ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53, 55, 18 Lodder, Jerry, New Mexico State University Leibniz Cohomology for Differentiable Manifolds In this talk we extend Loday’s Leibniz cohomology, HL^{∗}, from a Lie algebra invariant to a new invariant for differentiable manifolds. The extension is achieved via a cochain complex of tensors (from differential geometry), and as such generalizes the de Rham complex of differential forms. To be sure, HL^{∗} becomes a non-commutative version of Gelfand-Fuks cohomology. These new cohomology groups are diffeomorphism invariants, but fail to be homotopy invariants. The first obstruction to the homotopy invariance of HL^{∗} is the universal Godbillon-Vey invariant in dimension 2n + 1 for the Leibniz cohomology of Euclidean n-space. We present the calculation for HL^{∗}(R^{n}) in terms of (i) certain invariants of foliations and (ii) Loday’s product structure on HL^{∗}. Reference: Leibniz Cohomology for Differentiable Manifolds, to appear in Annales de l’Institut Fourier.