Approximation of 𝐺-maps by maps in equivariant general positions and imbeddings of 𝐺-complexes

Abstract

Let G be a finite group. In this paper we consider maps f : P → M f:P \to M from equivariant polyhedra into equivariant p.l. manifolds. We prove an equivariant general position result which shows how to approximate a given continuous proper equivariant (or isovariant) map f : P → M f:P \to M by a G-map which is in equivariant general position. We also apply this equivariant general position result to get a general G-imbedding theorem. Applied to the case of G-imbeddings of simplicial G-complexes into euclidean representation space this general G-imbedding theorem gives a result which provides a good hold on the required dimension of the euclidean representation space. For example in the case when G = Z m G = {Z_m} we prove that there exists a representation space R r ( k , m ) ( ρ ) {{\textbf {R}}^{r(k,m)}}(\rho ) with the property that any k-dimensional simplicial Z m {Z_m} -complex X admits a proper p.l. Z m {Z_m} -imbedding into R r ( k , m ) ( ρ ) {{\textbf {R}}^{r(k,m)}}(\rho ) and we also show that the dimension r ( k , m ) r(k,m) is best possible, i.e., one cannot find a euclidean representation space of lower dimension than r ( k , m ) r(k,m) with the same property as R r ( k , m ) ( ρ ) {{\textbf {R}}^{r(k,m)}}(\rho ) . Simple explicit expressions for the dimension r ( k , m ) r(k,m) are given. We also consider the case of semi-free actions with a given imbedding of the fixed point set into some euclidean space. Furthermore we show that the p.l. G-imbeddings of equivariant p.l. manifolds into euclidean representation space obtained by our G-imbedding results are in general equivariantly locally knotted although they are locally flat in the ordinary sense. This phenomenon can occur in arbitrarily high codimensions.