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
>>
>>
>>