  Recently 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. Davis determined many
topological invariants and manifold-theoretic properties of this space. The
main aim of this short note is to show that the $n$-Klein bottle is a real Bott
manifold and determine the corresponding Bott matrix. We also determine two
other classes of polygon spaces that are total spaces of an iterated $S^1$
bundle. In all these cases we compute the Betti numbers using a formula, due to
Suciu and Trevisan.
