# Splitting Theorems for Certain Equivariant Spectra

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*L. Gaunce Lewis, Jr.*

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 \(U\).

#### Table of Contents

# Table of Contents

## Splitting Theorems for Certain Equivariant Spectra

- Contents vii8 free
- Introduction 112 free
- Notational conventions 516 free
- Part 1. Geometrically Split Spectra 718
- Section 1. The notion of a geometrically split G-spectrum 819
- Section 2. Geometrically split G-spectra and G-fixed-point spectra 1122
- Section 3. Geometrically split G-spectra and II-fixed-point spectra 1526
- Section 4. Geometrically split spectra and finite groups 2132
- Section 5. The stable orbit category for an incomplete universe 2536

- Part 2. A Toolkit for Incomplete Universes 2940
- Section 6. A vanishing theorem for fixed-point spectra 3041
- Section 7. Spanier-Whitehead duality and incomplete universes 3243
- Section 8. Change of group functors and families of subgroups 3344
- Section 9. Change of universe functors and families of subgroups 3849
- Section 10. The geometric fixed-point functor Φ[sup(Λ)] for incomplete universes 4051
- Section 11. The Wirthmüller isomorphism for incomplete universes 4455
- Section 12. An introduction to the Adams isomorphism for incomplete universes 5061

- Part 3. The Longer Proofs 5566
- Section 13. The proof of Proposition 3.10 and its consequences 5667
- Section 14. The proofs of the main splitting theorems 6172
- Section 15. The proof of the sharp Wirthmüller isomorphism theorem 6576
- Section 16. The proof of the Adams isomorphism theorem for incomplete universes 6879
- Section 17. The Adams transfer for incomplete universes 7586

- Acknowledgments 8798
- Bibliography 8899