Leibniz Homology, Characteristic Classes and K-theory

Abstract

In this paper we identify many striking elements in Leibniz (co)homology which arise from characteristic classes and K-theory. For a group G a field k of characteristic zero, it is shown that all primary characteristic classes, i.e. H^*(BG; k), naturally inject into certain Leibniz cohomology groups via an explicit chain map. Moreover, if f: A \to B is a homomorphism of algebras or rigns, the relative Leibniz homology groups HL_*(f) are defined, and if in addition f is surjective with nilpotent kernel, A and B algebras over the rationals, then there is a natural surjection HL_{*+1}(gl(f)) \to HC_*(f), where HC_*(f) denotes relative cyclic homology, and gl(f): gl(A) \to gl(B) is the induced map on matrices. Here again, the surjection is realized via an explicit chain map, and offers a relation between Leibniz homology and K-theory.