Eta cocycles

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the classic Atiyah-Patodi-Singer index formula for $(X,\F)$. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairing of $K$-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form $0\to J \to A \to B \to 0$ with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data.

💡 Research Summary

The paper establishes a Godbillon‑Vey index formula for longitudinal Dirac operators on a foliated bundle ((X,\mathcal F)) with boundary, thereby introducing a secondary invariant—the Godbillon‑Vey eta invariant—associated to the boundary foliation. The authors start by recalling that for a codimension‑(q) foliation (\mathcal F) the Godbillon‑Vey class (\operatorname{GV}(\mathcal F)\in H^{3}(X;\mathbb R)) measures the non‑triviality of the transverse geometry. While the classical Atiyah‑Patodi‑Singer (APS) index theorem expresses the index of a Dirac operator on a manifold with boundary as a bulk term plus a boundary eta correction, this framework breaks down for type III foliations because the usual trace on the foliation C(^*)-algebra does not exist.

To overcome this obstacle the authors work inside the foliation C(^)-algebra (\mathcal A=C^{}(X,\mathcal F)) and select a dense, holomorphically closed ideal (J\subset\mathcal A) that captures the longitudinal pseudodifferential calculus. The quotient (B=\mathcal A/J) depends only on the boundary data. This yields an exact sequence of Banach algebras \