Zainab Hamed | Searcharxiv

arXiv2011

Topological Homotopy Groups

D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups.

math.AT↗

arXiv2010

On Topological Homotopy Groups of $n$-Hawaiian like spaces

By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\displaystyle{\varprojlim (Y_i^{(n)},y_i^*)}$, where $(Y_i^{(n)},y_i^*)=\bigvee_{j\leq i}(X_j^{(n)},x_j^*)$ is the wedge of $X_j^{(n)}$'s in which $X_j^{(n)}$'s are $(n-1)$-connected, locally $(n-1)$-connected, $n$-semilocally simply connected and compact CW spaces. In this paper, first we show that the natural homomorphism $\displaystyle{β_n:π_n(X,*)\rightarrow \varprojlim π_n(Y_i^{(n)},y_i^*)}$ is bijection. Second, using this fact we prove that the topological $n$-homotopy group of an $n$-Hawaiian like space, $π_n^{top}(X,x^*)$, is a topological group for all $n\geq 2$ which is a partial answer to the open question whether $π_n^{top}(X,x^*)$ is a topological group for any space $X$ and $n\geq 1$. Moreover, we show that $π_n^{top}(X,x^*)$ is metrizable.

math.AT↗