# A geometric approach to K-homology for Lie manifolds

## TL;DR

This paper introduces a geometric framework for computing the Fredholm index of elliptic operators on Lie manifolds by reducing the problem to Dirac operators and developing a specialized geometric K-homology theory.

## Contribution

It adapts Baum-Douglas geometric K-homology to Lie manifolds and defines a comparison map to relative K-theory, enabling index computations via geometric cycles.

## Key findings

- Reduction of index computation to Dirac operators
- Development of a geometric K-homology variant for Lie manifolds
- Establishment of a comparison map to relative K-theory

## Abstract

We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our framework ideas coming from Baum-Douglas geometric $K$-homology and in particular we introduce a notion of geometric cycles, that can be categorized as a variant of the famous geometric $K$-homology groups, for the specific situation here. We also define a comparison map between this geometric $K$-homology theory and a relative $K$-theory group, directly associated to a fully elliptic pseudodifferential operator.