Jan McGarry Furriol: Fibrations with fibre S^n times S^n

Abstract:

We study characteristic classes of oriented fibrations with fibre a product \(S^{n} \times S^{n}\) of odd-dimensional spheres. We provide a collection of such fibrations \(\Theta_{\gamma}\), indexed by cyclic subgroups of \(\mathrm{SL}_{2}(\mathbb{Z})\), which detects any given non-zero characteristic class. We prove that certain such classes \(\mathcal{E}_{2k}\) are non-zero as characteristic classes for oriented smooth \(S^{3} \times S^{3}\)-bundles.