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Search: MSC category 57M60 ( Group actions in low dimensions )

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1. CJM 2001 (vol 53 pp. 212)

Puppe, V.
 Group Actions and Codes 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, codesCategories:55M35, 57M60, 94B05, 05E20

2. CJM 1998 (vol 50 pp. 620)

Sjerve, Denis; Yang, Qing Jie
 The Eichler trace of $\bbd Z_p$ actions on Riemann surfaces We study $\hbox{\Bbbvii Z}_p$ actions on compact connected Riemann surfaces via their associated Eichler traces. We determine the set of possible Eichler traces and determine the relationship between 2 actions if they have the same trace. Categories:30F30, 57M60
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