Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids (Q1033874)

From the abstract: The index of the classical Hirzebruch signature operator on a manifold \(M\) is equal to the signature of the manifold. The examples of \textit{G. Lusztig} [J. Differ. Geom. 7, 229--256 (1972; Zbl 0265.57009)] and \textit{M. Gromov} [Prog. Math. 132, 1--213 (1996; Zbl 0945.53022)] present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In a previous paper [\textit{J. Kubarski}, DGA 2007, Olomouc, Czech Republic, August 27--31, 2007. Hackensack, NJ: World Scientific. 317--328 (2008; Zbl 1165.58011)], a signature operator for the cohomology of transitive Lie algebroids was given. In this paper, firstly, the authors present a general approach to the signature operator, and the above examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid \(A\) over a compact oriented manifold for which the top group of the real cohomology of \(A\) is nontrivial, the authors notice that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Hence, a second signature operator for Lie algebroids is obtained.