Group Actions and Codes

A Z2-action with "maximal number of isolated fixed points" (i.e., with only isolated fixed points such that dimk(⊕i Hi(M; k)) = |M Z2|, k = F2) on a 3-dimensional, closed manifold deter- mines a binary self-dual code of length=|M Z2|. In turn this code determines the cohomology algebra H � (M; k) and the equivariant cohomology H � Z2 (M; k). Hence, from results on binary self-dual codes one gets information about the cohomology type of 3-manifolds which admit involutions with max- imal 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 F2) of closed 3-manifolds do not admit a non-trivial involution.