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

Andrei V. Prasolov
University of Tromsø, Norway

On quasishape and quasihomology

Let X be a topological space, and let $$ \textbf{SX}:=(N\mathcal{U},\mathcal{U}\in NCOV(X)) $$ be a pro-space consisting of Vietoris nerves $N\mathcal{U}$, where $\mathcal{U}$ runs over all normal coverings of X. It is well-known (due to Bernd Günther) that X and SX are strong shape equivalent. Consider now the following pro-space $$ \textbf{QX}:=(N\mathcal{U},\mathcal{U}\in COV(X)) $$ where COV(X) is the set of all coverings. The pro-space QX represents a class [QX] in the homotopy category $$ pro-CW\left[SSE^{-1}\right] $$ where SSE is the class of strong shape equivalences of pro-spaces.

Definition. The class [QX] will be called the quasishape of X.

Definition. The quasishape category QSh is the full subcategory of $pro-CW[SSE^{-1}]$ having classes [QX] as objects.

Remark. Since the strong homology $\overline{H}_{\ast}$ is well-defined on the category $pro-CW[SSE^{-1}]$, it is equally well-defined on the category QSh. The corresponding homology will be called quasihomology and denoted by $QH_{\ast}$.

Remark. It is clear that the quasishape and quasihomology of X is equivalent to the strong shape and strong homology of X, when X is paracompact.

Examples.
1. If X is a finite topological space, then its quasishape is that of a compact polyhedron.

2. If X is a locally finite topological space (i.e. every point has a finite open neighborhood), then its quasishape is that of a polyhedron.

3. Let X be the 4-point circle (the space with 4 points, weakly equivalent to a circle), and let $Y=\bigvee X$ be the wedge of countably many copies of X. Then Y has the quasishape of the Hawaiian earring.

3'. $$ QH_{\ast}(Y)\approx \prod_{i=1}^{\infty}H_{\ast}(S^1). $$
Theorem. $QH_{\ast}$ satisfies the Eilenberg-Steenrod axioms and the wedge axiom, i.e. $$ QH_{\ast}(\bigvee X_{\alpha}) \approx \prod QH_{\ast}(X_{\alpha}). $$