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Topological and algebraic applications of the coarse shape theory

The coarse shape theory functorially generalizes the shape theory. A category frame for this theory is the (pointed) coarse shape category Sh (Sh? ), having (pointed) topological spaces as objects and having the (pointed) shape category Sh (Sh?) as a subcategory. There exist metrizable continua having the same coarse shape type and di¤erent shape types. The coarse shape preserves many important topological and shape invariants as connectedness, movability, strong movability, n-movability, shape dimension, triviality of shape, stability. Recently a notion of the coarse shape path connectedness has been introduced. This new topological invariant, on the class of metrizable compacta, coincides with the connectedness. On the other hand the shape path connectedness, which is introduced by generalizing the notion of joinability (which was considered by J. Krasinkiewicz and P. Minc), strictly implies the coarse shape path connectedness. In this talk we also consider isomorphisms that coarse shape paths induce between coarse shape groups (and homotopy pro-groups, as well) at the di¤erent base points of topological space.

coarse shape; homotopy groups; coarse shape groups; coarse shape path connectedness

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