.g. [58 ]. Whitehead [64 ] noticed that
Hen(A; ß) = lim[SN+n , K(ß, N) ^ A]o.
!
Here lim!is the direct limit of the sequence
[SN+n , K(ß, N) ^ A]o ! [SN+n+1 , SK(ß, N) ^ A]o
"*! [SN+n+1 , K(ß, N) ^ A]o
(see [19 , Ch 18] or [48 , II.3.24] for greater details). Since "n is an
nequivalence, and because of Theorem 11.1, we conclude that
Hen(A; ß) = [SN+n , K(ß, N) ^ A] for N large enough .
So, again because of Theorem 11.1,
eHn(A; ß) = {SN+n , K(ß, N) ^ A}
for N large enough.
Now, consider a dduality u : Sd ! A ^ A? . Fix i and choose N
large enough such that
Hei(A? ; ß) = {SN A? , K(ß, N + i)},
eHdi(A; ß) = {SN+d , K(ß, N + i) ^ A}.
By suspending the domain and the range, we get a duality (denoted
also by u)
u : SN+d ! A ^ SN A? .
This duality yields the desired isomorphism
Hi(u; ß) : = uK(i,N+i) : eHi(A? ; ß) = {SN A? , K(ß, N + i)}
! {SN+d , K(ß, N + i) ^ A} = eHdi(A; ß)
__
__
PL STRUCTURES 41
11.4. Definition. Dualizing 4.4, we say that a pointed map a : A !
Sk (or its stable homotopy class a 2 {A, Sk}) is a coreduction if the
induced map
a* : eHi(Sk) ! eHi(A)
is an isomorphism for i k.
11.5. Proposition. Let u : Sd ! A ^ A? be a dduality between two
finite CW spaces, and let k d. A class ff 2 {A? , Sk} is a coreduction
k dk
if and only if the class fi := uS ff 2 {S , A} is a reduction.
Proof. Let Hi(u) : eHi(A? ) ! eHdi(A) be the isomorphism as in 11.3.
Notice that the standard homeomorphism v : Sd ! Sk ^ Sdk is a
dduality. It is easy to see that the diagram
Hei(A? ) Hi(u)!eHdi(A)
x x
ff*?? ??fi*
Hei(Sk) Hi(v)!eHdi(Sdk)
commutes. In particular, the left vertical arrow is an isomorphism if __
and only if the right one is. __
Consider a closed connected ndimensional PL manifold M and em
bed it in RN+n+k with N large enough. Let ' : SN+n+k ! T N+k be a
collapse map as in 4.5, and let
J : ( N+k )o = ( N )o y oek ! ( N )o ^ oek
be the morphism as in 10.4.
11.6. Theorem. The map
SN+n+k  '! T N+k TJ! T N ^ oek
is an (N + n + k)duality map.
Proof. This is actually proved in [10 ]. For greater detail, see [48 ,
V.2.3(i)] (where in the proof the reference 2.8(a) must be replaced by__
2.8(b)). __
11.7. Corollary. The collapse map ' : SN+n ! T N is a reduction.
Proof. Recall that T oek = (M x Sk)=M = Sk(M+ ). Consider a sur
jective map e : M+ ! S0 and define " = Ske : T oek ! Sk. Since the
map
Sk' : SkSN+n = SN+n+k ! T N+k = SkT
42 YULI B. RUDYAK
can be written as
k' N+k N k 1^" N k k
SN+n+k S! T = T ^ T oe ! T ^ S = S T ,
where the composition of first two maps is the duality from 11.6, we
conclude that ' is dual to " with respect to duality (11.2). Clearly, "_is
a coreduction. Thus, the result follows from 11.5. __
For technical reasons, it will be convenient for us to consider the
duality
(11.1) SN+n+2k ! T N+2k TJ!T N+k ^ T oek.
This duality yields an isomorphism
(11.2)
k k k N+n+2k k N+k N+n+k N+k
D := uS : {T oe , S } ! {S , S ^ T } = {S , T }.
11.8. Proposition. For every automorphism ' : oek ! oek the follow
ing diagram commutes up to homotopy:
T(1y')^1 N+k k
SN+n+2k  '! T N+2k TJ! T N+k ^ T oek   !T ^ T oe
fl fl fl
fl fl fl
fl fl fl
1^T' N+k k
SN+n+2k   ! T N+2k TJ! T N+k ^ T oek  ! T ^ T oe
__
Proof. This follows from 10.5 . __
Every automorphism ' : oek ! oek yields a homotopy equivalence
T (1 y ') : T N+k = T ( N y oek)! T ( N y oek) = T N+k
and hence an isomorphism
T (1 y ')* : {SN+n+k , T N+k } ! {SN+n+k , T N+k }.
So, we have the aut oekaction
a : aut oek x {SN+n+k , T N+k }! {SN+n+k , T N+k },
a (', ff) = T (1 y ')*(ff).
Similarly, every automorphism ' of oek induces a homotopy equivalence
T oek ! T oek, and therefore we have the action
aff: aut oek x {T oek, Sk} ! {T oek, Sk}.
11.9. Theorem. The diagram
aut oek x {T oek, Sk} aoe! {T oek, Sk}
? ?
1xD ?y ?yD
aut oek x {SN+n+k , T N+k }a! {SN+n+k , T N+k }
PL STRUCTURES 43
commutes.
__
Proof. This follows from 11.8 and the definition of D, a and aff. __
Because of Theorem 11.1, for k large enough we have
{T oek, Sk} = ßk(T oek) and {SN+n+k , T N+k } = ßN+n+k (T N+k ).
Then we can rewrite the diagram from Theorem 11.9 as
aoe
aut oek x ßk T oek  ! ßk T oek
? ?
(11.3) 1xD?y ?yD
a
autoek x ßN+n+k T N+k  ! ßN+n+k T N+k
Let R 2 ßN+n+k (T N+k ) be the set of reductions, and let C 2
ßk(T oek) be the set of coreductions. Then, clearly, a (R) R and
aff(C) C. Therefore, in view of Proposition 11.5, the diagram (11.3)
yields the commutative diagram
aut oek x C aoe!C
? ?
(11.4) 1xD ?y ?yD
aut oek x R a! R
11.10. Theorem. For every ff, fi 2 C there exists an automorphism '
of oek such that aff(', ff) = fi. Moreover, this ' is unique up to fiberwise
homotopy. In other words, the action aff: aut oek x C ! C is free and
transitive.
Proof. Recall that T oek = (M xSk)=M. So, for every m 2 M, a pointed
map f : T oek ! Sk yields a pointed map fm : Skm ! Sk where Skm is
the fiber over m. Furthermore, f represents a coreduction if and only
if all maps fm belong to Fk. In other words, every coreduction for T oek
yields a homotopy class M ! Fk. Moreover, it is easy to see that, in
view of Proposition 10.1, the action affcoincides with the map
[M, Fk] x [M, Fk] ! [M, Fk]
__
induced by the product in Fk, and the result follows. __
Since D is an isomorphism, Theorem 11.10 yields the following corol
lary.
