Abstract view
Published:2001-02-01
Printed: Feb 2001
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.