The equivariant Hurewicz map
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- by L. Gaunce Lewis PDF
- 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
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 329 (1992), 433-472
- MSC: Primary 55Q91; Secondary 54H15, 55M35, 55N10, 55N91, 55P42, 55P91, 57S15
- DOI: https://doi.org/10.1090/S0002-9947-1992-1049614-9
- MathSciNet review: 1049614