ON SULLIVAN'S THEOREM OF THE NORMAL
INVARIANT OF HOMEOMORPHISM
Yuli B. Rudyak
Abstract. In [S1], [S2] Sullivan indicated the proof of the following the*
*orem:
Let h : M ! N be a homeomorphism of closed piecewise linear manifolds. *
*Then
the normal invariant of h is trivial provided H3(M) has no 2-torsion. The*
* goal of
this paper is to give a relative simple proof of this theorem in a partic*
*ular case of man-
ifolds M such that ss1(M) and Hi(M) are free abelian groups. Motivation: *
*Kirby-
Siebenmann [KS] proved that T OP=P L is the Eilenberg-Mac Lane space K(Z=*
*2; 3),
and it actually solves the Hauptvermutung for manifolds. The proof by Ki*
*rby-
Siebenmann uses a special case of the Normal Invariant Homeomorphism Theo*
*rem
when M is a product T kx Sn.
Introduction
In [S, theorem H] Sullivan indicated the proof of the following theorem:
Let h : M ! N be a homeomorphism of closed piecewise linear manifolds. Then
the normal invariant of h is trivial provided H3(M) has no 2-torsion.
For the definition of the normal invariant, see x5 of this paper. The goal *
*of
this paper is to give a relative simple (and clear, I hope) proof of this theor*
*em
in a particular case of manifolds M such that ss1(M) and Hi(M) are free abelian
groups. Now I motivate the importance of this work.
Kirby-Siebenmann [KS] proved that
(*) T OP=P L ' K(Z=2; 3);
and it actually solves the Hauptvermutung for manifolds. In the proof of the
equivalence (*) Kirby-Siebenmann use the result of Hsiang-Shaneson [HS] about
piecewise linear structures on manifolds which are homotopy equivalent to T kxS*
*n ,
______________
1991 Mathematics Subject Classification. 57Q25.
1
2 YULI B. RUDYAK
where T kis the k-dimensional torus and Sn is the n-dimensional sphere, see [KS,
the proof of V.4.5, p.249]. The result of Hsiang-Shaneson, in turn, uses the ab*
*ove
formulated Sullivan theorem for M = T kx Sn , see [HS, x10, p.42]. So, it is
reasonable to have a detailed proof of this special case of the Sullivan theore*
*m.
The paper is organized as follows. Sections 1 and 2 contains some preliminar*
*ies,
in x3 we prove a simple version of Sullivan's characteristic variety theorem, i*
*n x4
we prove that the composition
[M; T OP=P L] ! [M; G=P L] -s!Pi
is trivial for every closed smooth manifold with ss1(M) free abelian, in x5 we *
*prove
the main theorem.
We follow Browder [B, Ch. II, x2] in the definition of transversality.
Given a closed connected n-dimensional manifold M, we denote by [M]2 the
generator of the group Hn(M; Z=2) = Z=2. Furthermore, if M is orientable then
we denote by [M] a generator (on of two) of the group Hn(M) = Z.
Given a prime p, we denote by Z[p] the subgroup of Q which consists of all
irreducible fractions whose denominators are prime to p. Furthermore, we denote
by Z[1=p] the subgroup of all fraction of the form m=pk. We denote by X[p], res*
*p.
X[1=p] the Z[p]-localization, resp. Z[1=p]-localization of a space X. Finally*
*, X[0]
denotes the Q-localization of X.
Given a direct product X x Y of two spaces, we denote by p1 : X x Y !
X; p1(x; y) = x and p2 : X x Y ! Y; p2(x; y) = y, the canonical projections.
We use the abbreviation PL for "piecewise linear".
We use the sign ' for homotopy of maps.
x1 Preliminaries on classifying spaces
For definition of spaces BP Ln, BT OPn and BGn, see [MM, Ch.1]. We define
BG to be the homotopy direct limit of the sequence
. . .! BGn ! BGn+1 ! . . .
where the map BGn ! BGn+1 is induced by the standard inclusion Gn Gn+1 ,
see [MM, p.45]. Similarly, BP L (resp. BT OP ) is defined to be the homotopy di*
*rect
NORMAL INVARIANT OF HOMEOMORPHISM 3
limit of the sequence {. . .! BP Ln ! BP Ln+1 ! . .}., (resp. {. . .! BT OPn !