11.11. Corollary. The action a : aut oek x R ! R is free and tran __
sitive. __
44 YULI B. RUDYAK
Now we can finish the proof of Theorem 4.6. Assuming dim j = N +k
to be large enough, we conclude that o and jo are homotopy equivalent
over M, see Atiyah [2, Prop. 3.5]. (Notice that Atiyah works with
nonsectioned bundles, but there is no problem to adapt the proof for
sectioned ones.) Choose any such FN+k equivalence ' : jo ! o and
consider the induced homotopy equivalence T ' : T j ! T . Clearly,
the composition
T'
fi : SN+n+k  ff!T j  ! T
is a reduction. So, by 11.11, there exists an FN+k equivalence ~ : o !
jo over M with (T ~)*(fi) = '. Now, we define ~ : o ! jo to be the
fiber homotopy inverse to ~'. (The existence of an inverse equivalence
can be proved following Dold [9], cf. [32 ]). Clearly, ~*' = ff. This
proves the existence of the required equivalence ~.
Furthermore, if there exists another equivalence ~0 : jo ! o, then
~0O~1(') = ', and so ~ and ~0 are homotopic over M. This proves the__
uniqueness of ~. Thus, Theorem 4.6 is proved. __
12. Normal maps and F=P L
Throughout the section we fix a closed PL manifold Mn .
12.1. Definition ([5, 31]). A normal map at M is a commutative di
agram of PL maps
bb
E   ! E0
? ?
? ?
y y
V  b! M
where V = {E ! V } is the normal PL RN bundle over a closed PL
manifold V n, , = {E0 ! M} is a PL RN bundle over M and, finally,
bbinduces a PL isomorphism of fibers and preserves the sections. In
other words, a normal map is a bundle morphism ' : V ! ,.
A normal map is called reducible if the map
collapse T'
SN+n    !T V  ! T ,
is a reduction.
Because of the Thom Isomorphism Theorem, a normal map is re
ducible whenever b is a map of degree 1. (One can prove that , is
orientable if V and M are.)
PL STRUCTURES 45
12.2. ConstructionDefinition. Given a map (homotopy class) f :
M ! F=P L, we represent it by an FN morphism ' : oM ! (flNPL)o
with N large enough, see 3.5 and/or 3.8. We denote the base of ' by b
and set , = b*flNPL. Then the correcting FN morphism oM ! ,o yields
a commutative diagram
g 0 o
Uo   ! U
? ?
(12.1) q?y ?yp
M _______ M
where M = {q : U ! M}, , = {p : U0 ! M} and Uo, U0oare fiberwise
onepoint compactifications of U and U0, respectively.
We consider M as the zero section of ,, M U0 and deform g to a
map t : Uo ! U0o which is transversal to M. Set V = t1(M). We can
always assume that V U. So, we get a morphism of PL RN bundles
bb
E   ! E0
? ?
(12.2) ? ?
y y
V  b! M
where {E0 ! M} is a normal bundle of M in U0 and {E ! V } is a
normal bundle of V in U. Here b = tV . Clearly, , := {E0 ! M}
is the bundle of M in U0. Furthermore, since U is a total space of a
normal bundle, the normal bundle of U is trivial. Thus, {E0 ! M} is
the normal bundle V of V . In other words, the diagram (12.2) is a
normal map at M. We say that the normal map (12.2) is associated
with a map (homotopy class) f : M ! F=P L.
Clearly, there any many normal maps associated with a given map
f : M ! F=P L.
12.3. ConstructionDefinition. Let
(12.3) ' : V ! ,
be a reducible normal map at M and assume that dim V is large.
Consider a collapse map (homotopy class) ' : SN+n ! T M as in 4.5.
Since the map
collapse T'
ff : SN+n   ! T V   ! T ,
is a reduction, there exists, by Theorem 4.6, a unique FN morphism
~ : oM ! ,o with ~*(') = ff. Now, the morphism
~* o classif N o
oM  ! ,  ! (flPL )
46 YULI B. RUDYAK
is a homotopy PL structuralization of M . Thus, it gives us a map
f' : M ! F=P L.
12.4. Proposition. The normal map (12.3) is associated with the__
map f' : M ! F=P L. __
Recall that a closed manifold is called almost parallelizable if it be
comes parallelizable after deleting of a point. Notice that every al
most parallelizable manifold is orientable (e.g., because its first Stiefel
Whitney class is equal to zero).
12.5. Proposition. Let V m be an almost parallelizable PL manifold.
Then every map b : V m ! Sm of degree 1 is the base of a reducible
normal map at Sm .
fiP
Proof. We regard Sm = {(x1, . .,.xm+1 ) fi x2i= 1} as the union of
two discs, Sm = D+ [ D , where
D+ = {x 2 Sm xm+1 0}, D = {x 2 Sm xm+1 0}.
We can always assume (deforming b if necessary) that there is a small
closed disk D0 in V such that b+ := bD0 : D0 ! D+ is a PL isomor
phism. We set W = V \(Int D0). Since W is parallelizable, there exists
a morphism ' : V W ! `D of PL bundles such that bW : W ! D
is the base of '. Furthermore, since b+ is a PL isomorphism, there ex
ists a morphism '+ : V D0 ! `D+ over b+ such that '+ and '
coincide over b@W : @W ! Sm1 . Together '+ and ' give us a
morphism ' : V ! , where , is a PL bundle over Sm . Clearly, ' is a __
normal map with the base b, and it is reducible because deg b = 1. __
13. The Sullivan map s : [M, F=P L] ! PdimM
We define the groups Pi by setting
8
< Z if i = 4k,
Pi = Z=2 if i = 4k + 2,
: 0 if i = 2k + 1
where k 2 N.
Given a closed connected ndimensional PL manifold M (which is
assumed to be orientable for n = 4k), we define a map
(13.1) s : [M, F=P L] ! Pn
PL STRUCTURES 47
as follows. Given a homotopy class f : M ! F=P L, consider a normal
map (12.2)
bb
E   ! E0
? ?
? ?
y y
V  b! M
associated with f.
For n = 4k, let _ be the symmetric bilinear intersection form on
Ker {b* : H2k(Z; Q) ! H2k(M; Q)}.
We define s(u) = ff(_)_8where oe(_) is the signature of _. It is well
known that oe(_) is divisible by 8, (see e.g. [5]), and so s(u) 2 Z.
Also, it is easy to see that
oe(Z)  oe(M)
oe(u) = ______________
8
where oe(M), oe(Z) is the signature of the manifold M, Z, respectively.
For n = 4k + 2, we define s(u) to be the Kervaire invariant of the
normal map (b,bb). The routine arguments show that s is welldefined,
i.e. it does not depend on the choice of the associated normal map.
See [5, Ch. III, x4] or [40 ] for details.
In particular, if b is a homotopy equivalence then s(u) = 0.
One can prove that every map s is a homomorphism, where the
abelian group structure on [M, F=P L] is given by an Hspace structure
on F=P L.
