Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 583849
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Richard J. Allen, Equivariant embeddings of 𝑍_{𝑝}-actions in Euclidean space, Fund. Math. 103 (1979), no. 1, 23–30. MR 535832
  • [2] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
  • [3] A. H. Copeland Jr. and J. de Groot, Linearization of a homeomorphism, Math. Ann. 144 (1961), 80–92. MR 0133805
  • [4] A. L. Edmonds, Triangulation of piecewise linear group actions (unpublished note).
  • [5] A. Flores, Über n-dimensionale Komplexe, die im $ {R_{2n\, + \,1}}$ absolut selbstverschlungen sind, Ergebnisse eines Math. Kolloq. 6 (1933/34), 4-7.
  • [6] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844
  • [7] Sören Illman, Approximation of simplicial 𝐺-maps by equivariantly non degenerate maps, Transformation groups (Proc. Conf., Univ. Newcastle upon Tyne, Newcastle upon Tyne, 1976) Cambridge Univ. Press, Cambridge, 1977, pp. 279–283. London Math. Soc. Lecture Note Series, No. 26. MR 0474322
  • [8] Sören Illman, Smooth equivariant triangulations of 𝐺-manifolds for 𝐺 a finite group, Math. Ann. 233 (1978), no. 3, 199–220. MR 0500993
  • [9] J. M. Kister and L. N. Mann, Equivariant imbeddings of compact abelian Lie groups of transformations, Math. Ann. 148 (1962), 89–93. MR 0141727
  • [10] G. D. Mostow, Equivariant embeddings in Euclidean space, Ann. of Math. (2) 65 (1957), 432–446. MR 0087037
  • [11] G. de Rham, Reidemeister's torsion invariant and rotations of $ {S^n}$, in Differential Analysis, Tata Inst. and Oxford Univ. Press, London, 1964, pp. 27-36.
  • [12] T. Benny Rushing, Topological embeddings, Academic Press, New York-London, 1973. Pure and Applied Mathematics, Vol. 52. MR 0348752
  • [13] Reinhard Schultz, On the topological classification of linear representations, Topology 16 (1977), no. 3, 263–269. MR 0500964
  • [14] M. H. Schwartz, Théorème d’approximation simpliciale lié à une stratification, Mathematica (Cluj) 13(36) (1971), 301–334 (French). MR 0346800
  • [15] Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 0250324
  • [16] E. C. Zeeman, Unknotting combinatorial balls, Ann. of Math. (2) 78 (1963), 501–526. MR 0160218

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57Q65, 57Q35

Retrieve articles in all journals with MSC: 57Q65, 57Q35


Additional Information

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