Group Actions and Codes

Puppe, V.

Canadian journal of mathematics, Tome 53 (2001) no. 1, pp. 212-224

Résumé

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.

Détail

DOI : 10.4153/CJM-2001-009-0

Mots-clés : 55M35, 57M60, 94B05, 05E20, Involutions, 3-manifolds, codes

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