Real Bott manifold structure of n-dimensional Klein bottle and its rational Betti numbers

Abstract

Donald Davis initiated the study of an n-dimensional analogue of the Klein bottle. This generalized Klein bottle occurs as a moduli space of planar polygons for a certain choice of side lengths. In this paper, we show that the n-dimensional Klein bottle is a real Bott manifold and determines the corresponding Bott matrix. We determine the small cover structure on two other classes of moduli spaces of planar polygons. As an application, we compute the rational Betti numbers of these spaces using a formula, due to Suciu and Trevisan.