;

;

is defined and non-trivial; (ii) if k 2 then each of the Massey products

;

;

is defined and non-trivial; (iii) if k 3 then the Massey product

is defined and non-trivial.
Proof. We prove the first case of (ii) only, all the other cases can be cons*
*idered
similarly. Let p# : C*(B) ! C*(E) denote the induced homomorphism of singular
THOM SPACES AND MASSEY PRODUCTS 5
cochains. Take elements a; b; c 2 C*(B) with [a]_= ff; [b] = fi; [c] = fl. Then*
* there
are cochains x; y 2 C*(B) with dx = __ab; dy = bc and such that [__ay + __xc] =*
*2(ff; fi).
Take a cochain s 2 C2(E) with_[s] = . It is clear that d(s2p# x) = (sp# __a)(sp*
*# b)
and that d(sp# (y) = (sp# b)(p# c). So, it suffices to prove that [s2(p# (__ay *
*+ __xc)] =2
(p*ff; p*fl), i.e, that 2p*[__ax + __yc] =2(p*ff; p*fl).
Suppose the contrary. Then
2p* [__ay + __xc] = (p*ff)(u0 + u1 + : :+:kuk) + (p*fl)(v0 + v1 + : :+:kvk)
= (p*ff)(u0 + 2u1 + : :+:k+1 uk) + (p*fl)(v0 + 2v1 + : :+:k+1 vk)
Xk
where ui; vi 2 p*H*(B). By expanding k+1 = ii with i 2 H*(B), we have
i=0
2p*[__ay + __xc] = (p*ff)(a0 + a1 + : :+:kak) + (p*fl)(b0 + b1 + : :+:kbk)
where ai; bi 2 p*H*(B). Now, because of 2.1 and since k 2, we conclude that
p*[__ay+__xc] = p*(ff)a2+p*(fl)b2, and so [__ay+__xc] 2 (ff; fl). This is a con*
*tradiction.
3. Thom spaces and Massey products
We need some preliminaries on Thom spaces of normal bundles. Standard refer-
ences are [B], [R]. Let M, X be two closed smooth manifolds, and let i : M ! X *
*be
a smooth embedding. Let be a normal bundle of i : M X, dim = d, and let
T be the Thom space of . We assume that is orientable, choose an orientation
of and denote by U 2 Hd(T ) the Thom class of .
Let N be a closed tubular neighborhood of i(M) in X. Let V be the interior o*
*f N,
and set @N = N \V . The Thom space T can be identified with X=X \V = N=@N.
We denote by
(3.1) c : X -quotient----!X=(X \ V ) = T
the standard collapsing map.
Recall that, for every two pairs (Y; A); (Y; B) of topological spaces, there*
* is a
natural cohomology pairing
Hi(Y; A) Hj(Y; B) ! Hi+j(Y; A [ B)
see [D]. In particular, we have the pairings
' : Hi(T ) Hj(M) ! Hi+j(T )
of the form
Hj(N; @N) Hi(N) ! Hi+j(N; @N)
and the pairing
: Hi(T ) Hj(X) ! Hi+j(T )
of the form
Hj(X; X \ V ) Hi(X) ! Hi+j(X; X \ V )
6 YULI RUDYAK AND ALEKSY TRALLE
It is well known and easy to see that the diagram
H*(T ) H*(M) ---'-! H*(T )
x fl
1i* ?? flfl
(3.2) H*(T ) H*(X) ----! H*(T )
? ?
1c* ?y ?yc*
H*(X) H*(X) ----! H*(X)
commutes; here = X is induced by the diagonal X ! X x X. As usual, for the
sake of simplicity we denote each of the products '(a b), (a b) and (a b)
by ab.
Finally, we recall that the Euler class O = O() of is defined as O := z*U 2
Hd(M) where z : M ! T is the zero section of the Thom space. Furthermore,
(3.3) z*(Ua) = Oa for every a 2 H*(M)
see e.g. [R, Prop. V.1.27].
3.4. Lemma. Let ff; fi; fl 2 H*(M) be such that the Massey product __
is defined and 0 =2 .
Proof. Clearly, OffOfi = 0 = OfiOfl, and so the Massey product __