Jan Kubarski

Poincaré duality for transitive unimodular invariantly oriented Lie algebroids

Abstract Fubinis formula for an oriented bundle suggests a definition of the integration of forms of maximal degree in a transitive Lie algebroid A (using the fibre integral ∫A in A, defined and investigated by the author in his previous paper). In the case of a unimodular and invariantly oriented transitive Lie algebroid, this integral enables us to define the Poincare scalar product. The purpose of this paper is to investigate the fundamental properties of this product.

Fibre Integral in Regular Lie Algebroids

The idea of the fibre integral / in an oriented bundle is adapted to a regular Lie algebroid. It is based on the well-known result expressing the fibre integral of right-invariant differential forms on a principal bundle via some substitution operator. The object of this article is to define the integration operator f A over the adjoint bundle of Lie algebras g in a regular Lie algebroid A over a foliated manifold (M, F) with respect to a cross-section ɛ ∈ Sec ⋀ n g, n = rank g, and to demonstrate its main properties.

Bott’s vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: — differential groupoids, — principal bundles, — vector bundles, — actions of Lie groups on manifolds, — transversally complete foliations, — nonclosed Lie subgroups, — Poisson manifolds, — some complete closed pseudogroups. We carry over the idea of Bott’s Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

The Evens-Lu-Weinstein representation (QA, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (QAor, Dor) by tensoring by orientation flat line bundle, QAor=QA⊗or (M) and Dor=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (QAor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.

Transitive Lie algebroids of rank 1 and locally conformal symplectic structures

Abstract The aim of the paper is the study of transitive Lie algebroids with the trivial 1-rank adjoint bundle of isotropy Lie algebras g ≅M× R . We show that a locally conformal symplectic (l.c.s.) structure defines such a Lie algebroid, so our algebroids are a natural generalisation of l.c.s. structures. We prove that such a Lie algebroid has the Poincare duality property for the Lie algebroid cohomology (TUIO-property) if and only if the top-dimensional cohomology space is non-trivial. Moreover, if an algebroid is defined by an l.c.s. structure, then this algebroid is a TUIO-Lie algebroid if and only if the associated l.c.s. structure is a globally conformal symplectic structure.

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

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ([10], 1972) and Gromov ([4], 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In [6], we gave a signature operator for the cohomology of transitive Lie algebroids.In this paper, firstly, we present a general approach to the signature operator, and the above four 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, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

Algebroid nature of the characteristic classes of flat bundles

The following two homotopic notions are important in many domains of differential geometry: — homotopic homomorphisms between principal bundles (and between other objects), — homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be — in a natural way — expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the ChernWeil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat bundles) are invariants of Lie algebroids of these bundles. This enables one to construct the characteristic homomorphism of a flat regular Lie algebroid, measuring the incompatibility of the flat structure with a given subalgebroid. For two given Lie subalgebroids, these homomorphisms are equivalent if the Lie subalgebroids are homotopic. Some new examples of applications of this characteristic homomorphism to a transitive case (for TC-foliations) and to a non-transitive case (for a principal bundle equipped with a partial flat connection) are pointed out (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation which leads to the nontrivial characteristic homomorphism is obtained.