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On small homotopies of loops

Authors:	G Conner M Meilstrup D Repov&#x; A Zastrow M eljko

Institution:	^aDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA ^bInstitute for Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, Ljubljana 1001, Slovenia ^cInstitute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, Gdańsk 80-952, Poland

Abstract:	Two natural questions are answered in the negative: ? “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?” ? “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 π₁-shape injective. Keywords: Peano continuum; Path space; Shape injective; Homotopically Hausdorff; 1-ULC

Keywords:	Peano continuum Path space Shape injective Homotopically Hausdorff 1-ULC
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