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Group Actions and Codes

Published online by Cambridge University Press:  20 November 2018

V. Puppe
V. Puppe*
Affiliation:
Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany. email: [email protected]
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Abstract

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A ${{\mathbb{Z}}_{2}}$-action with “maximal number of isolated fixed points” (i.e., with only isolated fixed points such that ${{\dim}_{k}}\left( {{\oplus }_{i}}{{H}^{i}}\left( M;k \right) \right)\,\,=\,\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|,\,k\,=\,\left. {{\mathbb{F}}_{2}} \right)$ on a 3-dimensional, closed manifold determines a binary self-dual code of $\text{length}\,\text{=}\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|$. In turn this code determines the cohomology algebra ${{H}^{*}}\,\left( M;k \right)$ and the equivariant cohomology $H_{{{\mathbb{Z}}_{2}}}^{*}\,\left( M;k \right)$. 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, e.g., one gets that “most” cohomology types (over ${{\mathbb{F}}_{2}}$) of closed 3-manifolds do not admit a non-trivial involution.

Keywords

55M3557M6094B0505E20Involutions3-manifoldscodes
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 1 , 01 February 2001 , pp. 212 - 224
Copyright
Copyright © Canadian Mathematical Society 2001

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