BT OPn+1 ! . .}.). There is a sequence of forgetful maps
BP L -u!BT OP -v!BG:
The homotopy fiber of u is denoted by T OP=P L, the homotopy fiber of vu is
denoted by G=P L. Regarding both maps u and v as fibrations, we have a canonical
inclusion
(1.1) a : T OP=P L ! G=P L:
1.2. Definition. Recall that a map u : U ! V is called proper if u-1 (K) is
compact for every compact subset K of V . Given a commutative diagram
A ----! B
? ?
p?y ?yq
X ________X
we say that ' is a proper homotopy equivalence over X if there are maps : B !*
* A,
F : A x I ! A, and H : B x I ! B; with the following properties:
(i) pF (a; t) = p(a) for every (a; t) 2 AxI, qH(b; t) = q(b) for every (b; t*
*) 2 B xI;
(ii) F |A x {0} = ', F |A x {1} = 1A , H|B x {0} = ' , H|B x {1} = 1B ;
(iii) both maps F and H are proper.
Let (G=P L)N denote the fiber of the forgetful map BP LN ! BGN , and let
flN be the universal PL RN bundle over BP LN . The inclusion i : (G=P L)N !
BP LN of the (homotopy) fiber induces the bundle i*flN over (G=P L)N . Since *
*the
composition
(G=P L)N i-!BP L -! BGN
is trivial, there is a commutative diagram
D ----! (G=P L)N x RN
? ?
(1.3) p?y ?yp1
(G=P L)N ________ (G=P L)N
4 YULI B. RUDYAK
where i*flN = {p : D ! (G=P L)N } and is a proper homotopy equivalence over
(G=P L)N . The map is not unique, but we fix such a map from now on.
1.4. Definition. A homotopically trivialized PL RN bundle over X is a commu-
tative diagram of the form
E ---'-! X x RN
? ?
(1.5) ss?y ?yp1
X ________ X
where ss : E ! X is a PL RN -bundle and ' is a proper homotopy equivalence over
X. Two such diagrams (E0; ss0; '0) and (E1; ss1; '1) are equivalent iff there e*
*xists
a commutative diagram
bE ----! X x I x RN
? ?
(1.6) ?y ?ybp
X x I ________ X x I
where is a proper homotopy equivalence over X, bp(x; t; r) = (x; t), : bE! X *
*xI
is a PL RN -bundle and, moreover, bE|X x {i} = Ei, |Ei = 'i, and |Ei = ssi; i =
0; 1. Every such an equivalence class is called a homotopy triangulation of the*
* trivial
RN -bundle over X.
Given a finite polyhedron X, we construct a correspondence
( )
homotopy triangulations of
(1.7) [X; G=P L] ---! N ; N >> dim X
the trivial R -bundle over X
as follows. The maps BP Ln ! BGn yield a (non-unique) map (G=P L)N ! G=P L
which, in turn, induces a bijection
[X; (G=P L)N ] ! [X; G=P L]
for N >> dim X, (see e.g. [HW, Theorem 2] and [B, I.4.10]). Furthermore, any map
f : X ! (G=P L)N induces the diagram of the form (1.5) (a homotopy trivialized
PL RN -bundle) from (1.3). Now, it is easy to see that homotopic maps induce
equivalent trivializations.
NORMAL INVARIANT OF HOMEOMORPHISM 5
1.8. Theorem. The correspondence (1:7) is bijective.
Proof. See [B, Ch. II, x4].
We define the topologically trivialized PL RN bundle as a commutative diagr*
*am
like (1.5) where, however, ' must be a homeomorphism, and say that two such di-
agrams are equivalent if there is a diagram like (1.6) where is a homeomorphis*
*m.
Similarly to the above, every such an equivalence class is called a topological*
* trian-
gulation of the trivial RN -bundle over X, and we have a bijection
( )
topological triangulations of
(1.9) [X; T OP=P L] ---! N ; N >> dim X:
the trivial R -bundle over X
The map a in (1.1) induces the map a* : [X; T OP=P L] ! [X; G=P L] which,
under bijections (1.7) and (1.9), regards a homeomorphism as a proper homotopy
equivalence.
