engleski

Isomorphisms of the coarse shape groups induced by the coarse shape paths

The coarse shape theory was founded by introducing the (pointed) coarse shape category Sh^{;∗}; (Sh_{;⋆};^{;∗};), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory. Its isomorphisms classify topological spaces strictly coarser than the shape type classification and they isomorphisms preserve some important topological invariants as connectedness, (strong) movability, shape dimension and stability. In this talk we introduce an algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X, ⋆) and for every k∈N₀, the coarse shape group π_{;k};^{;∗};(X, ⋆), having the standard shape group π_{;k};(X, ⋆) for its subgroup, is defined. Furthermore, a functor π_{;k};^{;∗};:Sh_{;⋆};^{;∗};→Grp is constructed. The coarse shape and shape groups already differ on the class of polyhedra. The coarse shape groups give us more information than the shape groups and they are even more suitable then homotopy pro groups pro-π_{;k};. Recently a notion of the coarse shape path connectedness has been introduced. This new coarse shape invariant and, consequently, topological, homotopy and shape invariant, implies the connectedness. Moreover, on metrizable compacta the connectedness and coarse shape path connectedness coincide. 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 consider isomorphisms that coarse shape paths induce between coarse shape groups (and homotopy pro-groups, as well) at the different base points of topological space.

coarse shape path connectedness; (coarse) shape groups

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