SOME BASIC RESULTS FOR PROPER FREE G-MANIFOLDS, WHERE G IS A DISCRETE GROUP

Abstract

Let $G$ be a countable discrete group and let $M$ be a proper free $C^r$ $G$-manifold and $N$ a $C^r$ $G$-manifold, where $1\leq r\leq\omega$. We prove that if $G$ acts properly and freely also on $N$ and if $\dim(N)\geq2\dim(M)$, then equivariant immersions form an open dense subset in the space $C^r_G(M,N)$ of all equivariant $C^r$ maps from $M$ to $N$. The space $C^r_G(M,N)$ is equipped with a topology, which coincides with the Whitney $C^r$ topology if $G$ is finite and is suited to studying equivariant maps. We also prove an equivariant version of Thom’s transversality theorem and show that $C^\omega_G(M,N)$ is dense in $C^r_G(M,N)$, for $1\leq r\leq\infty$.

AMS 2000 Mathematics subject classification: Primary 57S20