Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary

TL;DR

This paper extends the theory of higher Atiyah-Patodi-Singer indices to manifolds with boundary acted upon by certain Lie groups, defining higher genera and establishing index formulas under specific geometric and algebraic conditions.

Contribution

It introduces higher Atiyah-Patodi-Singer indices for manifolds with boundary under new assumptions on Lie groups and derives a higher index formula, generalizing previous results to boundary cases.

Findings

Defined higher C^*-indices for manifolds with boundary.

Established a higher index formula relating indices to group cocycles.

Introduced higher genera for manifolds with boundary.

Abstract

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.