On small homotopies of loops

TL;DR

This paper investigates properties of small homotopies of loops in topological spaces, addressing questions about limits of null-homotopic loops and the effects of adding arcs on essential curves, with implications for homotopically Hausdorff spaces.

Contribution

It provides negative answers to two natural questions about small homotopies, clarifying the relationship between homotopically Hausdorff spaces and $$-shape injectivity.

Findings

Limits of null-homotopic loops may not be null-homotopic.

Adding arcs can turn essential curves into null-homotopic.

Clarifies the link between homotopically Hausdorff spaces and shape injectivity.

Abstract

Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\pi_1$-shape injective.