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In 1968-70, Israel Gelfand and Dimitri Fuchs did pioneering work on the cohomology of the Lie algebra of smooth vector fields on manifolds, nowadays known as Gelfand-Fuchs cohomology. The dimensions of the corresponding cohomology spaces for a given manifold M of dimension n can be computed in principle from the cohomology H(W_n) of the Lie algebra of formal vector fields in n variables for which a basis (Vey basis) is known. There exists a manifold X_n whose singular cohomology is isomorphic to H(W_n). For manifolds M with vanishing Pontryagin classes, Gelfand-Fuchs cohomology is then isomorphic to the singular cohomology of Map(M,X_n) (Theorem of Haefliger 1974, Bott-Segal 1977). For n=1, X₁=S³ (the standard 3-sphere) and well-known explicit generators of the cohomology spaces are available. For n>1, this is (to our knowledge) not the case.

We will review Gelfand-Fuchs cohomology along these lines and then come to our recent computations for some generators of the Gelfand-Fuchs cohomology spaces for surfaces (i.e. n=2).

There will be a dinner for the speaker.