4-manifolds \(X\) over \(BS\text{U}(2)\) and the corresponding homotopy types Map\((X,BS\text{U}(2))\)
Let \(X\) be an oriented simply connected closed 4-manifold, \(P_{k,X}\) the principal \(\text{SU}(2)\) bundle over \(X\) with \(c_2(P_{k,X})= k\in\mathbb{Z}\), and \(\text{Map}_k(X, \text{BSU}(2))\) the connected component of \(\text{Map}(X, \text{BSU}(2))\) including the map inducing \(P_{k,X}\) where \(\text{Map}(X, \text{BSU}(2))\) is the space of continuous maps from \(X\) to \(\text{BSU}(2)\). The authors search for a condition for \(\text{Map}_k(X, \text{BSU}(2))\simeq \text{Map}_l(X, \text{BSU}(2))\). In this paper the case of \(X\) with positive second Betti number is considered. The following theorem is obtained: Theorem. \(\text{Map}_k(X, \text{BSU}(2))\) is homotopy equivalent to \(\text{Map}_l(X, \text{BSU}(2))\) if and only if \(|k|=|l|\) if \(X\) admits an orientation reserving homotopy equivalence, \(k=l\) otherwise. Let \({\mathcal G}_{k,X}\) be the gauge group of \(P_{k,X}\) and \(B{\mathcal G}_{k,X}\) the classifying space of \({\mathcal G}_{k,X}\). As a corollary to the Theorem is obtained Corollary. \(B{\mathcal G}_{k,X}\simeq B{\mathcal G}_{l,X}\) if and only if \(|k|=|l|\) if \(X\) admits an orientation reserving homotopy equivalence, \(k=l\) otherwise. These results extend some results of the second author from the case \(X= S^4\) and about the relation \({\mathcal G}_{k,X}\cong{\mathcal G}_{l,X}\) [Topology Appl. 87, No. 3, 173-187 (1998; Zbl 0933.58005); Homotopy type of \(\text{Map}(S^4, \text{BSU}(2))\) and classifying spaces of gauge groups, preprint].
scientific article; zbMATH DE number 1523640
scientific article