Dubrovnik VI – Geometric Topology September 30 – October 7, 2007
	ABSTRACTS	pdf-version of this abstract

Tuomas Korppi
University of Helsinki, Finland

A non-standard homology theory with some nice properties

Let K be an arbitrary small subcategory of the category of pairs (X,A), where X is paracompact and A is closed in X and continuous maps between such pairs. Let G be an Abelian group, and let ^*G be its suitable elementary extension.

We present a microsimplicial homology theory for spaces in K with coefficients in ^*G, related to the McCord homology theory. Our homology theory, unlike McCord homology, is based on non-near standard microsimplices as well as near-standard microsimplices.

This homology theory has the following properties:

• The homology theory satisfies all the Eilenberg-Steenrod axioms including exactness.
• The homology theory is continuous with respect to resolutions of spaces.
• For compact spaces the homology theory coincides with Čech homology. (Note that the homology theory is not defined for an arbitrary coefficient group!)
• For simplicial pairs (K,L) we have a characterization of the homology groups.
• Let P be a one-point space, and X a space in K. Then $f: P \to X$ and $g: P \to X$ induce the same map in homology if and only if f(P) and g(P) lie in the same quasi-component of X.