𝑆¹-fibred nil Bott surfaces and the fundamental groups

Abstract

Torus and Klein Bottle are compact quotient manifolds R²/Γ by an action of a torsionfree discrete properly discontinuous normal subgroup Γof R²⋊O(2) on the Euclidean plane R². It is well known that a fundamental group of a compact quotient manifold R²/Γ is isomorphic to Γ. In this paper, I verify that the fundamental group of Klein Bottle has a faithful representation to R²⋊ O(2) in terms of an 𝑆¹-fibred nil Bott manifold.