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

Abstract

By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\underleftarrow{\lim }(Y_i^{(n)},y_i^{\ast })$, where $(Y_i^{(n)},y_i^{\ast })={\bigvee }_{j\le i}(X_j^{(n)},x_j^{\ast })$ 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 ${\beta }_n:{\pi }_n(X,\ast )\to \underleftarrow{\lim }{\pi }_n(Y_i^{(n)},y_i^{\ast })$ is bijection. Second, using this fact we prove that the topological $n$-homotopy group of an $n$-Hawaiian like space, ${\pi }_n^{top}(X,x^{\ast })$, is a topological group for all $n\ge 2$ which is a partial answer to the open question whether ${\pi }_n^{top}(X,x^{\ast })$ is a topological group for any space $X$ and $n\ge 1$. Moreover, we show that ${\pi }_n^{top}(X,x^{\ast })$ is metrizable.

arXiv comment8 pages

References in corpus (1)

• The Hawaiian earring group and metrizability

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