Subject: Re: Question about G-equivariant Eilenberg-Mac Lane spaces
From: Justin Smith
Date: Mon, 17 Jul 2006 10:30:49 -0400
>> Subject: Question about G-equivariant Eilenberg-Mac Lane spaces
>> From: "Boccellari"
>> Date: Sat, 15 Jul 2006 16:27:39 +0200
>>
>> I have a question about G-equivariant Eilenberg-Mac Lane spaces.
>> I don't know if the answer is well known or not.
>> Are there Eilenberg-Mac Lane spaces for G-equivariant cohomology and
>> their
>> cohomology and homology is known and competely described as for
>> classical
>> Eilenberg-Mac Lane spaces?
>> Are there books or papers about this subject?
>>
>> Thank you for your kind attention.
>>
>> Sincerely,
>> Tommaso Boccellari
>>
Partial answer:
I have a paper that gives a G-equivariant model for G-equivariant
Eilenberg MacLane spaces that is of finite type, allowing one to read
off homological k-invariants in many cases. The reference is:
``Equivariant Eilenberg-MacLane Spaces I --- The Z-torsion free case,''
/Proceedings of the A. M. S.,/ *101* (1987), 731-737.
(There never was a part II)
____________________________________________________________________
Subject: RE: Question about G-equivariant Eilenberg-Mac Lane spaces
From: "John Greenlees"
Date: Tue, 18 Jul 2006 09:01:03 +0100
The first thing to get clear is what you mean by G-equivariant
cohomology.
If you mean `ordinary' cohomology (ie satisfying
the dimension axiom) then the cohomology was constructed by
Bredon [Bredon, Glen E. Equivariant cohomology theories.
Lecture Notes in Mathematics, No. 34 Springer-Verlag, Berlin-New York
1967 vi+64 pp.]
(for finite groups) and he studies the representing
spaces. As for their cohomology, one wouldn't expect an answer
for general coefficient systems (for explanation of which,
see Bredon's book), but various things are known in special cases.
If, by G-equivariant cohomology, you mean Bredon cohomology
then the representing spaces are map(EG, K(M,n)), where K(M,n)
is the Eilenberg-MacLane space representing n-th Bredon cohomology
with coefficient system M. Comments on cohomology as above.
There's lots more.
John Greenlees