Let Y be the homotopy fiber of the map ffiSq2 : K(Z=2; 2) ! K(Z[2]; 5). We c*
*an
equip Y with a structure of an H-space by setting Y = Z where Z is the homotopy
fiber of ffiSq2 : K(Z=2; 3) ! K(Z[2]; 6). Recall that G=P L is an infinite loop*
* space
and hence a homotopy commutative H-space, [BV]. Let BO = BOx{1} BOxZ
be an H-space where the H-structure is induced by the tensor multiplication of
virtual vector bundles of dimension 1.
1.10. Theorem (Sullivan). We have ss4i(G=P L) = Z for every i > 1 and
ss4i+2(G=P L) = Z=2, ss2i+1(G=P L) = 0 for every i > 0. Furthermore, there are
H-equivalences
Y
(1.11) G=P L[2] ' Y x (K(Z[2]; 4i) x K(Z=2; 4i - 2))
i>1
and
(1.12) G=P L[1=2] ' BO [1=2]
Proof. See [MM, Ch. 4].
6 YULI B. RUDYAK
It is easy to see that H4(Y ; Z[2]) = Z[2]. Let -4 2 H4(Y ; Z[2]) = Z[2] be*
* an
element which is not divisible by 2, and let -2 2 H2(Y ; Z=2) = Z=2 be the non-*
*zero
element. Consider the map
Y *
* p1
g : G=P L -localization------!G=P L[2] ' Y x (K(Z[2]; 4i) x K(Z=2; 4i - 2*
*))-! Y
i>1
and define the classes K2 2 H2(G=P L; Z=2) and K4 2 H4(G=P L; Z[2]) as the maps
K2 : G=P L -g!Y --2! K(Z=2; 2); K4 : G=P L -g!Y --4! K(Z[2]; 4):
Furthermore, for every i > 1 consider the maps (where -proj-!is the obvious pro*
*jec-
tion)
Y proj
p4i-2 : G=P L[2]' Y x (K(Z[2]; 4i) x K(Z=2; 4i - 2))--! K(Z=2; 4i - 2);
i>1
Y proj
p4i: G=P L[2]' Y x (K(Z[2]; 4i) x K(Z=2; 4i - 2))--! K(Z[2]; 4i)
i>1
and define the classes K4i-2 2 H4i-2(G=P L; Z=2) and K4i 2 H4i(G=P L; Z[2]) via
the maps
K4i-2 : G=P L -localization------!G=P L[2] -p4i-2--!K(Z=2; 4i - 2);
(1.13)
K4i: G=P L -localization------!G=P L[2] -p4i-!K(Z=2; 4i):
Notices that the classes K4i and K4i-2 are not unique: they depends on (and
determine) the direct product decomposition (1.11).
1.14. Lemma. Let X be a finite CW -space such that the group H*(X) is torsion
free. Then the group [X; G=P L[1=2]] is torsion free.
Proof. Because of (1.12), it suffices to prove that [X; BO [1=2]] is torsion f*
*ree. To
prove this, it suffices to prove, in turn, that [X; BO [p]] is torsion free fo*
*r every
odd prime p. Let BO be the H-space where the H-structure is given by the
Whitney sum of virtual 0-dimensional vector bundles. It is known that, for every
odd prime p, there is an H-equivalence BO [p] ' BO [p], see [AS] or [MM, 9.15*
*].
So, it suffices to prove that KO*(X) Z[p] is torsion free. Considering the Ati*
*yah
-Hirzebruch spectral sequence for KO*(X) Z[p], we conclude that its initial te*
*rm
is torsion free because, by Bott Periodicity Theorem, the groups KO*(pt) Z[p]
are torsion free. Hence, the spectral sequence degenerates, and thus the group
KO*(X) Z[p] is torsion free.
NORMAL INVARIANT OF HOMEOMORPHISM 7
1.15. Proposition. Let X be a finite CW -space such that the group H*(X) is
torsion free. Let f : X ! G=P L be a map such that f* K4n = 0 and f* K4n+2 = 0
for every n 1. Then f is null-homotopic.
Proof. Consider the commutative square
G=P L ---l1-! G=P L[2]
? ?
l2?y ?yl3
G=P L[1=2] ---l4-! G=P L[0]
where the horizontal maps are the Z[2]-localizations and the vertical maps are *
*the
Z[1=2]-localizations. Because of 1.10, [X; G=P L] is a finite generated abelian*
* group,
and so it suffices to prove that both l1Of and l2Of are null-homotopic. First,*
* we
remark that l2Of is null-homotopic provided l1Of is. Indeed, since H*(X) is tor*
*sion
free, the group [X; G=P L[1=2]] is torsion free by 1.14. Now, if l1Of is null-h*
*omotopic
then l3Ol1Of is null-homotopic, and hence l4Ol2Of is null-homotopic. Thus, l2O*
*f is
null-homotopic since [X; G=P L[1=2]] is torsion free.
