Analogues of eta invariants for even dimensional manifolds

The eta invariant is a secondary geometric invariant, introduced by Atiyah, Patodi and Singer about forty years ago. Ever since, it has been the object of extensive research activity and has found applications in several areas of mathematics and physics. The two results among the rich literature that are most relevant to this thesis are the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles and the higher Atiyah-Patodi-Singer index theorem of Getzler and Wu. The former concerns a variational formula for eta invariants on odd dimensional manifolds and its topological interpretation; the latter is a generalization of the Atiyah-Patodi-Singer L²-index theorem for even dimensional manifolds with boundary in the context of cyclic cohomology and its pairing with K-theory.

In this thesis, we shall prove an analogue for each of these two theorems for the case of manifolds of the opposite dimensional parity. Accordingly, the thesis will consist of two parts.

In the first part, we prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles over odd dimensional closed manifolds. The corresponding eta invariant is shown to coincide with the one introduced by Dai and Zhang. The fact that this eta invariant is an appropriate analogue of the usual eta invariant for the odd dimensional case is made clear in the second part of the thesis. We conclude the first part by establishing an explicit relative pairing index formula for odd dimensional manifolds with boundary, which is the analogue of the pairing formula by Lesch, Moscovici and Pflaum. In the second part we prove an analogue for odd dimensional manifolds with boundary of the higher Atiyah-Patodi-Singer index theorem of Getzler and Wu, where the eta invariant is obtained via the Connes pairing between cyclic cohomology and cyclic homology. The eta invariant defined this way is a natural analogue of that for the odd dimensional case. By comparing our formula with the Toeplitz index formula of Dai-Zhang, we show that, up to an integer, this eta invariant is equivalent to the one defined in the first part of the thesis.