Given a map f : M ! F=P L, it is useful to introduce the notation
s(M, f) := s([f]) where [f] is the homotopy class of f.
13.1. Theorem. (i) The map s : [S4i, F=P L] ! Z is surjective for all
i > 1,
(ii) The map s : [S4i2, F=P L] ! Z=2 is surjective for all i > 0 .
(iii) The image of the map s : [S4, F=P L] ! Z is the subgroup of
index 2.
Proof. (i) For every k > 1 Milnor constructed a parallelizable 4k
dimensional smooth manifold W 4kof signature 8 and such that @W
is a homotopy sphere, see [5, V.2.9]. Since, by Theorem 3.11, every
homotopy sphere of dimension 5 is PL isomorphic to the standard
one, we can form a closed PL manifold
V := W [S4k1D4k
48 YULI B. RUDYAK
of the signature 8. Because of Proposition 12.5, there exists a reducible
normal map with the base V 4k! S4k. Because of Proposition 12.4,
this normal map is associated with a certain map (homotopy class)
f : S4k ! F=P L. Thus, s(S4k, f) = 1.
(ii) The proof is similar to that of (i), but we must use (4k+2)
dimensional parallelizable Kervaire manifolds W , @W = S4k+1 of the
Kervaire invariant one, see [5, V.2.11].
(iii) The Kummer algebraic surface [26 ] gives us an example of 4
dimensional almost parallelizable smooth manifold of the signature 16.
So, Im s 2Z.
Now suppose that there exists f : S4 ! F=P L with s(S4, f) = Z.
Then there exists a normal map with the base V 4! S4 and such that
V has signature 8. Since normal bundle of V is induced from a bundle
over S4, we conclude that w1(V ) = 0 = w2(V ). But this contradicts__
the Rokhlin Theorem 7.2. __
13.2. Theorem. For every closed simplyconnected PL manifold M
of dimension 5, the sequence
jF s
0  ! SPL (M)  ! [M, F=P L]  ! PdimM
is exact, i.e. jF is injective and Im jF = s1(0).
Proof. See [5, II.4.10 and II.4.11]. Notice that the map ! in loc. cit
is the zero map because, by Theorem 3.11, every homotopy sphere of __
dimension 5 is PL isomorphic to the standard sphere. __
13.3. Corollary. We have ß4i(F=P L) = Z, ß4i2(F=P L) = Z=2, and
ß2i1(F=P L) = 0 for every i > 0. Moreover, the map
s : [Sk, F=P L] ! Pk
is an isomorphism for k 6= 4 and has the form
Z = ß4(F=P L)  s! P4 = Z, a 7! 2a
for k = 4.
Proof. First, if k > 4 then, because of the Smale Theorem 3.11,
SPL (Sk) is the onepoint set. Now the result follows from 13.2 and
13.1.
If k 4 then ßk(P L=O) = 0, cf. Remark 6.7. So, ßk(F=P L) =
ßk(F=O). Moreover, the forgetful map ßk(BO) ! ßk(BF ) coincides
with the Whitehead Jhomomorphism. So, we have the long exact
sequence
. .!.ßk(F=O) ! ßk(BO) J! ßk(BF ) ! ßk1(F=O) ! . ...
PL STRUCTURES 49
For k 5 all the groups ßk(BO) and ßk(BF ) are known (notice that
ßk(BF ) is the stable homotopy group ßk+N1 (SN )), and it is also
known that J is an epimorphism for k = 1, 2, 4, 5, see e.g. [1]. Thus,_
ßk(F=O) ~=Pi for k 4. The last claim follows from 13.1. __
14. The homotopy type of F=P L[2]
As usual, given a space X and an abelian group ß, we do not dis
tinguish elements of Hn (X; ß) and maps (homotopy classes) X !
K(ß, n). For example, regarding a Steenrod cohomology operation
Sq2 as an element Sq2 2 Hk+2 (K(Z=2, k); Z=2), we can treat it as a
map Sq2 : K(Z=2, k) ! K(Z=2, k + 2).
Given a prime p, let Z[p] be the subring of Q consisting of all irre
ducible fractions with denominators relatively prime to p, and let Z[1=p]
be the subgroup of Q consisting of the fractions m=pk, m 2 Z. Given
a simplyconnected space X, we denote by X[p] and X[1=p] the Z[p]
and Z[1=p]localization of X, respectively. Furthermore, we denote by
X[0] the Q localization of X. For the definitions, see [21 ].
Consider the short exact sequence
j*
0  ! Z[2] 2! Z[2]  ! Z=2  ! 0
where 2 over the arrow means multiplication by 2 and æ is the modulo
2 reduction. This exact sequence yields the Bockstein exact sequence
j n
. . .! Hn (X; Z[2]) 2!Hn (X; Z[2])  ! H (X; Z=2)
(14.1) ffi!Hn+1 (X; Z[2]) ! . ...
Put X = K(Z=2, n) and consider the fundamental class
' 2 Hn (K(Z=2, n); Z=2).
Then we have the class ffi := ffi(') 2 Hn+1 (K(Z=2, n), Z[2]). Accord
ing to what we said above, we regard ffi as a map ffi : K(Z=2, n) !
K(Z[2], n + 1).
14.1. Proposition (Sullivan [56 , 57]). For every i > 0 there are co
homology classes
K4i2 H4i(F=P L; Z[2]), K4i22 H4i2(F=P L; Z=2)
such that
s(M4i, f) =
for every closed oriented PL manifold M, and
s(N4i2, f) = .
50 YULI B. RUDYAK
for every closed manifold N. Here [M] 2 H4i(M) is the orientation of
M, [N]2 2 H4i2(N; Z=2) is the modulo 2 fundamental class of N, and
<, > is the Kronecker pairing.
Proof. Let MSO*() denote the oriented bordism theory, see e.g [48 ].
Recall that if two maps f : M4i ! F=P L and g : N4i ! F=P L are
bordant (as oriented singular manifolds) then s(M, f) = s(N, g). Thus,
s defines a homomorphism
es: MSO4i(F=P L) ! Z.
It is well known that the SteenrodThom map
t : MSO*() Z[2] ! H*(; Z[2])
splits, i.e. there is a natural map v : H*(; Z[2]) ! MSO*() Z[2]
such that tv = 1 (a theorem of Wall [61 ], see also [55 , 48 , 3]. In
particular, we have a natural homomorphism
bs: H4i(F=P L; Z[2]) v! MSO4i(F=P L) Z[2] es! Z.
Since the evaluation map
ev : H*(X; Z[2]) ! Hom (H*(X; Z[2]), Z[2]), (ev(x)(y) =
is surjective for all X, there exists a class K4i2 H4i(F=P L; Z[2]) such
that ev(K4i) = bs. Now
s(M, f) = bs(f*[M]) = = .
So, we constructed the desired classes K4i.
The construction of classes K4i2is similar. Let MO*() denoted the
nonoriented bordism theory. Then the map s yields a homomorphism
es: MO4i2(F=P L) ! Z=2.
