Splitting theorems for certain equivariant spectra /

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Bibliographic Details
Author / Creator:Lewis, L. G. (L. Gaunce), 1949-
Imprint:Providence, R.I. : American Mathematical Society, 2000.
Description:ix, 89 p. ; 25 cm.
Language:English
Series:Memoirs of the American Mathematical Society, 0065-9266 ; no. 686
Subject:
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/4218373
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ISBN:082182046X (alk. paper)
Notes:"Volume 144, number 686 (fourth of 5 numbers)."
Includes bibliographical references.
Description
Summary:Let $G$ be a compact Lie group, $\Pi$ be a normal subgroup of $G$, $\mathcal G=G[LAMBDA]Pi$, $X$ be a $\mathcal G$-space and $Y$ be a $G$-space. There are a number of results in the literature giving a direct sum decomposition of the group $[\Sigma^\infty X,\Sigma^\infty Y]_G$ of equivariant stable homotopy classes of maps from $X$ to $Y$. Here, these results are extended to a decomposition of the group $[B,C]_G$ of equivariant stable homotopy classes of maps from an arbitrary finite $\mathcal G$-CW sptrum $B$ to any $G$-spectrum $C$ carrying a geometric splitting (a new type of structure introduced here). Any naive $G$-spectrum, and any spectrum derived from such by a change of universe functor, carries a geometric splitting.Our decomposition of $[B,C]_G$ is a consequence of the fact that, if $C$ is geometrically split and $(\mathfrak F',\mathfrak F)$ is any reasonable pair of families of subgroups of $G$, then there is a splitting of the cofibre sequence $(E\mathfrak F_+\wedge C)^\Pi \rarrow (E\mathfrak F'_+\wedge C)^\Pi \rarrow (E(\mathfrak F',\mathfrak F)\wedge C)^\Pi$ constructed from the universal spaces for the families. Both the decomposition of the group $[B,C]_G$ and the splitting of the cofibre sequence are proven here not just for complete $G$-universes, but for arbitrary $G$-universes.Various technical results about incomplete $G$-universes that should be of independent interest are also included in this paper. These include versions of the Adams and Wirthmuller isomorphisms for incomplete universes. Also included is a vanishing theorem for the fixed-point spectrum $(E(\mathfrak F',\mathfrak F)\wedge C)^\Pi$ which gives computational force to the intuition that what really matters about a $G$-universe $U$ is which orbits $G/H$ embed as $G$-spaces in $U$.
Item Description:"Volume 144, number 686 (fourth of 5 numbers)."
Physical Description:ix, 89 p. ; 25 cm.
Bibliography:Includes bibliographical references.
ISBN:082182046X