Leibniz Homology, Characteristic Classes and K-theory, by Jerry M. Lodder

In this paper we identify many striking elements in Leibniz (co)homology which arise from characteristic classes and K-theory. For a group G and 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 --> B is a homomorphism of algebras or rings, 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)) --> HC*(f), where HC*(f) denotes relative cyclic homology, and gl(f): gl(A) --> gl(B) is the induced map on matrices. Here again, the above surjection is realized via an explicit chain map, and offers a relation between Leibniz homology and K-theory.

Jerry M. Lodder <jlodder@nmsu.edu>