Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is proved by
Crabb--Sutherland and the second author that the number of $A_n$-types of the gauge groups of $P$ is finite if $n<infty$ and $K$ is a finite complex. We show that the number of $A_infty$-types of the gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely many $P$.
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 p
rove 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 $Grightarrowmathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $mathrm{Map}_0(B_nG,BG)rightarrowmathrm{Map}(B_nG,BG)rightarrow BG$ 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.
A Lie group is called $p$-regular if it has the $p$-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into $p$-regular $mathrm{SO}(2n)_{(p)}$ is determined, which completes t
he list of (non)triviality of such Samelson products in $p$-regular simple Lie groups. As an application, we determine the homotopy normality of the inclusion $mathrm{SO}(2n-1)tomathrm{SO}(2n)$ in the sense of James at any prime $p$.
We construct the homotopy pullback of $A_n$-spaces and show some universal property of it. As the first application, we review the Zabrodskys result which states that for each prime $p$, there is a finite CW complex which admits an $A_{p-1}$-form but
no $A_p$-form. As the second application, we investigate $A_n$-types of gauge groups. In particular, we give a new result on $A_n$-types of the gauge groups of principal $mathrm{SU}(2)$-bundles over $S^4$, which is a complete classification when they are localized away from 2.
Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show that the number of gauge groups of principal $G$-bundles over $B$ is finite up to $A_n$-equivalence for $n<infty$. As an example, we give a lower bound of the number of $A_
n$-equivalence types of gauge groups of principal $SU(2)$-bundles over $S^4$.
Connected components of $Map(S^4,BSU(2))$ are the classifying spaces of gauge groups of principal $SU(2)$-bundles over $S^4$. Tsukuda [Tsu01] has investigated the homotopy types of connected components of $Map(S^4,BSU(2))$. But unfortunately, the pro
of of Lemma 2.4 in [Tsu01] is not correct for $p=2$. In this paper, we give a complete proof. Moreover, we investigate the further divisibility of $epsilon_i$ defined in [Tsu01]. In [Tsu], it is shown that divisibility of $epsilon_i$ have some information about $A_i$-equivalence types of the gauge groups.
