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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 Involutions, $3$-manifolds, codes
MSC Classifications: 55M35, 57M60, 94B05, 05E20 show english descriptions Finite groups of transformations (including Smith theory) [See also 57S17]
Group actions in low dimensions
Linear codes, general
Group actions on designs, geometries and codes
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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