Re: Topology on cohomology groups

[Andrew Stacey <andrew.stacey@math.ntnu.no>]

On Fri, Jun 19, 2009 at 10:26:13AM +0100, Steve Vickers wrote:
> A cohomology group can easily be an infinite power of the coefficient
> group. But such a group has a natural non-discrete topology, namely the
> compact-open (which in this case is also the product topology).
>
> Are there approaches to cohomology that, as part of the process, also
> supply topologies on the cohomology groups?
>

I'm not sure if this is quite what you are looking for, but the topology on
cohomology theories is given as an inverse limit (if I have my limits the
correct way round) over the finite skeleta.  This has an impact, for example,
in the correct statement of the Kunneth theorem on cohomology of products (one
has to complete the tensor product with respect to the topology).

A fairly comprehensive and detail account is in Boardman and
Boardman+Johnson+Wilson in the Handbook of Algebraic Topology: MR1361889 and
MR1361900 (though it was known well before that).  These papers are available
online from Steve Wilson's homepage:

http://www.math.jhu.edu/~wsw/

(scan right down to the bottom).

Sarah Whitehouse and I also look at this in our paper 'The Hunting of the Hopf
Ring' (arxiv:0711.3722, to appear in HHA).

Andrew Stacey

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