Mapping spaces from projective spaces

We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid having the homotopy type of a CW complex and $G'$ a well-pointed grouplike topological monoid. We prove that there is a natural weak equivalence between the pointed mapping space $\mathrm{Map}_0(B_nG,BG')$ and the space $\mathcal{A}_n(G,G')$ of all $A_n$-maps from $G$ to $G'$. Moreover, if we suppose $G=G'$, then an appropriate union of path-components of $\mathrm{Map}_0(B_nG,BG)$ is delooped.

This fact has several applications. As the first application, we show that the evaluation fiber sequence $\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(B_nG,BG)\rightarrow BG$ extends to the right. As other applications, we investigate higher homotopy commutativity, $A_n$-types of gauge groups, $T_k^f$-spaces and homotopy pullback of $A_n$-maps. The concepts of $T_k^f$-space and $C_k^f$-space were introduced by Iwase–Mimura–Oda–Yoon, which is a generalization of $T_k$-spaces by Aguadé. In particular, we show that the $T_k^f$-space and the $C_k^f$-space are exactly the same concept and give some new examples of $T_k^f$-spaces.

Keywords

mapping space, homotopy fiber sequence, $A_n$-space, higher homotopy commutativity, gauge group

2010 Mathematics Subject Classification

Primary 54C35. Secondary 18D50, 55P45.

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Published 31 May 2016