Homology of formally infinite chains

[Jamma]

Homology of "formally infinite chains"

Hello all.

To define Borel Moore homology (http://en.wikipedia.org/wiki/Borel-Moore_homology), one can allow formally infinite singular chains (as opposed to the usual finite ones) which satisfy the property that given any compact subset of the space you are probing, it will intersect with the support of only finitely many of the chains.

This is a useful gadget - if you are looking at manifolds, you can define a "fundamental class" and show that the Borel Moore homology is equal to the usual singular cohomology (with a regrading, as in Poincare duality). It is functorial over proper maps (i.e. maps for which the inverse image of a compact subset is compact).

My question is: what if we simply allow formally infinite chains, without the condition that a compact subset must only intersect with the support of finitely many chains? What do we get? We surely get a homology theory back. I'd imagine that the homology groups are likely to be isomorphic to the Borel Moore homology groups, but I'm not sure. Perhaps we could make the groups a little more tame by only allowing countably infinite chains, or something like that.

This homology theory would be functorial with respect to all continuous maps, not just proper ones.

 

[Jamma]

Never mind, there's an obvious problem:

There can be infinitely many chains with the same boundary, so the boundary map isn't well defined. Perhaps you could try to resurrect it by asserting that you can only pick formal sums of chains which give a well defined boundary (i.e. don't give an infinite coefficient on some chain), but by that point you are probably going a bit too far (and this still won't be defined on all continuous maps).