So, it remains to prove that l1Of is null-homotopic. We set
Y Y
:= (K(Z[2]; 4i) x K(Z=2; 4i - 2))
i>1
Clearly, the equalities f* K4i= 0 and f* K4i-2 = 0, i > 1, imply that the map
Y p2 Y
X ! G=P L -l1!G=P L[2] ' Y x -!
is null-homotopic. So, it remains to prove that the map
Y p1
g : X -f!G=P L -l1!G=P L[2] ' Y x -! Y
is null-homotopic. We have the following diagram with the exact row:
H4(X; Z[2]) --i*--! [X; Y ] -(-2)*---!H2(X; Z=2)
??
y(-4)*
H4(X; Z[2])
Notice that (-4)*i* : Z[2] ! Z[2] is the multiplication by 2" where " is an inv*
*ertible
element of the ring Z[2]. Since (-2)*(g) = 0, we conclude that g = i*(a) for so*
*me
a 2 H4(X; Z[2]). Now,
0 = (-4)*(g) = (-4)*(i*a) = 2a":
But H*(X; Z[2]) is torsion free, and thus a = 0.
8 YULI B. RUDYAK
x2. The Sullivan map [X; G=P L] ! PdimX
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 Z, i.e., Pi is the Wall group of the trivial group. Given a closed co*
*nnected
n-dimensional PL manifold X (which is assumed to be orientable for n = 4k), we
define a map
(2.1) s : [X; G=P L] ! Pn
as follows. According to 1.8, every element u 2 [X; G=P L] is the equivalence c*
*lass
of a commutative diagram
E ---'-! X x RN
? ?
ss?y ?yp1
X ________ X
where ss : E ! X is a PL RN -bundle over X and ' is a proper homotopy equivalen*
*ce
over X. We deform (properly) ' to a map : E ! X x RN which is transversal to
X, and put Z = -1 (X). We set h := |Z : Z ! X. Then the normal bundle Z of
Z has the form h*j for some PL bundle j over X (which turns out to be properly
homotopy equivalent to X ) over X), see e.g. [MM, 2.23]. In other words, there *
*is
a commutative diagram
bh
A ----! B
? ?
pZ?y ?ypX
Z ---h-! X
where Z = {pZ : A ! Z}, j = {pX : B ! X}, and where the map bhyields PL
isomorphisms of fibers. For n = 4k, let be the symmetric bilinear intersecti*
*on
form on
Ker{h* : H2k(Z : Q) ! H2k(X; Q)}:
We define s(u) = oe(_)8where oe( ) is the signature of . For n = 4k + 2, we de*
*fine
s(u) to be the Kervaire invariant of the normal map (h; bh), see [B, Ch. III, *
*x4],
NORMAL INVARIANT OF HOMEOMORPHISM 9
[N1] for details. In particular, if h turns out to be a homotopy equivalence t*
*hen
s(u) = 0.
Given a map f : X ! G=P L with X as above, it is useful to introduce the
notation s(M; f) := s([f]) where [f] is the homotopy class of f.
Recall that every smooth manifold can be regarded canonically as a PL manifo*
*ld.
2.2. Theorem (Sullivan). We can choose the classes K4i2 H4i(G=P L; Z[2]) and
K4i+2 2 H4i+2(G=P L; Z=2) in (1:13) so that, for every map f : M ! G=P L of a
closed connected smooth manifold M, the following hold:
(i) If dim M = 4k then
* +
X
s(M; f) = L(o M) f* (K4i); [M]
i
where L(o M) is the total Hirzebruch class of the tangent bundle o M of M.
(ii) If dim M = 4k + 2 then
* +
X
s(M; f) = V 2(o M) f* (K4i+2); [M]2
i
where V 2 is the square of the total Wu class.
