Mapping spaces from projective spaces

Abstract

We denote the $n$-th projective space of a topological monoid $G$ by $B_{n}G$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid of the homotopy type of a CW complex and $G^{\prime }$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space ${\mathrm{Map}}_0(B_{n}G,BG)$ and the space of all $A_n$-maps from $G$ to $G^{\prime }$. This fact has several applications. As the first application, we show that the connecting map $G\to {\mathrm{Map}}_0(B_{n}G,BG)$ of the evaluation fiber sequence ${\mathrm{Map}}_0(B_{n}G,BG)\to \mathrm{Map}(B_{n}G,BG)\to BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups, $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon and homotopy pullback of $A_n$-maps. 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.

26 pages, 3 figures; the appendix in v3 is deleted since its argument was incomplete

Cited by in corpus (2)

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