On gauge groups over high dimensional manifolds and self-equivalences of $H$-spaces

Abstract

Let $Y$ be a pointed space and let $\mathcal{E}(Y^r)$ be the group of based self-equivalences of $Y^r$, $r\ge 2$. For $Y$ a homotopy commutative $H$-group we construct a subgroup ${\mathcal{E}}_{\mathrm{Mat}}(Y^r)$ of $\mathcal{E}(Y^r)$ which has a group structure isomorphic to either $G{L}_r(\mathbb{Z})$, or $G{L}_r({\mathbb{Z}}_d)$, $d\ge 2$. We classify principal bundles over connected sums of $q$-sphere bundles over $n$-spheres and use the group ${\mathcal{E}}_{\mathrm{Mat}}(Y^r)$ to obtain homotopy decompositions of their gauge groups. Using these decompositions we give an integral classification, up to homotopy, of the gauge groups of principal $SU(2)$-bundles over certain 2-connected 7-manifolds with torsion-free homology.

30 pages