Furthermore, there exists a natural map H*(; Z=2) ! MO*() which
splits the SteenrodThom homomorphism, and so we have a homomor
phism
bs: H4i2(F=P L; Z=2)  ! MO4i2(F=P L) Z[2] es!Z=2
with bs(f*([M]2) = s(M, f). Now we can complete the proof similarly __
to the case of classes K4i. __
We set
Y
(14.2) := (K(Z[2], 4i) x K(Z=2, 4i  2)).
i>1
PL STRUCTURES 51
Together the classes K4i : F=P L ! K(Z[2], 4i), i > 1 and K4i2 :
F=P L ! K(Z[=2, 4i  2), i > 1 yield a map
K : F=P L ! .
14.2. Lemma. The map
(14.3) K[2] : F=P L[2] !
induced an isomorphism of homotopy groups in dimensions 5.
__
Proof. This follows from Theorem 13.1 and Corollary 13.3. __
14.3. Proposition. The Postnikov invariant
 2 H5(K(Z=2, 3), Z[2])
of F=P L[2] is nonzero. Moreover,  = ffiSq2 2 H5(K(Z=2, 3); Z[2]).
Proof. Let h : ß4(F=P L) ! H4(F=P L) be the Hurewicz homomor
phism. Suppose that  = 0. Then
H4(F=P L; Z[2]) = Z[2] H4(K(Z=2, 2; Z[2],
and therefore the homomorphism
H4(F=P L; Z[2]) ev! Hom (H4(F=P L) Z[2], Z[2])
h*! Hom (ß
4(F=P L) Z[2], Z[2])
must be surjective. But this contradicts 13.1(ii). Thus,  6= 0.
Furthermore, since F=P L is an infinite loop space,  has the form
N a for some a 2 HN+5 (K(Z=2, N +3), Z[2]). But, for general reasons,
for N large enough the last group consists of elements of the order 2.
Thus,  has the order 2. It is easy to see that H5(K(Z=2, 3), Z[2] = __
Z=4 = {x} with 2x = ffiSq2, see [48 ]. Thus,  = ffiSq2. __
14.4. Remark. It is interesting to have a geometrical description of
the class z 2 H4(F=P L) with = 1. Let j denote the canonical
complex line bundle over CP2. One can prove that 24j is fiberwise
homotopy trivial. So, there exist a map f : CP2 ! F=P L such that
the map
f
CP2  ! F=P L  ! BP L
classifies 24j. Since p1(j) = 1, we conclude that p1(24j) = 24, and
therefore L1(24j) = 8 (here L1 denotes the first Hirzebruch class), see
[38 ]. Thus, s(CP2, f) = 8=8 = 1, and therefore = 1.
52 YULI B. RUDYAK
Let Y be the homotopy fiber of the map
ffiSq2 : K(Z=2, 2) ! K(Z[2], 5).
In other words, there is a fibration
p
(14.4) K(Z[2], 4) i! Y  ! K(Z=2, 2)
with the characteristic class ffiSq2.
Because of Proposition 14.3, the space Y is the Postnikov 4stage of
F=P L[2]. In particular, we have a map
_ : F=P L[2] ! Y
which induces an isomorphism of homotopy groups in dimension 4.
Together with the map K[2] from 14.3, this map yields a map OE :
F=P L[2] ! Y x .
14.5. Theorem. The map
(14.5) OE : F=P L[2] ' Y x
is a homotopy equivalence.
__
Proof. This follows from 14.3 and what we said about _. __
14.6. Lemma. Let X be a finite CW space such that the group H*(X)
is torsion free. Then the group [X, F=P L[1=2]] is torsion free.
Proof. It suffices to prove that [X, F=P L[p]] is torsion free for every
odd prime p. Notice that F=P L[p] is an infinite loop space since F=P L
is. So, there exists a connected plocal spectrum E such that
eE0(Y ) = [Y, F=P L[p]] = [Y, F=P L] Z[p].
Moreover, Ei(pt) = ßi(E) = ßi(F=P L) Z[p], So, because of the
isomorphism Ee0(X) ~= [X, F=P L[p]], it suffices to prove that E*(X)
is torsion free. Consider the AtiyahHirzebruch spectral sequence for
E*(X). Its initial term is torsion free because E*(pt) is torsion free.
Hence, the spectral sequence degenerates, and thus the group E*(X) __
is torsion free. __
14.7. Proposition. Let X be a finite CW space such that the group
H*(X) is torsion free. Let f : X ! F=P L be a map such that f*K4n =
0 and f*K4n+2 = 0 for every n 1. Then f is nullhomotopic.
PL STRUCTURES 53
Proof. Consider the commutative square
F=P L l1! F=P L[2]
? ?
l2?y ?yl3
F=P L[1=2] l4! F=P L[0]
where the horizontal maps are the Z[2]localizations and the verti
cal maps are the Z[1=2]localizations. Because of 14.5, [X, F=P L] is
a finitely generated abelian group, and so it suffices to prove that
both l1Of and l2Of are nullhomotopic. First, we remark that l2Of
is nullhomotopic whenever l1Of is. Indeed, since H*(X) is torsion
free, the group [X, F=P L[1=2]] is torsion free by 14.6. Now, if l1Of
is nullhomotopic then l3Ol1Of is nullhomotopic, and hence l4Ol2Of is
nullhomotopic. Thus, l2Of is nullhomotopic since [X, F=P L[1=2]] is
torsion free.
So, it remains to prove that l1Of is nullhomotopic.
Clearly, the equalities f*K4i= 0 and f*K4i2= 0, i > 1, imply that
the map
p2
X  ! F=P L l1! F=P L[2] ' Y x  !
is nullhomotopic. So, it remains to prove that the map
f l1 p1
g : X   ! F=P L   ! F=P L[2] ' Y x   ! Y
is nullhomotopic.
It is easy to see that H4(Y ; Z[2]) = Z[2]. Let u 2 H4(Y ; Z[2]) be a
free generator of the free Z[2]module H4(Y ; Z[2]). The fibration (14.4)
gives us the following diagram with the exact row:
p* 2
H4(X; Z[2]) i*! [X, Y ]  ! H (X; Z=2)
?
?
yu*
H4(X; Z[2])
Notice that
u*i* : Z[2] ! Z[2]
is the multiplication by 2" where " is an invertible element of the ring
Z[2]. Since f*K2 = 0, we conclude that p*(g) = 0, and so g = i*(a) for
some a 2 H4(X; Z[2]). Now,
0 = u*(g) = u*(i*a) = 2a".
__
But H*(X; Z[2]) is torsion free, and thus a = 0. __
54 YULI B. RUDYAK
For completeness, we mention also that F=P L[1=2] ' BO[1=2]. So,
there is a Cartesian square (see [31 , 57])
F=P L   ! x Y
? ?
? ?
y y
ph Q
BO[1=2]   ! K(Q, 4i)
where ph is the Pontryagin character.
