Lie semigroups, homotopy, and global extensions of local homomorphisms

Abstract

For a finite dimensional connected Lie group G with Lie algebra g, we consider a Lie-generating Lie wedge W [...] g. If S is a Lie subsemigroup of G with subtangent wedge W we give sufficient conditions for S to be free on small enough local semigroups U [...] S in the sense that continuous local homomorphisms extend to global ones on S. The constructions involve developing a homotopy theory of U [...] S -directed paths. We also consider settings where the free construction leads to a simply connected covering of S.