论文标题
关于拓扑同型组和与夏威夷群体的关系
On Topological Homotopy Groups and Relation to Hawaiian Groups
论文作者
论文摘要
通过将晶须拓扑概括为$ n $ th同型尖式空间$（x，x_0）$，由$π_n^{wh}（x，x_0）$表示，我们表明$π_n^{wh}（x，x_0）$是一个拓扑组，如果$ n \ ge ge 2 $。另外，我们为$π_n^{wh}（x，x_0）$呈现一些必要和充分的条件，即离散，Hausdorff和Indiscrete。然后，我们证明$ l_n（x，x_0）$夏威夷组$ \ MATHCAL {h} _n（x，x_0）$的自然表达图像等于$π_n^{WH}（x，x_0）$中的所有收敛序列的所有类别的融合序列。结果，我们表明$ l_n（x，x_0）\ cong l_n（y，y_0）$如果$π_n^{wh}（x，x，x_0）\congπ_n^{wh}（wh}（y，y_0）$，但总体上不存在相反的条件。另外，我们表明，在某些类别的空间中，例如半$ n $ n $ simply连接的空间和$ n $ hawaiian，例如空间，晶须拓扑以及由$ n $ loop太空的紧凑型拓扑造成的拓扑结构。最后，我们表明$ n $ -slt路径可以转移$π_n^{wh} $，因此$ l_n $沿其点上异构化。
By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $π_n^{wh}(X, x_0)$, we show that $π_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $π_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $π_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $π_n^{wh}(X, x_0) \cong π_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $π_n^{wh}$ and hence $L_n$ isomorphically along its points.
