Splitting theorems for certain equivariant spectra
L. Gaunce Lewis, Jr.
AMS Subject Classification:
Primary 55M35, 55N91, 53P42, 55P91, 57S15
Secondary 55N20 55Q10 55Q91 55R12
permanent address:
L. Gaunce Lewis, Jr.
Math. Dept.
Syracuse University
Syracuse NY 13244
98-99 academic year address:
L. Gaunce Lewis, Jr.
Math. Dept. Rm 2-130
MIT
Cambridge MA 02139-4307
Email: lglewis@syr.edu
Note: The dvi file spltspec.dvi uses the yswab font
Let $G$ be a compact Lie group, $N$ be a normal subgroup of $G$,
$X$ be a $G/N$-space and $Y$ be a $G$-space. There are a number
of results in the literature giving a direct sum decomposition
of the group of equivariant stable homotopy classes of maps from
$X$ to $Y$. Here, these results are extended to a decomposition
of the group of equivariant stable homotopy classes of maps
from an arbitrary finite $G/N$-CW spectrum $B$ 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 direct sum decomposition is a
consequence of the fact that, if $C$ is geometrically split
and $(\mathcal{F}',\mathcal{F})$ is any reasonable pair of families
of subgroups of $G$, then there is a splitting of the cofibre sequence
\begin{equation*}
(E\mathcal{F}_+ \smsh C)^N \lrarrow (E\mathcal{F}'_+ \smsh C)^N
\lrarrow (E(\mathcal{F}',\mathcal{F}) \smsh C)^N
\end{equation*}
constructed from the universal spaces for the families. Both the
decomposition of the stable homotopy groups 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 Wirthm\"{u}ller isomorphisms for
incomplete universes. Also included is a vanishing theorem for the
fixed-point spectrum $(E(\mathcal{F}',\mathcal{F})\smsh C)^N$ 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 $U$.
This is a revised and expanded version of a paper submitted to the
archives several years ago.