On the topological contents of eta invariants

Abstract

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $\rho$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.

85 pages, 5 figures Revisions: Definition 4.2 of a geometrisation is corrected since the cohomological character may not be unique (former Lemma 4.2 is not correct). A Section 6. on the K-theory of $K(\pi ,n)$-bundles was added. Proposition 6.1 replaces the former Proposition 5.15 whose proof is not correct. The construction of a geometrisation of a string manifold is revised correspondingly

References in corpus (3)

• On Ramond-Ramond fields and K-theory
• Computation of the homotopy of the spectrum tmf
• Uniqueness of smooth extensions of generalized cohomology theories