The homotopy types of \(U(n)\)-gauge groups over \(S^4\) and \(\mathbb{C}P^2\) (Q1707275)

Let \(G\) be a compact connected Lie group. For a principal \(G\)-bundle \(\xi_k\) over \(S^4\) with the second Chern class \(k\in\mathbb Z=H^4(S^4,\mathbb Z)\), let \(\mathcal G_k(G)\) denote the gauge group of \(\xi_k\). Similarly, for a principal \(G\)-bundle \(\xi_{(k,l)}\) over \(\mathbb CP^2\) with the first and second Chern classes \((k,l)\in \mathbb Z^2=H^2(\mathbb CP^2,\mathbb Z)\oplus H^4(\mathbb CP^2,\mathbb Z)\), let \(\mathcal G^{(k,l)}(G)\) (resp. \(\mathcal G^{(k,l)}_*(G)\)) be the gauge group (resp. the based gauge group) of \(\xi_{(k,l)}\). Let \(\mathcal G^l(\mathrm{SU}(n))\) denote the gauge group of the principal \(\mathrm{SU}(n)\)-bundle over \(\mathbb CP^2\) with the second Chern class \(l\in\mathbb Z=H^4(\mathbb CP^2,\mathbb Z)\). In this paper, the author studies the homotopy types of the gauge groups \(\mathcal G_k(G)\), \(\mathcal G^{(k,l)}(G)\) and \(\mathcal G_*^{(k,l)}(G)\). Firstly he considers the group \(\mathcal G_k(G)\). He shows that there is an isomorphism of principal \(\mathcal G(\mathrm{SU}(n))\)-bundles over \(S^1\), \(\mathcal G_k(U(n))\cong\mathcal G_k(\mathrm{SU}(n))\times S^1\) for \(n\geq 3\) and that there is an isomorphism of principal \(\mathcal G_{2l}(\mathrm{SU}(2))\)-bundles over \(S^1\), \(\mathcal G_{2l}(U(2))\cong\mathcal G_{2l}(\mathrm{SU}(2))\times S^1\) for \((n,k)=(2,2l)\). Moreover, he proves that there is an isomorphism of principal \(\mathcal G_{2l+1}(\mathrm{PU}(2))\)-bundles over \(\mathcal G_{2l+1}(\mathrm{PU}(2))\), \(\mathcal G_{2l+1}(U(2))\cong\mathcal G_{2l+1}(\mathrm{PU}(2))\times S^1\) for \((n,k)=(2,2l1)\). As an application, by using results due to Kono and Theriault etc., he also obtains that there is a homotopy equivalence \(\mathcal G_k(U(2))\simeq\mathcal{G}_l(U(2))\) iff \((12,k)=(12,l)\) and that there is a homotopy equivalence \(\mathcal G_k(U(3))\simeq\mathcal G_l(U(3))\) iff \((24,k)=(24,l)\). Furthermore, he also obtains that there is a \(p\)-local homotopy equivalence \(\mathcal G_k(U(5))\simeq_p\mathcal G_l(U(5))\) iff \((k,120)=(l,120)\) for any prime \(p\) or \(p=0\) (the rational case). Secondly, he considers the groups \(\mathcal G^{(k,l)}(G)\) and \(\mathcal G_*^{(k,l)}(G)\). He proves that there is a homotopy equivalence \[ \mathcal G^{(k,l)}(U(2))\simeq\begin{cases} \mathcal G^{(0,l^{\prime})} & k\equiv 0\text{ mod }2 \\ \mathcal G^{(1,l^{\prime})} & k\equiv 1\text{ mod }2\end{cases} \] for a suitable integer \(l^{\prime}\), and that there is a homotopy equivalence \(\mathcal G^{(k,l)}(U(2))\simeq\mathcal G^{(k,l+12)}(U(2))\). Moreover, he also proves that there is an isomorphism of principal \(\mathcal G^l(\mathrm{SU}(n))\)-bundles over \(S^1\), \(\mathcal G^{(0,l)}(U(n))\cong S^1\times\mathcal G^l(\mathrm{SU}(n))\), and that there is a homotopy equivalence \(\mathcal G^{(1,l)}(U(2))\simeq S^1\times\mathcal G^{4l-1}(\mathrm{SU}(2))\) when localized away from \(2\). Furthermore, he proves that the two spaces \(\mathcal G^{(0,l)}(U(2))\) and \(\mathcal G^{(1,l^{\prime})}(U(2))\) (resp. \(\mathcal G^{(0,l)}_*(U(2))\) and \(\mathcal G^{(1,l^{\prime})}_*(U(2))\)) have different homotopy types for any integers \(l\) and \(l^{\prime}\). His proof is based on the standard homotopy calculations and the homotopy equivalence obtained by \textit{D. H. Gottlieb} [Ann. Math. (2) 87, 42--55 (1968; Zbl 0173.25901)].