The equivariant Hurewicz map

Lewis, L. Gaunce

doi:10.1090/S0002-9947-1992-1049614-9

The equivariant Hurewicz map
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Trans. Amer. Math. Soc. 329 (1992), 433-472 Request permission

Abstract:

Let $G$ be a compact Lie group, $Y$ be a based $G$-space, and $V$ be a $G$-representation. If $\pi _V^G(Y)$ is the equivariant homotopy group of $Y$ in dimension $V$ and $H_V^G(Y)$ is the equivariant ordinary homology group of $Y$ with Burnside ring coefficients in dimension $V$, then there is an equivariant Hurewicz map \[ h:\pi _V^G(Y) \to H_V^G(Y).\] One should not expect this map to be an isomorphism, since $H_V^G(Y)$ must be a module over the Burnside ring, but $\pi _V^G(Y)$ need not be. However, here it is shown that, under the obvious connectivity conditions on $Y$, this map induces an isomorphism between $H_V^G(Y)$ and an algebraically defined modification of $\pi _V^G(Y)$. The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.

References

Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
J. Caruso and S. Waner, An approximation theorem for equivariant loop spaces in the compact Lie case, Pacific J. Math. 117 (1985), no. 1, 27–49. MR 777436
Tammo tom Dieck and Ted Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 (1978), no. 3, 273–287. MR 501372, DOI 10.1007/BF01579214
Andreas W. M. Dress, Contributions to the theory of induced representations, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 183–240. MR 0384917
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
H. Hauschild, Äquivariante Homotopie. I, Arch. Math. (Basel) 29 (1977), no. 2, 158–165 (German). MR 467774, DOI 10.1007/BF01220390
H. Hauschild, Zerspaltung äquivarianter Homotopiemengen, Math. Ann. 230 (1977), no. 3, 279–292 (German). MR 500966, DOI 10.1007/BF01367581
Sören Illman, Equivariant singular homology and cohomology for actions of compact Lie groups, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 403–415. MR 0377858
Sören Illman, Equivariant singular homology and cohomology. I, Mem. Amer. Math. Soc. 1 (1975), no. issue 2, 156, ii+74. MR 375286, DOI 10.1090/memo/0156

An introduction to Mackey functors

The ordinary

graded cohomology of a point for cyclic

groups

The

graded equivariant ordinary cohomology of complex projective spaces with linear

actions

G. Lewis, J. P. May, and J. McClure, Ordinary $RO(G)$-graded cohomology, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 208–212. MR 598689, DOI 10.1090/S0273-0979-1981-14886-2
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778

Transformation groups and algebraic

theory

Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Takao Matumoto, Equivariant cohomology theories on $G$-$\textrm {CW}$ complexes, Osaka Math. J. 10 (1973), 51–68. MR 343259
Unni Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (1983), no. 2, 431–460. MR 701504, DOI 10.1090/S0002-9947-1983-0701504-9
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375

complexes

Stefan Waner, Mackey functors and $G$-cohomology, Proc. Amer. Math. Soc. 90 (1984), no. 4, 641–648. MR 733419, DOI 10.1090/S0002-9939-1984-0733419-0
George W. Whitehead, On spaces with vanishing low-dimensional homotopy groups, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 207–211. MR 28027, DOI 10.1073/pnas.34.5.207