Some results on orientation preserving involutions

Gibbs, David E.

doi:10.1090/S0002-9947-1976-0410770-5

Some results on orientation preserving involutions
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Trans. Amer. Math. Soc. 218 (1976), 321-332 Request permission

Abstract:

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

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