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THE C1-Invariance of the Godbillon-Vey Map in Analytical K-Theory
Published online by Cambridge University Press: 20 November 2018
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An action α of a discrete group Γ on the circle S1 as orientation preserving C∞-diffeomorphisms gives rise to a foliation on the homotopy quotient S1Γ, and its Godbillon-Vey invariant is, by definition, a cohomology class of S1Γ([1]). This cohomology class naturally defines an additive map from the geometric K-group K0(S1, Γ) into C, through the Chern character from K0(S1, Γ) to H*(S1, Γ Q).
Using cyclic cohomology, Connes constructed in [2] an additive map, GV(α), which we shall call the Godbillon-Vey map, from the K0-group of the reduced crossed product C*-algebra C(S1) ⋊ αΓ into C. He showed that GV(α) agrees with the geometric Godbillon-Vey invariant through the index map μ from K0(S1, Γ) to K0(C(S1) ⋊ αΓ).
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- Copyright © Canadian Mathematical Society 1987
References
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Connes, A., Cyclic cohomology and the transverse fundamental class of a foliation., Proceedings of the Japan-U.S. Seminar on Operator Algebras, Kyoto (1983), Geometric methods in operator algebras, (Longman, London, 1986), 52–144.Google Scholar
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14 (Academic Press, London, 1979).Google Scholar
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