Proof. See [MM, Ch. 4].
x3. Detecting families
Given a CW -space X, consider a connected closed smooth singular manifold
fl : M ! X in X. Then, for every map f : X ! G=P L, the invariant s(M; ffl) 2
PdimM is defined. Clearly, if f is null-homotopic then s(M; ffl) = 0.
3.1 Definition Let {fli : Mi ! X} be a family of closed connected smooth singul*
*ar
manifolds in X. We say that the family {fli : Mi ! X} is a detecting family for
X if, for every map f : X ! G=P L, the validity of the equalities s(Mi; ffli) =*
* 0
implies that f is null-homotopic.
Notice that G=P L is an H-space, and hence the collection s(Mi; ffli) unique*
*ly
determines the homotopy class of any map f : X ! G=P L for every detecting
family {fli : Mi ! X}.
10 YULI B. RUDYAK
The concept of detecting family is related to Sullivan's "characteristic var*
*iety",
but it is not precisely the same. If a family F of singular manifolds in X con-
tains a detecting family, the F itself is a detecting family. On the contrary,*
* the
characteristic variety is in some sense "minimal" detecting family.
3.2. 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} where each *
*Mi
is orientable.
Proof. Since H*(X) is torsion free, every homology class in H*(X) can be realiz*
*ed
by a closed connected smooth oriented singular manifold, see e.g. [C]. Let {fl*
*i :
Mi ! X} be a family of smooth oriented closed connected singular manifolds such
that the elements (fli)*[Mi] generate all the groups H2k(X).
We prove that {fli : Mi ! X} is a detecting family. Consider a map f : X !
G=P L such that si(Mi; ffli) = 0 for all i. We must prove that f is null-homoto*
*pic.
Because of 1.15, it suffices to prove that f* K4n = 0 and f* K4n-2 = 0 for e*
*very
n 1. Furthermore, H*(X) = Hom (H*(X); Z) because H*(X) is torsion free. So,
it suffices to prove that
(3.3) = 0 for everyx 2 H4n(X)
and
(3.4) = 0 for everyx 2 H4n-2 (X; Z=2):
First, we prove (3.3). Since the classes (fli)*[Mi]; dim Mi = 4n generates the *
*group
H4n(X), it suffices to prove that
= 0 whenever dim Mi = 4n :
We prove this by an induction on n.
First, let n = 1. By 2.2(i), we have
0 = s(Mi; ffli) = <(ffli)*K4; [Mi]>= :
Suppose by induction that = 0 whenever dim Mi = 4m < 4n.
Since the classes (fli)*[Mi] generates all the 4m-dimensional homology classes *
*with
m < n, we conclude that f* K4m = 0 for m < n. Now, because of 2.2(i), for every
4n-dimensional Mi we have
0 = s(Mi; ffli) = <(ffli)*K4n; [Mi]>= :
NORMAL INVARIANT OF HOMEOMORPHISM 11
This completes the proof of the equality 3.3.
Passing to the case n = 4k + 2, notice that the group H2k(X; Z=2) is generat*
*ed
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, just using 2.2(ii) instead*
* of
2.2(i).
x4. Splitting theorems
4.1. Definition. Let Mn ; Ak and W n+k be three smooth connected manifolds
where M is assumed to be compact. We say that a map f : W n+k ! Mn x Ak
splits along Mn if there is a homotopy
ft : W n+k ! Mn x Ak; t 2 I
such that:
(i) f0 = f;
(ii) there is a compact subset K of W such that ft|W \ K = f|W \ K for every
t 2 I;
(iii) the map g := f1 is transversal to M;
(iv) g|g-1 (M) : g-1 (M) ! M is a homotopy equivalence.
4.2. Theorem. Let Mn ; n 5 be a closed connected n-dimensional smooth man-
ifold such that ss1(M) is a free abelian group. Then every proper homotopy equi*
*va-
lence h : W n+1 ! Mn x R splits along Mn .
Proof. Because of the transversality theorem, there is a homotopy ht : W ! M xR
which satisfies conditions (i)-(iii) of 4.1. Because of a theorem of Novikov [*
*N2,
Theorem 3], there is a sequence of interior surgeries of the inclusion f-1 (M) *
* W
in W such that the final result of these surgeries is a homotopy equivalence V *
* W .
Using the Pontryagin-Thom construction, we can realize this sequence of surgeri*
*es
via a homotopy ft such that ft satisfies conditions (i)-(iii) of 4.1 and f-11(M*
*) =
V .
