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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The equivariant Hurewicz map


Author: L. Gaunce Lewis
Journal: Trans. Amer. Math. Soc. 329 (1992), 433-472
MSC: Primary 55Q91; Secondary 54H15, 55M35, 55N10, 55N91, 55P42, 55P91, 57S15
MathSciNet review: 1049614
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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

$\displaystyle h:\pi _V^G(Y) \to H_V^G(Y).$

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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1049614-9
PII: S 0002-9947(1992)1049614-9
Keywords: Hurewicz map, Hurewicz isomorphism, Whitehead theorem, equivariant ordinary homology, Freudenthal Suspension Theorem
Article copyright: © Copyright 1992 American Mathematical Society