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

TL;DR

This paper proves that the n-dimensional Klein bottle is a real Bott manifold, determines its structure, and computes its rational Betti numbers, expanding understanding of moduli spaces of planar polygons.

Contribution

It establishes the real Bott manifold structure of the n-dimensional Klein bottle and related moduli spaces, providing explicit Bott matrices and Betti number calculations.

Findings

The n-dimensional Klein bottle is a real Bott manifold.

Explicit Bott matrices for these manifolds are determined.

Rational Betti numbers of the moduli spaces are computed.

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 determine 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.