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.

我们将小n-夏威夷环定义为小指向地图的特例，并研究由小n-夏威夷环类\（{\ mathcal {H}} _ n ^ s（X，x_0）\）组成的组。作为具有非平凡的1-Hawaiian小环的空间的示例，我们将第一组夏威夷群岛群岛作为一维夏威夷耳环的一族的特殊商群。当且仅当\（{\ mathcal {H}} _ n ^ s（X，x）\）不重要时，我们才在第n个夏威夷组上定义晶须拓扑。我们还证明，在度量空间上，当且仅当自然同态映射\（{\ mathcal {H}} _ n（X，x_0）\）时，零同伦可以变得足够小。\（L_n（X，x_0）\）仅且仅当\（{\ mathcal {H}} _ n ^ s（X，x_0）\）不重要。因此，在没有非平凡的小n-夏威夷回路的空间中，n -SLT路径沿点同构地传递\（{\ mathcal {H}} _ n \）。我们显示谐波群岛是两个不同点上的非平凡的1-Hawaiian小环和非同构的第一夏威夷群的示例。然后，我们定义n -SHLT条路径，该路径沿点同构地转移第n个夏威夷族。