engleski

The coarse shape groups

Coarse shape isomorphisms preserve some important topological invariants as connectedness, (strong) movability, shape dimension and stability. There are also several new algebraic coarse shape invariants. In this talk 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∈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. 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_{; ; k}; ; (X, ⋆)=0 does not imply π_{; ; k}; ; ^{; ; ∗}; ; (X, ⋆)=0 (e.g. for solenoids), but from pro-π_{; ; k}; ; (X, ⋆)=0 follows π_{; ; k}; ; ^{; ; ∗}; ; (X, ⋆)=0. Moreover, for pointed metric compacta (X, ⋆), the n-shape connectedness is characterized by π_{; ; k}; ; ^{; ; ∗}; ; (X, ⋆)=0, for every k≤n.

shape; coarse shape; shape group; hmotopy pro-group

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