Splitting Theorems for Certain Equivariant Spectra
2000
American Mathematical Society (Verlag)
978-0-8218-2046-9 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-2046-9 (ISBN)
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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 $Y$. This book extends these results to a decomposition of the group $[B,C]_G$ of equivariant stable homotopy classes of maps.
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 $U$.
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 $U$.
Introduction Notational conventions Part 1. Geometrically Split Spectra: Part 2. A Toolkit for Incomplete Universes: Part 3. The Longer Proofs: Acknowledgments Bibliography.
| Erscheint lt. Verlag | 1.5.2000 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Zusatzinfo | bibliography |
| Verlagsort | Providence |
| Sprache | englisch |
| Gewicht | 198 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 0-8218-2046-X / 082182046X |
| ISBN-13 | 978-0-8218-2046-9 / 9780821820469 |
| Zustand | Neuware |
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