| Abstract: | If B is a compact connected Lie group and N a finite central subgroup, let \({f\colon B\to B/N}\) be the associated covering morphism. The mapping cylinder \({{\mathrm{MC}}(f)}\) is a compact monoid which we call a covering space semigroup. A prominent example is the classical Möbius band \({\mathbb{M}^2}\). An (L)-semigroup is a compact n-manifold X with connected boundary B together with a monoid structure on X such that B is a subsemigroup of X. Every covering space semigroup with \({|N|=2}\) is an (L)-semigroup, and every nonorientable (L)-semigroup is a covering space semigroup. Here \({\mathbb{M}^2}\) is a guiding example. In general, a covering space semigroup X is not a manifold but does have a well-defined manifold boundary. The study of covering space semigroups leads to the following Theorem. Let B be a compact connected Lie group with a central circle group as a direct factor. Then there exist infinitely many pairwise nonisomorphic covering space semigroups with boundary B, and each such semigroup is a retract of a compact connected Lie group. |
