Subject: Re: 2 confs and question From: "Hadi Zare" Date: Fri, 27 Apr 2007 21:59:16 +0100 (BST) Dear Johannes, If you know all (mod p) (co-)homology, then you might be able to use the Bockstein spectral sequence. I think that it would be too complicated looking for a general description. There is a paper by Cohen-Peterson "On the homology of certain spaces looped beyond their connectivity" regarding homology of $QS^{-1}$. It contains some mistakes and later on is detected in a paper by Soren Galatius "Mod $p$ homology of the stable mapping class group". Although the description there also is not very explicit and still all of what I said is with $Z/2$-coefficients. The descriptions for odd primes p is less clear. There is also a paper by Thomas Hunter for homology of $\Omega^{n+1}S^n$ and again localised at primes. All of this arguments are based on Bar resolutions and Eilenberg-Moore spectral sequences. About the image of $\eta$ one would guess that it will be the desuspension of the image of $\eta:QS1--->QS0$ modulo decomposables. I think in this special case the image of $\eta:QS1--->QS0$ is known with integer coefficients, but I don't remember it now. I will be very happy to see more clear facts about homology of $QS^{-1}$. Best, Hadi p.s. that email address is not working! >> Subject: A question for the list >> From: Johannes Ebert >> Date: Wed, 25 Apr 2007 14:10:21 +0200 (CEST) >> >> Hello, >> >> I have the following question for you. I want to know explicitly the first >> few (co)homology groups of the spaces QS0 and QS^{-1} = \Omega QS0. I >> would >> be pleased if I knew the (co)homology up to degree 5 or 6, but I >> definitely >> need the integral homology (and not the mod p homology, which is >> well-known). I >> also like to know the effect of the Hopf map QS0 \to QS^{-1} on the >> (co)homology. I am sure that this is written down somewhere, but I cannot >> find a reference. >> Can someone of you help me? >> >> Best regards, >> >> Johannes Ebert >> >> >>