On Small n-Hawaiian Loops

Abstract

We define small n-Hawaiian loop as a special case of small pointed map, and study the group consisting of classes of small n-Hawaiian loops, $${\mathcal {H}}_n^s(X, x_0)$$ . As an example of spaces with non-trivial small 1-Hawaiian loops, we present 1st Hawaiian group of harmonic archipelago space as a special quotient group of 1st Hawaiian group of 1-dimensional Hawaiian earring. We define whisker topology on nth Hawaiian group which is Hausdorff if and only if $${\mathcal {H}}_n^s(X, x)$$ is trivial. We also prove that on metric spaces, null-homotopies can become small enough if and only if natural homomorphism maps $${\mathcal {H}}_n(X, x_0)$$ isomorphically on $$L_n(X, x_0)$$ if and only if $${\mathcal {H}}_n^s(X, x_0)$$ is trivial. Thus in spaces without non-trivial small n-Hawaiian loop, n-SLT paths transfer $${\mathcal {H}}_n$$ isomorphically along the points. We show that harmonic archipelago is an example with non-trivial small 1-Hawaiian loop and non-isomorphic 1st Hawaiian groups at two different points. Then we define n-SHLT paths which transfers nth Hawaiian group isomorphically along the points.