Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some results on orientation preserving involutions

Author: David E. Gibbs
Journal: Trans. Amer. Math. Soc. 218 (1976), 321-332
MSC: Primary 57D85
MathSciNet review: 0410770
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57D85

Retrieve articles in all journals with MSC: 57D85

Additional Information

Keywords: Equivariant bordism, Euler characteristic, orientation preserving involution, representation, Witt ring
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society