Subject: RE: two postings
From: "John Greenlees"
Date: Mon, 9 Jan 2006 15:33:05 -0000
Dear John,
I think this is covered by a theorem of Gaunce Lewis's
in `Splitting theorems for certain equivariant spectra.'
Mem. Amer. Math. Soc. 144 (2000), no. 686, x+89 pp.
In effect you are asking for a splitting for the incomplete
H-universe U consisting of sums of restrictions of representations
of G. One then gets (the expected) splitting indexed by
H-conjugacy classes of subgroups K of H so that H/K embeds in U
(this is Lewis's Theorem 2.4, since Q_GY is `geometrically U-split'.
More precisely, the U-suspension H-spectrum of Y is so, and Q_GY is its
0th space).
John
-----Original Message-----
Subject: tom Dieck splittinq question
From: "John R. Klein"
Date: Sun, 8 Jan 2006 00:40:49 -0500
Here's a question for the list.
Let G be a finite group, with normal subgroup H.
Let Y a based space with H action.
Form Q_G Y = space representing the G-equivariant
stable homotopy of Y.
(A point in Q_G Y is a colimit of mapping spaces of
maps of the form S^V --> S^V ^ Y, where V ranges over
a complete G-universe. )
Note: usually, one forms Q_G Y when Y is a based
G-space, but the construction makes
sense for any based space Y. When H acts on Y,
then Q_G Y has an H action.
Question: is there an analog of the tom Dieck
splitting for the fixed points (Q_G Y)^H ?
Remark: there is one when the H action on Y
extends to a G action. This result in
this case is stated in Peter May's Alaska
notes on the top of p. 207.
Does the statement appearing there generalize
to the case when Y only has an H action?
jk
John R. Klein, Professor
Department of Mathematics
Wayne State University