Models for classifying spaces for $${\mathbb {Z}\rtimes \mathbb {Z}}$$ Z ⋊ Z

Abstract

We construct two models for the classifying space for the family of infinite cyclic subgroups of the fundamental group of the Klein bottle. These examples do not fit in general constructions previously done, for example, for hyperbolic groups.Then for n = 1:and we conclude thatwhere J ′ is a countably infinite set. If n = 2 in (23), thenAnd at last, for i > 3, H i (X * K) = 0.