engleski

Alternative approach to the coarse shape groups

The coarse shape theory is a relatively new theory, introduced about ten years ago as a generalisation of the shape theory, giving a coarser tool for classifying localy bad topological spaces. In this talk we give a new characterisation of coarse shape groups which are essential invariants in this theory. Coarse shape groups showed to be very interesting algebraic homotopy, shape and coarse shape invariant. Namely, the coarse shape groups provide information on pointed spaces better than the shape groups, especially when the corresponding shape groups vanish. Further, for a pointed metric compact space (pointed compactum) (X ; x0), unlike the shape groups, coarse shape groups t into a long exact sequence. Comparing the coarse shape groups of metric compacta with corresponding homotopy pro-groups, one can notice that coarse shape groups have many advantages. Namely, for a pointed compactum (X ; x0), pro-k (X ; x0) ' 0 is equivalent to k (X ; x0) = 0 ; for every k 2 N ; but on the other hand, pro-k (X ; x0) does not have an algebraic structure of a group while k (X ; x0) does. Elements of the coarse shape groups are coarse shape morphisms between a pointed k-dimenisonal sphere Sk and a pointed topological space represented by morphisms of pro-Grp. A reduction functor eR from pro-Grp to pro-Grp is de fied, which represents morphisms in pro-category as morphisms in pro-category between more complex objects. The benefit is obvious since it is much eas- ier to analyse and construct isomorphisms in pro-category than in pro- category, even when we have more complicated terms in inverse systems.

coarde shape group, pro category, homotopy

nije evidentirano