15. Splitting theorems
15.1. Definition. Let An+k and W n+k be two connected PL mani
folds (without boundaries), and let Mn be a closed submanifold of A.
We say that a map g : W n+k! An+k splits along Mn if there exists a
homotopy
gt : W n+k! An+k , t 2 I
such that:
(i) g0 = g;
(ii) there is a compact subset K of W such that gtW \ K = gW \ K
for every t 2 I;
(iii) the map g1 is transversal to M;
(iv) the map b := (g1)g11(M) : g11(M) ! M is a homotopy equiv
alence.
An important special case is when An+k = Mn x Bk for some con
nected manifold Bk. In this case we can regard M as submanifold
M x {b0}, b0 2 B of A and say that g : W ! A splits along M if it
splits along M x {b0}. Clearly, this does not depend on the choice of
{b0}, i.e. g splits along M x {b0} if and only if it splits along M x {b}
with any other b 2 B.
Recall that a map f is called proper if f1 (C) is compact whenever C
is compact. A map f : X ! Y is called a proper homotopy equivalence
if there exists a map g : Y ! X and the homotopies F : gf ' 1X ,
G; fg ' 1Y such that all the four maps f, g, F : X x I ! X and
G : Y x I ! Y are proper.
15.2. Theorem. Let Mn , n 5 be a closed connected ndimensional
PL manifold such that ß1(M) is a free abelian group. Then every proper
homotopy equivalence h : W n+1! Mn x R splits along Mn .
Proof. Because of the transversality theorem, there is a homotopy
ht : W ! M x R which satisfies conditions (i)(iii) of 15.1. We let
f = h1. Because of a crucial theorem of Novikov [41 , Theorem 3], there
PL STRUCTURES 55
is a sequence of interior surgeries of the inclusion f1 (M) W in W
such that the final result of these surgeries is a homotopy equivalence
V W . (This is the place where we use the fact that ß1(M) is free.)
Using the PontryaginThom construction, we can realize this sequence
of surgeries via a homotopy ft such that ft satisfies conditions (i)(iii)_
of 15.1 and f11(M) = V . __
15.3. Theorem. Let Mn be a manifold as in 15.2. Then every homo
topy equivalence W n+1! Mn x S1 splits along Mn .
Proof. This follows from results of Farrell and Hsiang [12 , Theorem __
2.1] since Wh (Zm ) = 0 for every m. __
15.4. Corollary. Let Mn be a manifold as in 15.2. Let T k denote
the kdimensional torus. Then every homotopy equivalence W n+k !
Mn x T ksplits along Mn .
__
Proof. This follows from 15.3 by induction. __
15.5. Theorem. Let Mn be a manifold as in 15.2. Then every home
omorphism h : W n+k! Mn x Rk of a PL manifold W n+k splits along
Mk.
Proof. We use the Novikov's torus trick. The canonical inclusion T k1x
R Rk yields the inclusion M x T k1x R M x Rk. We set W1 :=
h1(M x T k1x R). Notice that W1 is a smooth manifold since it is an
open subset of W . Now, set u = hW1 : W1 ! M x T k1x R. Then,
by 15.2, u splits along M x T k1, i.e. there is a homotopy ut as in
15.1. We set f := u1, V := f1 (M x T k1) and g := fV . Because of
15.4, g : V ! M x T k1splits along M. Hence, f splits along M, and
therefore u splits along M. Let __utbe the homotopy which realizes this
splitting as in 15.1. Now, we define the homotopy ht : W ! M xRn by
setting htW1 := __utW1 and htW \ W1 := hW \ W1. Notice that {ht}
is a welldefined and continuous family since the family {__ut} satisfies
15.1(ii). It is clear that ht satisfies the conditions (i)_(iii) of 15.1_and
that h1 extends f on the whole W , i.e. h splits along M. __
15.6. Remark. The above used theorems of Novikov and Farrell
Hsiang were originally proved for smooth manifolds, but they are valid
for PL manifolds as well, because there is an analog of the Thom
Transversality Theorem for PL manifolds, [65 ].
15.7. Lemma. Suppose that a map g : W ! A splits along a sub
manifold M of A. Let , = {E ! A} be a PL bundle over A, let
56 YULI B. RUDYAK
g*, = {D ! W }, and let k : g*, ! , be the gadjoint bundle mor
phism. Finally, let l : D ! E be the map of the total spaces induced by
k. Then l splits over M. (Here we regard A as the zero section of ,,
and so M turns out to be a submanifold of E).
Proof. Let G x I ! A be a homotopy which realizes the splitting of g.
Recall that g*, x I is equivalent to G*,. Now, the morphism
IG,,
g*, x I ~= G*,  ! ,
__
gives us the homotopy D x I ! E which realizes a splitting of l. __
15.8. Lemma. Let M be a manifold as in 15.2. Consider two PL RN 
bundles , = {U ! M} and j = {E ! M} over M and a topological
morphism ' : , ! j over M of the form
g
U   ! E
? ?
? ?
y y
M _______M.
Then there exists k such that the map
g x 1 : U x Rk ! E x Rk
splits along M, where M is regarded as the zero section of j.
Proof. Take a PL Rm bundle i such that j i = `N+m and let W be
the total space of , i. Then the map
' 1 : , i ! j i = `N+m
yields a map
(15.1) : W ! M x RN+m
of the total spaces. Because of Theorem 15.5, the map splits along
M. Now, considering the morphism
' 1 1 : , i j ! j i j
and passing to the total spaces, we get a map
g x 1 : U x R2N+m ! E x R2N+m .
In view of Lemma 15.7, this map splits over M because does. So,__
we can put k = 2N + m. __
Now, let a : T OP=P L ! F=P L be a map as in (2.5).
PL STRUCTURES 57
15.9. Theorem. Let M be as in 15.2. Then the composition
[M, T OP=P L] a*! [M, F=P L] s! PdimM
is trivial, i.e., sa*(v) = 0 for every v 2 [M, T OP=P L]. In other words,
s(M, af) = 0 for every f : M ! T OP=P L.
Proof. In view of 3.4, every element v 2 [M, T OP=P L] is a concordance
class of a topological morphism
' : NM ! flNPL
of PL RN bundles. Passing from the class v 2 [M, T OP=P L] to the
class a*v 2 [M, F=P L], we must consider the equivalence class of FN 
morphism 'o : ( M )o ! (flNPL)o. Now, following the definition of the
map s : [M, F=P L] ! PdimM , we get a commutative diagram
g 0 o
Uo   ! U
? ?
(15.2) q?y ?yp
M _______ M
like (12.1). However, here we know that g is a homeomorphism. Thus,
g(U) = U0, and so we get the diagram
g 0
U  ! U
? ?
