On the twisted -homology of simple Lie groups

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 .