Approximation of $G$-maps by maps in equivariant general positions and imbeddings of $G$-complexes
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- by Sören Illman PDF
- Trans. Amer. Math. Soc. 262 (1980), 113-157 Request permission
Abstract:
Let G be a finite group. In this paper we consider maps $f:P \to M$ 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 $f:P \to M$ 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 $G = {Z_m}$ we prove that there exists a representation space ${{\textbf {R}}^{r(k,m)}}(\rho )$ with the property that any k-dimensional simplicial ${Z_m}$-complex X admits a proper p.l. ${Z_m}$-imbedding into ${{\textbf {R}}^{r(k,m)}}(\rho )$ and we also show that the dimension $r(k,m)$ is best possible, i.e., one cannot find a euclidean representation space of lower dimension than $r(k,m)$ with the same property as ${{\textbf {R}}^{r(k,m)}}(\rho )$. Simple explicit expressions for the dimension $r(k,m)$ 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.References
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Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 262 (1980), 113-157
- MSC: Primary 57Q65; Secondary 57Q35
- DOI: https://doi.org/10.1090/S0002-9947-1980-0583849-6
- MathSciNet review: 583849