Adjointness of Suspension and Shape Path Functors

Abstract

In this paper, we introduce a subcategory ${\widetilde{Sh}}_{\ast }$ of Sh${}_{\ast }$ and obtain some results in this subcategory. First we show that there is a natural bijection $Sh(\Sigma (X,x),(Y,y))\cong Sh((X,x),Sh((I,\dot{I}),(Y,y)))$, for every $(Y,y)\in {\widetilde{Sh}}_{\ast }$ and $(X,x)\in S{h}_{\ast }$. By this fact, we prove that for any pointed topological space $(X,x)$ in ${\widetilde{Sh}}_{\ast }$, ${\check{\pi }}_n^{top}(X,x)\cong {\check{\pi }}_{n-k}^{top}(Sh((S^k,\ast ),(X,x)),e_x)$, for all $1\le k\le n-1$.

11 pages. arXiv admin note: text overlap with arXiv:1401.2557

References in corpus (2)

• The fundamental group as a topological group
• The topological fundamental group and free topological groups