4.3. Theorem (Farrell-Hsiang). Let Mn be a manifold as in 4:2. Then every
homotopy equivalence W n+1 ! Mn x S1 splits along Mn .
Proof. This follows from results of Farrell and Hsiang [FH, Theorem 2.1] since
Wh (Zm ) = 0 for every m.
12 YULI B. RUDYAK
4.4. Corollary. Let Mn be a manifold as in 4:2. Let T kdenote the k-dimensional
torus. Then every homotopy equivalence W n+k ! Mn x T ksplits along Mn .
Proof. This follows from 4.3 by induction.
4.5. Theorem. Let Mn be a manifold as in 4:2. Then every homeomorphism
h : W n+k ! Mn x Rk of a PL manifold W n+k splits along Mk .
Proof. The canonical inclusion T k-1x R Rk yields the inclusion M x T k-1x R
M x Rk. We set W1 := h-1 (M x T k-1x R). Notice that W1 is a smooth manifold
since it is an open subset of W . Now, set u = h|W1 : W1 ! M x T k-1x R. Then,
by 4.2, u splits along M x T k-1, i.e., there is a homotopy ut as in 4.1. We s*
*et
f := u1; V := f-1 (M x T k-1) and g := f|V . Because of 4.4, g : V ! M x T k-1
splits along M. Hence, f splits along M, and therefore u splits along M. Let __*
*utbe
the homotopy which realizes this splitting as in 4.1. Now, we define the homoto*
*py
ht : W ! M x Rn by setting ht|W1 := __ut|W1 and ht|W \ W1 := h|W \ W1. Notice
that {ht} is a well-defined and continuous family since {__ut} satisfy 4.1(ii).*
* It is
clear that ht satisfies the conditions (i)_(iii) of 4.1 and that h1 extends f o*
*n the
whole W , i.e., h splits along M.
Now, let a : T OP=P L ! G=P L be a map as in (1.1).
4.6. Theorem. Let M be as in 4:2. Then the composition
[M; T OP=P L] -a*![M; G=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. According to (1.9), every element v 2 [M; T OP=P L] is an equivalence cl*
*ass
of a commutative diagram
W ---'-! M x RN
?? ?
y ?yp1
M ________ M
where W is a PL manifold and ' is a homeomorphism over M. Passing to the
element a*v, we must regard the diagram as an element of [M; G=P L]. Now, by 4.*
*5,
the homeomorphism ' is properly homotopic to a PL map g : W ! M x RN such
that g|g-1 (M) : g-1 (M) ! M is a homotopy equivalence. Thus, s(a*v) = 0.
Now we show that the condition dim M 5 in 4.6 is not necessary.
NORMAL INVARIANT OF HOMEOMORPHISM 13
4.7. Corollary. Let M be a closed connected smooth manifold such that ss1(M)
is a free abelian group. Then the composition
[M; T OP=P L] -a*![M; G=P L] -s!PdimM
is trivial.
Proof. Let CP 2 be 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 eve*
*ry
g : M ! G=P L, see [B, Ch. III, x5]. But, by 4.6, the composition
[M x CP 2; T OP=P L] -a*![M x CP 2; G=P L] -s!PdimM
is trivial, and so s(M; af) = 0 for every f : M ! T OP=P L.
x5. The triviality of the normal invariant of a
homeomorphism of manifolds with free abelian
fundamental group and torsion free homology groups
5.1. Definition. Given two PL RN -bundles = {p : A ! X} and j = {q : B !
X} over a space X, we define a G-equivalence : ! j to be a commutative
diagram
A ---'-! B
? ?
p?y ?yq
X ________X
where ' is a proper homotopy equivalence, see 1.2.
Let Xn be a closed connected orientable n-dimensional PL manifold which is
embedded in RN+n ; N >> n, and let X ; dim X = N, be a normal bundle of
this embedding. Let T X be the Thom space of X , and let c : SN+n ! T X
be the degree 1 map (the Browder-Novikov map) which collapces the complement
of a tubular neighbourhood of X in RN+n , see [B, II.2.11]. We denote by ff 2
ssN+n (T X ) the homotopy class of c.
