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# Group Actions and Codes

Published:2001-02-01
Printed: Feb 2001
• V. Puppe
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## Abstract

A $\mathbb{Z}_2$-action with maximal number of isolated fixed points'' ({\it i.e.}, with only isolated fixed points such that $\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$ on a $3$-dimensional, closed manifold determines a binary self-dual code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines the cohomology algebra $H^*(M;k)$ and the equivariant cohomology $H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual codes one gets information about the cohomology type of $3$-manifolds which admit involutions with maximal number of isolated fixed points. In particular, most'' cohomology types of closed $3$-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, {\it e.g.}, one gets that most'' cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do not admit a non-trivial involution.
 Keywords: Involutions, $3$-manifolds, codes
 MSC Classifications: 55M35 - Finite groups of transformations (including Smith theory) [See also 57S17] 57M60 - Group actions in low dimensions 94B05 - Linear codes, general 05E20 - Group actions on designs, geometries and codes

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