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Donald Davis initiated the study of an $n$-dimensional analogue of the Kleinbottle. This generalized Klein bottle occurs as a moduli space of planarpolygons 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 thecorresponding Bott matrix. We determine the small cover structure on two otherclasses of moduli spaces of planar polygons. As an application, we compute therational Betti numbers of these spaces using a formula, due to Suciu andTrevisan.

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