Local cross sections in locally compact groups

Abstract

Introduction. Let p be the natural projection of the topological group G, with subgroup H, onto the coset space G/H. The subgroup H is said to have a local cross section if there exists an open set U in G/H, and a continuous function f defined on U with values in G such that pf(x) x for x in U. The most general conditions on G and H under which such a function exists are not known. It has been conjectured [7, p. 33]2 that if G is compact and of finite dimension, then H has a local cross section. (For the infinite-dimensional case, there are examples of compact groups with closed subgroups not having a local cross section.) In this paper, we show that if G is locally compact, separable, metric, and of finite dimension, and H is a closed subgroup of G, then H has a local cross section. In ? 1, several elementary lemmas necessary for the proof are stated, along with certain properties of Lie groups and projective limits. In ?2, we prove the main theorem.