Celebratio Mathematica — Bott

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Works connected to Robert Osserman

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R. Bott: “On the Gelfand–Fuks cohomology,” pp. 357–364 in Differential geometry (Stanford, CA, 30 July–17 August 1973), part 1. Edited by S.-S. Chern and R. Osserman. Proceedings of Symposia in Pure Mathematics 27. American Mathematical Society (Providence, RI), 1975. MR 0380820 Zbl 0324.57013 incollection

Roughly four years ago Gelfand and Fuks had the inspired idea of considering the continuous cohomology of the Lie-algebra of \( C^{\infty} \)-vector fields on a manifold ([1969, 1970a, 1970b]). They showed that the answer was finite dimensional for all compact manifodls and that for \( S^1 \) this cohomology is a tensor product of an exterior algebra in dimension 3 and a polynomial ring with generator in dimension 2: \[ H_c(\mathcal{L}S^1) = E(\omega)\otimes \mathbf{R}[y]. \]

In this report I would like to present a new point of view towards their work which arose, on the one hand, out of the new insights into cohomology due to Guillemin and Trauber, and on the other out of conversations between G. Segal, A. Haefliger and myself during the past year. I will finally formulate a conjecture which would completely settle the homotopy theoretic nature of this cohomology. Although one finds evidence for its validity on every side, its proof has so far eluded us. I should also mention that when I recently showed this conjecture to Gelfand he told me that it had also, quite independently, been formulated by Fuks.

S. S. Chern	Related	Volume
Robert Osserman	Related

R. Bott, C. Earle, D. Hejhal, J. Jenkins, T. Jorgensen, A. Marden, and R. Osserman: “Lars Valerian Ahlfors (1907–1996),” Notices Am. Math. Soc. 45 : 2 (1998), pp. 248–255. Edited and with an introduction by Steven G. Krantz. MR 1601820 Zbl 0908.01015 article

Clifford John Earle, Jr.	Related
Lars Valerian Ahlfors	Related
Robert Osserman	Related
Steven Krantz	Related
Dennis Arnold Hejhal	Related
James Jenkins	Related
Troels Jørgensen	Related