Topological rigidity and H_1-negative involutions on tori

Details Topological rigidity and H_1-negative involutions on tori Topological rigidity and H_1-negative involutions on tori

2011-02-14

arxiv.org/abs/1102.2660v3

Mathematics

Geometric Topology

36 pages. Section 2 and Appendix A are generalized

Scientific paper

We prove there is only one involution (up to conjugacy) on the n-torus which acts as -Id on the first homology group when n is of the form 4k, is of the form 4k+1, or is less than 4. In all other cases we prove there are infinitely many such involutions up to conjugacy, but each of them has exactly 2^n fixed points and is conjugate to a smooth involution. The key technical point is that we completely compute the equivariant structure set for the corresponding crystallographic group action on R^n in terms of the Cappell UNil-groups arising from its infinite dihedral subgroups. We give a complete analysis of equivariant topological rigidity for this family of groups.

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