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

Let $X$ be a connected $CW$-complex and $G$ be a Lie group. The isomorphism classes of principal $G$-bundles over $X$ are in one-to-one correspondence with the set $[X,BG]$ of unpointed homotopy classes of maps from $X$ to the classifying space of $G$. Given a principal $G$-bundle $P$ over $X$, the gauge group of $P$, denoted $G^P\left(X\right)$, is the group of its bundle automorphisms covering the identity on $X$. The pointed gauge group $G_{\ast }^P\left(X\right)$ consists of all bundle automorphisms that pointwise fix the fibre at the base point. Endowed with the compact-open topology, $G^P\left(X\right)$ (resp. $G_{\ast }^P\left(X\right))$ is homotopy equivalent to the loop space of the connected component of the mapping space $Map(X,BG)$ (resp. ${Map}_{\ast }(X,BG))$ which contains the map that classifies the bundle [AB]. Although the set of isomorphism classes of principal $G$-bundles over a finite $CW$-complex $X$ might be infinite, there exist only finitely many distinct homotopy types among the gauge groups [CS].

The topology of gauge groups and their associated classifying spaces has received considerable attention due to their connections to mathematical physics and other areas in mathematics. Particular attention has been paid in counting the number of homotopy types of gauge groups over surfaces and 4-manifolds. Although new ideas coming from differential geometry and mathematical physics suggest the possibility of extending gauge theories to high dimensions [DT], the homotopy theory of gauge groups over high dimensional manifolds has been barely explored. Research done in this direction includes the study of gauge groups over high dimensional spheres [HKK, KKT, Spf], $S^3$-bundles over $S^4$ [IMS] and $(n-1)$-connected $2n$-manifolds [RH1].

The study of the group of self-homotopy equivalences $E\left(Y\right)$ of a topological space $Y$ has a long tradition in homotopy theory (see for instance [Ark],[Rut2]). Very little is known, however, about this group when $Y$ is a product or a wedge sum of spaces. Moreover, the applications of the group of self-equivalences require further investigations. In this work we present an application of the group $E\left(Y\right)$ to the homotopy classification of mapping spaces.

It is well known that there are homeomorphisms $Map({⨆}_{i=1}^r{X}_i,Z)\cong {\prod }_{i=1}^{r}Map(X_i,Z)$ and ${Map}_{\ast }({\bigvee }_{i=1}^r{X}_i,Z)\cong {\prod }_{i=1}^r{Map}_{\ast }(X_i,Z)$. Similar decompositions do not hold in general for $Map({\bigvee }_{i=1}^r{X}_i,Z)$. It has been shown that the loops on these kinds of mapping spaces appear in homotopy decompositions of gauge groups of principal $G$-bundles over 7-dimensional manifolds [IAM2]. For suitable spaces $X_i$ and $Z$, $Y={Map}_{\ast }(X_i,Z)$ is a homotopy commutative $H$-group and so is ${\prod }_{i=1}^r{Map}_{\ast }(X_i,Z)$. We construct a subgroup of the group of self-equivalences of the $r$-fold cartesian product of certain homotopy commutative $H$-groups $Y$. We show that this construction can be used to decompose the loop spaces of the connected components of $Map({\bigvee }_{i=1}^r{X}_i,Z)$. As a result, we provide full decompositions of the gauge groups associated to $q$-sphere bundles over $n$-spheres. Finally, we use these decompositions to give a complete classification, up to homotopy, of $SU\left(2\right)$-gauge groups over connected sums of certain $S^3$-bundles over $S^4$ with cross sections.

Let ${Mat}_r\left(Z\right)$ be the set of $r$-by-$r$ matrices with integer coefficients, and let $G{L}_r\left(Z\right)$ be the subset of invertible matrices. Given an $H$-group $Y$, and a matrix $A=\left(a_{ij}\right)\in {Mat}_r\left(Z\right)$, we define a self-map, called a matrix map, on the $r$-fold cartesian product of $Y$ as the composite