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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Sören Illman
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0583849-6
Keywords: Equivariant general position, equivariant imbeddings
Article copyright: © Copyright 1980 American Mathematical Society