On topological shape homotopy groups

Abstract

In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces which make them Hausdorff topological groups. We then exhibit an example in whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group induced by the kth shape loop space and show that it coincides with the above topology.

Topological coarse shape groups of compact metric spaces

“…Also, it would be interesting if one can somehow relate a topological coarse shape group of a subspace to the topological coarse shape group of the whole space. We will give a complete answer when a subspace is a retract, which will be an analogy of the "shape" case, already considered in [5], with a slight improvement. Definition 4.3.…”

“…The topological shape groupπ top k (X, x 0 ), constructed in [5], has for elements of its topological basis the sets…”

“…Recently, certain topology has been constructed on shape groups which turns them into topological groups (naturally called topological shape homotopy groups, see [5]) and it has been shown that they are new shape invariants, with ability to differ shape of some pairs of topological spaces which regular shape groups cannot differ. It is important to emphasize that topology shape (homotopy) groups of compact metric spaces are metrizable.…”

“…We will prove that, in the both cases, topological groups are obtained, which will be denoted byπ d k (X, x 0 ) andπ * d * k (X, x 0 ). In the case of shape groups, these topological groups are isomorphic to topological shape groups ( [5]) (according to this, no new name will be needed).…”

“…Let X be the Hawaiian earring. Then, by Example 4.5. of [5] we have thatπ d 1 (X) is not discrete. Therefore, by using Proposition 4.2 we infer thatπ * d * 1 (X) is not discrete (we are not specifying a base point since X is connected compact metric space).…”