Rigidity and crystallographic groups, I

Abstract

This is the first in a series of articles about rigidity theorems for crystallographic groups. The main theorem is (1.1). It is an "h-cobordism rigidity" statement. It implies (see (1.6)) that any crystallographic manifold, with odd order holonomy, which is h-cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it. In the second part of this work [9], we will exploit this result to prove that one can remove the word "h-cobordant" from the previous sentence. The main theorem concerns an involution on the group Wh~P'°(Mr). This is the group of G h-cobordisms of the flat torus Mr, of a crystallographic group F with holonomy group G. (Definitions of these terms appear in (1.4).) We prove a vanishing result of the Tate cohomology: