On the Twisted K-Homology of Simple Lie Groups

TL;DR

This paper characterizes the twisted K-homology of simply connected simple Lie groups, revealing its algebraic structure and linking it to representation theory and bordism groups.

Contribution

It provides a detailed algebraic description of the twisted K-homology for these Lie groups and relates it to representation dimensions and bordism relations.

Findings

Twisted K-homology forms an exterior algebra tensor a cyclic group.

The order of the cyclic group is described via irreducible representation dimensions.

Congruences for cyclic order lift to relations in twisted Spin-c bordism.

Abstract

We prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n-1 generators tensor a cyclic group. We give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted Spin-c bordism group of G.