Splitting theorems for certain equivariant spectra
About this Title
L. Gaunce Lewis, Jr.
Publication: Memoirs of the American Mathematical Society
Publication Year 2000: Volume 144, Number 686
ISBNs: 978-0-8218-2046-9 (print); 978-1-4704-0277-8 (online)
MathSciNet review: 1679450
MSC: Primary 55P42; Secondary 55M35, 55N20, 55P91, 55R12
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 $U$.
Graduate students and research mathematicians interested in algebraic topology.
Table of Contents
- Notational conventions
- Part 1. Geometrically split spectra
- Section 1. The notion of a geometrically split $G$-spectrum
- Section 2. Geometrically split $G$-spectra and $G$-fixed-point spectra
- Section 3. Geometrically split $G$-spectra and II-fixed-point spectra
- Section 4. Geometrically split spectra and finite groups
- Section 5. The stable orbit category for an incomplete universe
- Part 2. A toolkit for incomplete universes
- Section 6. A vanishing theorem for fixed-point spectra
- Section 7. Spanier-Whitehead duality and incomplete universes
- Section 8. Change of group functors and families of subgroups
- Section 9. Change of universe functors and families of subgroups
- Section 10. The geometric fixed-point functor $\Phi ^\Lambda $ for incomplete universes
- Section 11. The Wirthmüller isomorphism for incomplete universes
- Section 12. An introduction to the Adams isomorphism for incomplete universes
- Part 3. The longer proofs
- Section 13. The proof of Proposition 3.10 and its consequences
- Section 14. The proofs of the main splitting theorems
- Section 15. The proof of the sharp Wirthmüller isomorphism theorem
- Section 16. The proof of the Adams isomorphism theorem for incomplete universes
- Section 17. The Adams transfer for incomplete universes