The finite coarse shape groups

Abstract

In this paper we define a new pointed finite coarse shape invariant - k-th finite coarse shape group of a pointed topological space and compare it with the corresponding shape and coarse shape group. We also define a functor πˇk⊛ from the pointed finite coarse shape category Sh⋆⊛ to the category Grp of all groups (for k=0 to the category Set⋆ of all pointed sets) associating with every pointed topological space (X,x0) a k-th finite coarse shape group πˇk⊛(X,x0) and with every finite coarse shape morphism F⊛:(X,x0)→(Y,y0) a homomorphism (a base point preserving function) πˇk⊛(F⊛):πˇk⊛(X,x0)→πˇk⊛(Y,y0). We show that, in a case of a pointed compact polyhedral system, functor πˇk⊛ is continuous, i.e., commutes with the inverse limit. Finally, finite coarse shape groups of some interesting spaces (dyadic solenoid and arbitrary stable pointed space) are calculated.