Some results on orientation preserving involutions

Abstract

The bordism of orientation preserving differentiable involutions is studied by use of the signature-like invariant ab : O ∗ ( Z 2 ) → W 0 ( Z 2 ; Z ) {\text {ab}}: {\mathcal {O}_\ast }({Z_2}) \to {W_0}({Z_2};Z) . The equivariant Witt ring W 0 ( Z 2 ; Z ) {W_0}({Z_2};Z) is calculated and is shown to be isomorphic under ab to the effective part of O 4 ( Z 2 ) {\mathcal {O}_4}({Z_2}) . Modulo 2 relations are established between the representation of the involution on H 2 k ( M 4 k ; Z ) / torsion {H^{2k}}({M^{4k}};Z)/{\operatorname {torsion}} and χ 0 ( F ) {\chi _0}(F) and χ 2 ( F ) {\chi _2}(F) , where χ i ( F ) {\chi _i}(F) is the Euler characteristic of those components of the fixed point set with dimensions congruent to i modulo 4. For manifolds of dimension 4 k + 2 4k + 2 , it is shown that χ 0 ( F ) ≡ χ 2 ( F ) ≡ 0 ( mod 2 ) {\chi _0}(F) \equiv {\chi _2}(F) \equiv 0\;(\bmod 2) . Finally the ideal E 0 ( Z 2 ; Z ) {E_0}({Z_2};Z) consisting of those elements of W 0 ( Z 2 ; Z ) {W_0}({Z_2};Z) admitting a representative of type II is determined.