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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_
Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.
The $p$-local homotopy types of gauge groups of principal $G$-bundles over $S^4$ are classified when $G$ is a compact connected exceptional Lie group without $p$-torsion in homology except for $(G,p)=(mathrm{E}_7,5)$.
Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $kinmathbb{Z}$. It is known that the homotopy type of $mathcal{G}_k(M)$
Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles ov