"Lie Algebroids: Spectral Sequences and Signature"

(joint work with Alexandr Mishchenko, Matem. Sbornik (in print))
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We prove that for any transitive unimodular invariantly oriented Lie algebroid L on acompact, oriented and connected manifold with isotropy Lie algebra g and trivial monodromy the cohomology algebra is the Poincare algebra with trivial signature. In particular, the examples of such algebroids are when M is simply connected or when OutG=IntG, for simply connected Lie group G with the Lie algebra g, or when the adjoint Lie algebra bundle g is a trivial flat bundle. To investigate the signature of the Lie algebroid L, we use the technique of spectral sequences for the Cech-deRham complex for a manifold and apply the well-known methods and statements from the paper by Chern, S.S., Hirzebruch, F., Serre, J-P., On the index of a fibered manifolds Proc. AMS, 8 (1957), 587-596. We additionally prove the following proposition:
Let A be any DG-algebra with a decreasing filtration A_j and E_s^j,i its spectral sequence. Assume the following regularity A₀=A of the filtration A_j and that there exist natural numbers m and n with the following conditions: (a) E₂^j,i=0 for j>m and i>n, (2) E₂ is a Poincare algebra with respect to the total gradation and the top group E₂^(m+n)=E₂^m,n. Then H(A) is a Poincare algebra, dimH^m+n(A)=1, and SignE₂=SignH(A).
This paper is a complete version of the abridged paper [49]