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