The Signature mod 8

521

THEOREM 2. P{x2k) [M] = <t(M) modulo S for any spin Poincaré complex M"^*. For oriented Poincaré complexes, we can show

THEOREM 3. There is a characteristic class y^j, in Я'^''(BSG; Zg) such that y^k [^] = o"(M) modulo S for any oriented Poincaré complex.

Theorem 3 is the best possible 'Hirzebruch Signature Theorem' for Poincaré complexes; the integer 8 cannot be replaced by a larger integer.

The classes y^j, are related to the /^-invariants of the fibration F/Top -> BSTop -► ->BSG.

1 . The Spin Case

We begin by considering the Wu class V2k in Я^'^(BSpinG; Zj). We wish to show V2k is the mod 2 reduction of a Z4 cohomology class. Equivalently, we can show Sq^ (t;2jt) = 0, where Sq^ is the i-th Steenrod Square. (In fact, f2fc = 0 for к odd, but this is irrelevant to what follows.)

We recall one definition of the Wu classes ü^. Let у be the universal spherical tion on BSG(m), m large, let MSG (m) be the Thom space of 7, and let U be the Thom class. Then define

v , ==T - ' { x { SqV ) ) ,

where Tis the Thom isomorphism (Г:Я*(BSG(т))->Я*(MSG(/«)) and x is the anti-automorphism of the Steenrod Algebra. If Af" is a Poincaré complex, let A:Äf-> -►BSG (m) be the classifying map of the stable Spivak normal fibration of M. Define Vi (M) as h* (Vi). It follows from [6, Ch. Ill] that

{ Vi { M ) ux ) [ M - ] = Sq'(x)lM2

for all X in Я"" ^ (M; Z2). Thus this is an acceptable definition of the Wu classes.

Now note that Sq^{v2k-U) = Sq^{v2k)'U, since Sq\U) = 0. Thus Sq^{v2k) = 0 ifSq^(v2k'U) = 0,But

Sq' { v2k'U ) ^Sq' ( xSq'' ) U

= (xSq')(xSq'')U = x{Sq'W)U.

Now

Sq^'Sq' = Sq^Sq^'-' + aSq'Sq^\ aeZ2, by an Adem relation, so

xiSq^'Sq' ) = xiSq''-')xiSq') + ax{Sq'')xiSq').