The homotopy types of \(SU(3)\)-gauge groups over simply connected 4-manifolds
If \(G\) is a compact connected Lie group, \(M\) a connected finite \(CW\)-complex, and \(P\rightarrow M\) is a principal \(G\)-bundle, then the gauge group of this bundle is the group of \(G\)-equivariant self-maps of \(P\) covering the identity map on \(M\). In the paper under review, \(M\) is supposed to be a simply-connected closed \(4\)-dimensional manifold and \(G\) is a simple, simply-connected compact Lie group. Then it is known [\textit{C. H. Taubes}, J. Differ. Geom. 17, 139--170 (1982; Zbl 0484.53026)] that the isomorphism classes of principal \(G\)-bundles over \(M\) are in one-to-one correspondence with the set \([M,BG]=\mathbb Z\) of the homotopy classes of maps from \(M\) into the classifying space \(BG\) of \(G\). If \(P \rightarrow M\) is a principal bundle corresponding to \(k\in \mathbb Z\), then \(\mathcal G_ k(M,G)\) denotes the gauge group of this bundle. In its main result, the paper under review specializes to \(G = SU(3)\) and studies the number of homotopy types of gauge groups of principal \(SU(3)\)-bundles over \(M\). Given two integers \(a\) and \(b\), let \((a,b)\) be the greatest common divisor of \(| a|\) and \(| b|\). The main result is the following theorem: (a) If (in addition to the above mentioned properties) \(M\) is a spin manifold, then there is an integral homotopy equivalence \(\mathcal G_ k(M,G) \simeq \mathcal G_ {k^\prime}(M,G)\) if and only if \((24,k)=(24,k^\prime)\); (b) If \(M\) is a non-spin manifold, then an integral homotopy equivalence \(\mathcal G_ k(M,G) \simeq \mathcal G_ {k^\prime}(M,G)\) implies that \((12,k)=(12,k^\prime)\), while \((12,k)=(12,k^\prime)\) implies that there is a homotopy equivalence \(\mathcal G_ k(M,G) \simeq \mathcal G_ {k^\prime}(M,G)\) after localizing rationally or at any prime. The proof of the main result combines general methods and results with information specific to \(SU(3)\) and \(4\)-dimensional manifolds.
scientific article; zbMATH DE number 6098973
scientific article