## Topology & Geometric Group Theory Seminar

## Fall 2010

### 1:30 – 2:30, Malott 253

Tuesday, October 5

** Jim Davis**,
Indiana University

*Equivariant rigidity, Smith theory, and actions on tori*

I will state the problem of equivariant rigidity for a discrete group
G. A complete analysis will be made for the crystallographic group
G_{n} given by the semidirect product of Z/2 acting on
Z^{n} by multiplication by -1. I will discuss the theorem
that G_{n} satisfies equivariant rigidity when n is congruent
to 0 or 1 modulo 4 and the constructions of counterexamples when n > 3
is congruent to 2 or 3 modulo 4.

Equivalently, one classifies involutions on tori which induce
multiplication by -1 on the first homology.

Ingredients are Smith theory, surgery theory, and the
Farrell–Jones conjecture. A key aspect of the application of
Smith theory is the use of a theorem from point-set topology that a
connected complete metric space is path-connected.

This is joint work with Qayum Khan and Frank Connolly.

Back to seminar home page.