(15.3) q?y ?yp
M _______M
which is a topological morphism of PL bundles over M. We embed M
in U0 as the zero section. By the definition of the map s, we conclude
that s(M, a*v) = 0 if the map g : U ! U0 splits along M (because
in this case the associated normal map is a map over a homotopy
equivalence). Moreover, since, for every k, the topological morphisms
' and
(' 1) N k N+k
M `k  ! flPL `   ! flPL
represent the same element of [M, T OP=P L], it suffices to prove that
there exists k such that the map
g x 1 : U x Rk ! U0 x Rk
__
splits along M. But this follows from Lemma 15.8. __
Now we show that the condition dim M 5 in 15.9 is not necessary.
15.10. Corollary. Let M be a closed connected PL manifold such that
ß1(M) is a free abelian group. Then s(M, af) = 0 for every map
f : M ! T OP=P L.
58 YULI B. RUDYAK
Proof. Let CP 2denote the complex projective plane, and let
p1 : M x CP 2! M
be the projection on the first factor. Then s(M x CP 2, gp1) = s(M, g)
for every g : M ! F=P L, see [5, Ch. III, x5]. In particular, for every
map f; M ! T OP=P L we have
s(M, af) = s(M x CP2, (af)p1) = s(M x CP2, a(fp1)) = 0
__
where the last equality follows from Theorem 15.9. __
16. Detecting families
Recall the terminology: a singular smooth manifold in a space X is
a map M ! X of a smooth manifold.
Given a CW space X, consider a connected closed smooth singular
manifold fl : M ! X in X. Then, for every map f : X ! F=P L, the
invariant s(M, ffl) 2 PdimM is defined. Clearly, if f is nullhomotopic
then s(M, ffl) = 0.
16.1. Definition. Let {fli : Mi ! X}i2I be a family of closed con
nected smooth singular manifolds in X. We say that the family {fli :
Mi ! X} is a detecting family for X if, for every map f : X ! F=P L,
the validity of all the equalities s(Mi, ffli) = 0, i 2 I implies that f is
nullhomotopic.
Notice that F=P L is an Hspace, and hence, for every detecting
family {fli : Mi ! X}, the collection s(Mi, ffli) determine a map
f : X ! F=P L uniquely up to homotopy.
The concept of detecting family is related to Sullivan's "characteris
tic variety", but it is not precisely the same. If a family F of singular
manifolds in X contains a detecting family, then F itself is a detect
ing family. On the contrary, the characteristic variety is in a sense
"minimal" detecting family.
16.2. Theorem. Let X be a connected finite CW space such that the
group H*(X) is torsion free. Then X possesses a detecting family {fli :
Mi ! X} such that each Mi is orientable.
Proof. Since H*(X) is torsion free, every homology class in H*(X) can
be realized by a closed connected smooth oriented singular manifold,
see e.g. [8, 15.2] or [48 , 6.6 and 7.32]. Let {fli : Mi ! X} be a family
of smooth oriented closed connected singular manifolds such that the
elements (fli)*[Mi] generate all the groups H2k(X).
PL STRUCTURES 59
We prove that {fli : Mi ! X} is a detecting family. Consider a map
f : X ! F=P L such that si(Mi, ffli) = 0 for all i. We must prove that
f is nullhomotopic.
Because of 14.7, it suffices to prove that f*K4n = 0 and f*K4n2 = 0
for every n 1. Furthermore, H*(X) = Hom (H*(X), Z) because
H*(X) is torsion free. So, it suffices to prove that
(16.1) = 0 for everyx 2 H4n(X)
and
(16.2) = 0 for everyx 2 H4n2(X; Z=2).
First, we prove (16.1). Since the classes (fli)*[Mi], dim Mi = 4n gener
ates the group H4n(X), it suffices to prove that
= 0 whenever dim Mi = 4n .
But, because of 14.1, for every 4ndimensional Mi we have
0 = s(Mi, ffli) = <(ffli)*K4n, [Mi]>= .
This completes the proof of the equality (16.1).
Passing to the case n = 4k + 2, notice that the group H2k(X; Z=2)
is generated by the elements (fli)*[Mi]2, dim Mi = 2k, since H*(X) is
torsion free. Now the proof can be completed similarly to the case __
n = 4k. __
17. A special case of the theorem on the normal
invariant of a homeomorphism
17.1. Theorem. Let M be closed connected PL manifold such that
each of the group Hi(M) and ß1(M) is a free abelian group. Then
jF (x) = 0 whenever x 2 SPL can be represented by a homeomorphism
h : V ! M.
Proof. The maps jTOP and jF from section 3 can be included in the
following commutative diagram:
jTOP
TPL (M)  ! [M, T OP=P L]
? ?
? ?
(17.1) y ya*
jF
SPL (M)  ! [M, F=P L]
where the left arrow is the obvious forgetful map and a* is induced by
a as in (2.5).
Suppose that x can be represented by a homeomorphism h : V ! M.
Consider a map f : M ! T OP=P L such that jTOP (h) is homotopy
60 YULI B. RUDYAK
class of f. Then, clearly, the class jF (x) 2 [M, F=P L] is represented
by the map
f a
M  ! T OP=P L   ! G=P L.
By 16.2, M possesses a detecting family {fli : Mi ! M}. We can
assume (performing oriented surgeries of Mi if necessary) that ß1(Mi)
is a subgroup of ß1(M), and so ß1(Mi) is a free abelian group. Hence,
by 15.9 and 15.10, s(Mi, affli) = 0 for every i. But {fli : Mi ! M} is
a detecting family, and therefore af is nullhomotopic. Thus, jF (x))_=_
0. __
PL STRUCTURES 61
Chapter 3. Applications
18. Topological invariance of rational Pontryagin
classes
18.1. Lemma. The homotopy groups ßi(P L=O) are finite. Thus, the
space P L=O[0] is contractible.
__
Proof. See [48 , IV.4.27(iv)]. __
Recall that H*(BO; Q) = Q[p1, . .,.pi, . .].where pi, dim pi = 4i is
the universal rational Pontryagin class, see e.g. [38 ]. (In fact, pi is the
image of the integral Pontryagin class pi 2 H*(BO).)
18.2. Theorem. The forgetful map ff = ffOPL : BO ! BP L induces
an isomorphism
ff* : H*(BP L; Q) ! H*(BO; Q).
Proof. It suffices to prove that ff[0] : BO[0] ! BP L[0] is a homotopy
equivalence. But this holds because the homotopy fiber of ff[0] is the_
contractible space P L=O[0]. __
It follows from 18.2 that H*(BP L; Q) == Q[p01, . .,.p0i, . .].where
p0iare the cohomology classes determined by the condition ff*(p0i) = pi.
Now, given a PL manifold M, we define its rational Pontryagin classes
p0i(M) 2 H4i(M; Q) by setting
p0i(M) = t*p0i
where t : M ! BP L classifies the stable tangent bundle of M. Clearly,
if we regard a smooth manifold as a PL manifold then pi(N) = p0i(N).
18.3. Corollary (PL invariance of Pontryagin classes, [60 , 45]).Let
f : M1 ! M2 be a PL isomorphism of smooth manifolds. Then __
f*pi(M2) = pi(M1). __
18.4. Theorem. The forgetful map ff = ffPLTOP: BP L ! BT OP in
duces an isomorphism
ff* : H*(BT OP ; Q) ! H*(BP L; Q).
