The coarse shape groups

Abstract

The (pointed) coarse shape category Sh * ( Sh ⋆ * ), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space ( X , ⋆ ) and for every k ∈ N 0 , the coarse shape group π ˇ k * ( X , ⋆ ) , having the standard shape group π ˇ k ( X , ⋆ ) for its subgroup, is defined. Furthermore, a functor π ˇ k * : Sh ⋆ * → Grp is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, π ˇ k ( X , ⋆ ) = 0 does not imply π ˇ k * ( X , ⋆ ) = 0 (e.g. for solenoids), but from pro- π k ( X , ⋆ ) = 0 follows π ˇ k * ( X , ⋆ ) = 0 . Moreover, for pointed metric compacta ( X , ⋆ ) , the n-shape connectedness is characterized by π ˇ k * ( X , ⋆ ) = 0 , for every k ⩽ n .