Secondary Analytic Indices

We define real-valued characteristic classes of flat complex vector bundles, and flat real vector bundles with a duality structure. We construct pushforwards of such vector bundles with vanishing characteristic classes. These pushforwards involve the analytic torsion form in the first case and the eta form of the signature operator in the second case. We show that the pushforwards are independent of the geometric choices made in the constructions and hence are topological in nature. We give evidence that in the first case, the pushforwards are given topologically by the Becker-Gottlieb-Dold transfer.