Mapping spaces from projective spaces

TL;DR

This paper establishes a weak equivalence between certain mapping spaces and $A_n$-maps, with applications to homotopy theory, gauge groups, and $T_k^f$-spaces, providing new insights and examples in algebraic topology.

Contribution

It proves a fundamental weak equivalence linking mapping spaces and $A_n$-maps, and explores its implications for homotopy commutativity, gauge groups, and $T_k^f$-spaces.

Findings

Weak equivalence between $ ext{Map}_0(B_nG,BG)$ and $A_n$-maps.

Delooping of the connecting map in the evaluation fiber sequence.

Equivalence of $T_k^f$-spaces and $C_k^f$-spaces, with new examples.

Abstract

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 of the homotopy type of a CW complex and $G'$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the connecting map $G\rightarrow\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(B_nG,BG)\rightarrow BG$ 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.