Proof. This can be deduced from Theorem 8.8 just in the same manner_
as we deduced Theorem 18.2 from Theorem 18.1. __
Now we introduce the universal classes p00i2 H4i(BT OP ; Q) by the
equality
(ffPLTOP)*p0i= p00i.
62 YULI B. RUDYAK
Furthermore, given a topological manifold M, we set
p00i(M) = t*p00i
(where t classifies the stable tangent bundle of M) and get the following
corollary.
18.5. Corollary (topological invariance of Pontryagin classes, [41 ]).
Let f : M1 ! M2 be a homeomorphism of smooth manifolds. Then __
f*pi(M2) = pi(M1). __
19. The space F=T OP
It turns out to be that, in view of the Product Structure Theorem,
the Transversality Theorem holds for topological manifolds and bun
dles. I am not able to discuss it here, see [48 , IV.7.18] for the references.
Since we have the topological transversality, we can define the maps
s0: [M, F=T OP ] ! PdimM
which are obvious analogs of maps s defined in (13.1). However, here
we allow M to be a topological manifold. The following proposition
demonstrates the main difference between F=P L and F=T OP .
19.1. Proposition. The map s0: ß4(F=T OP ) ! Z is a surjection.
Proof. Notice that the Freedman manifold from Theorem 7.1 is almost
parallelizable and has signature 8. Now the proof can be completed __
just as 13.1(i). __
Recall that in (2.5)we described a fibration
T OP=P L a! F=P L  b! F=T OP.
19.2. Theorem. For i 6= 4 the map b : F=P L ! F=T OP induces an
isomorphism
b* : ßi(F=P L) ! ßi(F=T OP ).
The map
b* : Z = ßi(F=P L) ! Z = ßi(F=T OP )
is the multiplication by 2.
Proof. Recall that T OP=P L = K(Z=2, 3). So, the homotopy exact
sequence of the fibration
T OP=P L  a! F=P L  b! F=T OP
PL STRUCTURES 63
yields an isomorphism b* : ßi(F=T OP ) ~= ßi(F=P L) for i 6= 4. Fur
thermore, we have the commutative diagram
0 _______ß4(T OP=P L)
?
a*?y
Z _______ ß4(F=P L)  s! Z
? fl
b*?y flfl
0
ß4(F=T OP )  s! Z
?
?
y
ß3(T OP=P L) _______Z=2
?
?
y
ß3(F=P L) _______ 0
where the middle line is a short exact sequence. So, ß4(F=T OP ) = Z
or ß4(F T OP ) = Z Z=2. By 13.1, Im s is the subgroup 2Z of Z,
while s0 is a surjection by 19.2. Thus, ß4(F=T OP ) = Z and b* is the__
multiplication by 2. __
Now, following 14.1, we can introduce the classes
K04i2 H4i(F=T OP, Z[2]) and K04i22 H4i2(F=T OP, Z=2)
such that
s0(M4i, f) = and s0(N4i2, f) = .
However, here M and N are assumed to be topological (i,e, not neces
sarily PL) manifolds.
Together these classes yield the map
Y
K0 : F=T OP  ! (K(Z[2], 4i) x K(Z=2, 4i  2)).
i>0
19.3. Theorem. The map
Y
K0[2] : K0 : F=T OP [2] ! (K(Z[2], 4i) x K(Z=2, 4i  2))
i>0
is a homotopy equivalence.
Proof. Together 13.1 and 19.1 imply that the homomorphisms s0 :
ß2i(F=T OP ) ! P2iare surjective. Now, in view of 19.2, all the homo__
morphisms s0 are isomorphisms, and the result follows. __
64 YULI B. RUDYAK
So, the only difference between F=P L and F=T OP is that F=T OP [2]
has trivial Postnikov invariants while F=P L[2] has just one nontrivial
Postnikov invariant ffiSq2 2 H5(K(Z=2, 2); Z[2]).
20. The map a : T OP=P L ! F=P L
20.1. Proposition. The map a : T OP=P L ! F=P L is essential.
Proof. For general reasons, the fibration
T OP=P L  a! F=P L   ! F=T OP
yields a fibration
(F=T OP )  u! T OP=P L a! F=P L.
If a is inessential then there exists a map v : T OP=P L ! (F=T OP )
with uv ' 1. But this is impossible because ß3(T OP=P L) = Z=2 while
ß3( (F=T OP )) = ß4(F=T OP ) = Z.
Take an arbitrary map f : X ! T OP=P L. Let ` : F=P L !
F=P L[2] denote the localization map.
20.2. Proposition. The following three conditions are equivalent:
(i) the map
f a
X  ! T OP=P L   ! F=P L
is essential;
(ii) the map
f a `
X   ! T OP=P L  ! F=P L   ! F=P L[2]
is essential;
(iii) the map
f a ` projection
X  ! T OP=P L  ! F=P L  ! F=P L[2]    !Y
is essential.
Proof. It suffices to prove that (i) ) (ii) )(iii). To prove the first
implication, recall that a map u : X ! F=P L is inessential if both
localized maps
X u! F=P L ! F=P L[2], X  u! F=P L ! F=P L[1=2]
are inessential. Now, (i) ) (ii) holds since T OP=P L[1=2] is con
tractible.
To prove the second implication, notice that a map v : X ! F=P L[2]
is inessential if both maps (we use notation as in 14.5)
X v! F=P L[2] K ! , X  v! F=P L[2]  ! Y
PL STRUCTURES 65
are inessential. So, it suffices to prove that the map
`af
X  ! F=P L[2]  !
is inessential. This holds, in turn, because the map T OP=P L !
F=P L ! F=T OP is inessential and the diagram
K[2]
F=P L[2]  ! _______
? fl
? fl
y b[2] fl
K0[2]Q proj
F=T OP [2]  ! i>0(K(Z=2, 4i  2) x K(Z[2], 4i))  !
__
commutes. __
Consider the fibration
K(Z[2], 4) i! Y  ! K(Z=2.2)
as in (14.4).
20.3. Lemma. For every space X, the homomorphism
H4(X; Z[2]) = [X, K(Z[2], 4)] i*! [X, Y ]
is injective. Moreover, i* is an isomorphism if H2(X; Z=2) = 0.
Proof. The fibration (14.4) yields the exact sequence (see e.g. [39 ])
ffiSq24 i* 2
(20.1) H1(X; Z=2) ! H (X; Z[2]) ! [X, Y ] ! H (X; Z=2)
__
where ffiSq2(x) 0 (because ffiSq2(x) = 0 whenever deg x = 1). __
Let g : T OP=P L ! Y be the composition
proj
T OP=P L a! F=P L  `! F=P L[2]  ! Y.
Notice that g is essential because of 20.1 and 20.2.
20.4. Corollary. The map
T OP=P L = K(Z=2, 3) ffi!K(Z[2], 4) i! Y
is homotopic to g, i.e. g ' iffi.
