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Abstract:

In 1990 Michel Rumin published an article containing the construction of what is today called the “Rumin complex”, an intrinsically defined, fine resolution of the constant sheaf of a Contact manifold. The motivation behind this construction was a question asked by his thesis advisors Pierre Pansu and Mikhaïll Gromov, about the existence of a homogeneous differential through anisotropic dilations on Heisenberg groups. He was then asked to study the holonomy of a horizontal 1-form in the 3-dimensional Heisenberg group along small horizontal curves, which is where the middle degree, second order differential operator D in the complex appears.

Rumin later generalized this complex to Carnot groups, the local model for C-C manifolds. This construction is no longer intrinsic, as one has to assume an inner product on the tangent space at the identity, in order to define the partial inverse of a linear operator related to the exterior derivative. Being motivated by spectral problems, assuming an inner product was natural, and not viewed as an issue. This complex eventually led to the approximation of, and in some cases exact calculation of the Novikov-Shubin numbers of a Carnot group, a homotopy invariant related to the eigenvalues of the Laplacian of a manifold. The goal of these lectures is to go through the differences in the construction of this complex on 2-step Carnot groups with 1-dimensional second layer, s-step Carnot groups, graded Lie groups, and talk about the possibility of generalization to nilpotent Lie groups.