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Splitting Theorems for Certain Equivariant Spectra
L. Gaunce Lewis, Jr., Syracuse University, NY
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Memoirs of the American Mathematical Society
2000; 89 pp; softcover
Volume: 144
ISBN-10: 0-8218-2046-X
ISBN-13: 978-0-8218-2046-9
List Price: US$48 Individual Members: US$28.80
Institutional Members: US\$38.40
Order Code: MEMO/144/686

Let $$G$$ be a compact Lie group, $$\Pi$$ be a normal subgroup of $$G$$, $$\mathcal G=G/\Pi$$, $$X$$ 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 $$(E\mathfrak F_+ \wedge C)^\Pi \longrightarrow (E\mathfrak F'_+ \wedge C)^\Pi \longrightarrow (E(\mathfrak F', \mathfrak F) \wedge C)^\Pi$$ 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 Wirthmüller 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 $$U$$.

Readership

Graduate students and research mathematicians interested in algebraic topology.

Table of Contents

• Introduction
• Notational conventions
Part 1. Geometrically Split Spectra
Part 2. A Toolkit for Incomplete Universes
Part 3. The Longer Proofs
• Acknowledgments
• Bibliography
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