Proof. Since the sequence (20.1) is exact, the set [K(Z=2, 3), Y ] has
precisely two elements. Since both maps g and iOffi are essential (the_
last one because of Lemma 20.3), we conclude that g ' iffi. __
66 YULI B. RUDYAK
20.5. Theorem. Given a map f : X ! T OP=P L, the map
f a
X  ! T OP=P L   ! F=P L
is essential if and only if the map
f ffi
X   ! T OP=P L = K(Z=2, 3)  ! K(Z[2], 5)
is essential.
Proof. af is essential 20.2()gf is essential (20.4)iffif is essential (20.3)ff*
*if_
is essential. __
21. The theorem on the normal invariant of a
homeomorphism
21.1. Lemma. Let X be a finite CW space such that Hn(X) is 2
torsion free. Then the homomorphism
ffi : Hn (X; Z=2) ! Hn+1 (X; Z[2])
is zero.
Proof. Because of the exactness of the sequence (14.1)
Hn (X; Z=2)  ffi!Hn+1 (X; Z[2]) 2! Hn+1 (X; Z[2]),
it suffices to prove that Hn+1 (X; Z[2]) is 2torsion free. Since Hn(X)
is 2torsion free, we conclude that Ext (Hn(X), Z[2]) = 0. Thus,
Hn+1 (X; Z[2]) = Hom (Hn+1(X; Z[2]) Ext(Hn(X); Z[2])
= Hom (Hn+1(X; Z[2]),
__
and the result follows. __
21.2. Theorem. Let M be a closed PL manifold such that H3(M)
is 2torsion free. Then the normal invariant of any homeomorphism
h : V ! M is trivial.
Proof. Since h is a homeomorphism, the normal invariant jF (h) turns
out to be the homotopy class of a map
f a
M  ! T OP=P L  ! F=P L
where the homotopy class of f is jTOP (h). Because of 20.2 and 20.3, it
suffices to prove that the map
f ffi
M  ! T OP=P L = K(Z=2, 3)  ! K(Z[2], 4)
__
is inessential. But this follows from Lemma 21.1. __
PL STRUCTURES 67
21.3. Corollary. Let M, dim M 5 be a closed simplyconnected PL
manifold such that H3(M) is 2torsion free. Then every homeomor
phism h : V ! M is homotopic to a PL isomorphism. In particular,
V and M are PL isomorphic.
__
Proof. This follows from 13.2 and 21.2. __
21.4. Remark. Rourke [46 ] suggested another proof of 21.2, using the
technique of simplicial sets.
22. A counterexample to the Hauptvermutung, and
other examples
22.1. Example. Two manifolds which are homeomorphic but not PL
isomorphic.
Let RPn denote the real projective space of dimension n.
22.2. Lemma. For every homotopy equivalence h : RP5 ! RP5 we
have jF (h) = 0.
Proof. We triangulate RP4 and take the induced triangulation of the
covering space S4. Take the corresponding triangulation of the sus
pension SS4 = S5. Let r : S5 ! S5 be the reflection with respect
to the equator S4. Since r is an antipodal map, it yields a map
f : RP5 ! RP5. Clearly, f is a map of degree 1.
It follows from the obstruction theory that every homotopy equiva
lence RP5 ! RP5 is homotopic either to f or to the identity map. (For
the homotopy classification of maps RPn ! RPn, see [18 ].) Since f is_
a PL isomorphism, the lemma follows. __
Recall that jTOP : TPL (RP5) ! [RP5, T OP=P L] is a bijection. Con
sider a homeomorphism k : M ! RP5 such that
jTOP (k) 6= 0 2 [RP5, T OP=P L] = H3(RP5; Z=2) = Z=2.
Notice that
ffi : Z=2 = H3(RP5; Z=2) ! H4(RP5; Z=2) = Z=2
is an isomorphism, and hence ffi(jTOP (k)) 6= 0. So, by Theorem 20.5,
a*jTOP (k) 6= 0. In view of commutativity of the diagram (17.1),
jF (k) = a*jTOP (k), i.e. jF (k) 6= 0. Thus, in view of Lemma 22.2,
M is not PL isomorphic to RP5.
22.3. Example. A homeomorphism h : S3 x Sn ! S3 x Sn, n > 3
which is homotopic to a PL isomorphism but is not concordant to a PL
isomorphism.
68 YULI B. RUDYAK
Take an arbitrary homeomorphism f : V ! S3 x Sn. Then jF (f)
is trivial by Theorem 21.2. Thus, by 13.2, f is homotopic to a PL
isomorphism. In particular, V is PL isomorphic to S3 x Sn.
Now, we refine the situation and take a homeomorphism h : S3 x
Sn ! S3 x Sn such that
jTOP (h) 6= 0 2 TPL (S3 x Sn) = H3(S3 x Sn; Z=2) = Z=2.
So, h is not concordant to the identity map, and therefore h is not
concordant to a PL isomorphism. But, as we have already seen, h is
homotopic to a PL isomorphism.
Notice that the maps h and the identity map have the same domain
while they are not concordant. So, this example serves also the Remark
3.2(3).
22.4. Example. A topological manifold which does not admit any PL
structure.
Already constructed F x T n, see 7.3.
22.5. Example. A topological manifold which is homeomorphic to a
polyhedron but does not admit any PL structure.
Let M be a closed topological 7dimensional manifold, and let
(M) 2 H4(M; Z=2)
be the KirbySiebenmann invariant described in 8.10. Let
ffi : H4(M; Z=2) ! H5(M; Z)
be the Bockstein homomorphism. Scharlemann [49 ] proved that if
ffi(M) = 0 and if a quadruple suspension over a certain 3dimensional
homology sphere is homeomorphic to S7, then M admits a simplicial
triangulation. Cannon [6] proved that a double suspension over every 3
dimensional homology sphere is homeomorphic to S5. So, if (M) 6= 0
while ffi(M) = 0 then M is homeomorphic to a polyhedron but does
not admits a PL structure.
Take M = F x T 3. Then (M) 6= 0 by 7.3. On the other hand,
ffi(M) = 0 because F x T 3is torsion free, while 2ffi(x) = 0 for every x.
Epilogue
Certainly, it is useful (or even necessary) to write a book about this
subject. The paper on hand looks as a reasonable point to start. The
contents of the book is more or less clear (we can follow the graph
from the introduction). From my (maybe, personal) point of view, the
PL STRUCTURES 69
accurate treatment of PL manifolds and their normal bundles, and of
the Product Structure Theorem  is most difficult thing in this business
(while surgery exact sequence, fake tori, splitting theorems, etc. are
more or less well exposed in the literature). I appreciate any comments
and would be glad to collaborate with anybody who will be interested
in it.
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MaxPlanck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn,
Germany
Email address: rudyak@mpimbonn.mpg.de
Mathematisches Institut Universität Heidelberg, Im Neuenheimer
Feld 288, 69120 Heidelberg, Germany
Email address: rudyak@mathi.uniheidelberg.de