On certain homeomorphism groups

Abstract

Let R, Q, I and C denote respectively the reals, the rationals, the irrationals and the Cantor set endowed with their usual topologies. For any topological space X let G(X) denote the group of homeomorphisms of X. Let H(R) denote the group of orientation preserving homeomorphisms of R, S_ω the group of permutations of the set N of natural numbers and G₀(Rⁿ) the group of homeomorphisms of Rⁿ with compact support for any integer n≤1. We show that G*F can be imbedded in G when G is any one of the groups G(Q), G(I ), G( C), H( R), S_ω or G₀(R _n) where F is a free group of rank c=#R .