On exactness of the coarse shape group sequence

Abstract

The coarse shape groups are recently introduced. Gi-ven a pointed pair (X, X₀, x₀) and a k∈N, the relative coarse shape group π_{; ; ; k}; ; ; ^{; ; ; ∗}; ; ; (X, X₀, x₀), having the standard relative shape group π_{; ; ; k}; ; ; (X, X₀, x₀) for its subgroup, is defined. They establish a functorial relations of the topological, homotopy and (coarse) shape category to the category of groups. Therefore, the coarse shape groups are new algebraic topological, homotopy and (coarse) shape type invariants. For every pointed pair of metric compacta (X, X₀, x₀) and for every k>1, the boundary homomorphism ∂_{; ; ; k}; ; ; ^{; ; ; ∗}; ; ; :π_{; ; ; k}; ; ; ^{; ; ; ∗}; ; ; (X, X₀, x₀)→π_{; ; ; k-1}; ; ; ^{; ; ; ∗}; ; ; (X₀, x₀)=π_{; ; ; k-1}; ; ; ^{; ; ; ∗}; ; ; (X₀, {; ; ; x₀}; ; ; , x₀) is introduced which induces a natural transformation. The corresponding sequence of the coarse shape groups is exact, although the shape sequence generally failed to be exact. This exactness makes powerful tool for computing coarse shape groups of some particular pointed pairs of metric compacta.