The Atiyah-Milnor-Spanier duality yields an isomorphism
(5.2) Z = HN (T X ) ~=Hn(X) = Z:
In particular, every (one of two) orientation [X] 2 Hn(X) gives us a Thom class*
* u 2
HN (T X ). It turns out that h(ff) \ u = [X] where h : ssN+n (T X ) ! HN+n (T X*
* )
is the Hurewicz homomorphism., see e.g. [B, II.2.12], [Sw, 14.40].
14 YULI B. RUDYAK
5.3. Theorem (Browder, Novikov). Let j be an oriented PL RN -bundle over X,
and let v 2 HN (T j) be the Thom class. We are given fi 2 ssN+n (T j) such that
h(fi) \ v = [X]. Then there exist a G-equivalence : j ! X such that *fi = ff,
and such is unique up to proper homotopy over X
Proof. See [B, I.4.19]. Notice that Browder [B] formulates I.4.15 (which is use*
*d in
I.4.19) for simply connected X, but in fact we do not need X in I.4.15 to be si*
*mply
connected, see e.g. [Sw, 14.49].
5.4. Construction-Definition. Given a homotopy equivalence h : M ! X of
closed connected orientable PL manifolds, we define its normal invariant
NI (h) 2 [X; G=P L]
as follows. Let [M] 2 Hn(M) and [X] 2 Hn(X) be any two orientations with
h*[M] = [X]. Let M be a normal bundle of a certain embedding M RN+n ,
and let w 2 HN (T M ) be the Thom class which corresponds to [M] under (5.2).
Denoting by fl 2 ssN+n (T ) the homotopy class of the collapsing map SN+n !
T M , we conclude that fl \ w = [M]. Let g : M ! X be homotopy inverse to h.
By setting fi := h*fl; v := g*v, we conclude that b \ v = [X]. Now, by 5.3, (w*
*ith
j = g*M ), we get a G-equivalence : g*M ! X over X. Furthermore, the
G-equivalence
1 : g*M oX ! X oX
gives us the commutative diagram
E ---'-! X x RN
? ?
(5.5) p?y ?yp1
X ________ X
where g*M oX = {p : E ! X} and ' is a proper homotopy equivalence over X.
We define NI(h) 2 [X; G=P L], the normal invariant of h, to be the element which
is given by the diagram (5.5) via the correspondence (1.7) (which is bijective *
*by
1.8).
Similarly, but simpler, every homeomorphism h : M ! X yields a bundle mor-
phism g*M ! X which gives us a homeomorphism (over X) of total spaces. So,
we have a diagram of the form (5.5) where ' is a homeomorphism. Thus, we have
an element of [X; T OP=P L].
NORMAL INVARIANT OF HOMEOMORPHISM 15
So, for every closed connected orientable PL manifold X we have the commuta-
tive diagram
{homeomorphisms M ! X} ----! [X; T OP=P L]
? ?
(5.6) \?y ?ya*
{homotopy equivalences M ! X} --NI--! [X; G=P L]
where M runs over (all admissible) closed PL manifolds and the left arrow is the
obvious inclusion.
5.7. Theorem. Let X be closed connected PL manifold such that each of the group
Hi(X) and ss1(X) is a free abelian group. Then NI(h) = 0 for every homeomorphism
h : M ! X.
Notice that X is orientable since H1(X) is torsion free, and so the normal i*
*n-
variant NI(h) is defined.
Proof. Let f : X ! T OP=P L be a map such that [f] 2 [X; T OP=P L] is the
image of {h : M ! X} under the top map of (5.6). Then, clearly, the class
NI (h) 2 [X; G=P L] is represented by the map
X -f!T OP=P L -a!G=P L:
By 3.2, X possesses a detecting family {fli : Mi ! X}. We can assume (performing
surgeries of Mi if necessary) that ss1(Mi) is a subgroup of ss1(X), and so ss1(*
*Mi)
is a free abelian group. Hence, by 4.6 and 4.7, s(Mi; affli) = 0 for every i. *
* But
{fli : Mi ! X} is a detecting family, and therefore af is null-homotopic. Thus,
NI (h) = 0.
References
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16 YULI B. RUDYAK
[HW] A. Haefliger, C. T. C. Wall: Piecewise linear bundles in the stable ran*
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Universit"at-GH Siegen, FB6/Mathematik, 57068 Siegen, Germany
E-mail address: rudyak@mathematik.uni-siegen.de, july@mathi.uni-heidelberg.de