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Register a new userFiniteness of $A_n$-equivalence types of gauge groups
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Mitsunobu Tsutaya
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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 $S^4$.
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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 $G$ 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 $P$.
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 $mathcal{G}_k(M)$ is determined by the homotopy type of $mathcal{G}_k(mathbb{CP}^2)$. In this paper we investigate properties of $mathcal{G}_k(mathbb{CP}^2)$ when $G = SU(n)$ that partly classify the homotopy types of the gauge groups.
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