### Leibniz Cohomology for Differentiable Manifolds, by Jerry Lodder

The goal of this paper is to extend Loday's Leibniz cohomology
from a Lie algebra invariant to an invariant for differentiable
manifolds so that Leibniz cohomology is a non-commutative version of
de Rham cohomology. The non-commutativity arises by considering a
cochain complex of tensors (from differential geometry)
which are not necessarily skew-symmetric.
Leibniz cohomology, HL^*, is no longer a homotopy invariant---in
fact the first obstruction to the homotopy invariance of HL^*(R^n) is
the universal Godbillon-Vey invariant in dimension 2n+1. The main
calculational result of the paper is then a calculation of
HL^*(R^n) in terms of (i) certain universal invariants of
foliations, and (ii) Loday's product structure on HL^*. Unlike Lie
algebra or Gelfand-Fuks cohomology, the Leibniz cohomology of vector
fields on R^n contains infinite families of elements which support
non-trivial products.

Jerry Lodder <jlodder@nmsu.edu>