On small homotopies of loops

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.

arXiv comment12 pages, 5 figures

References in corpus (1)

• The fundamental groups of subsets of closed surfaces inject into their first shape groups

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