Sullivan constructions for transitive Lie algebroids - smooth case

Авторы: Mishchenko Aleksandr S., Oliveira Jose R.
Журнал: arXive e-print
Том: 1709
Номер: 07494
Год издания: 2017
Первая страница: 1
Последняя страница: 32
Аннотация: Let M be a smooth manifold, smoothly triangulated by a simplicial complex K, and $\cA$ a transitive Lie algebroid on M. The Lie algebroid restriction of $\cA$ to a simplex Δ of K is denoted by $\cA^{!!}_{\Delta}$. A piecewise smooth form of degree p on $\cA$ is a family ω=(ωΔ)Δ∈K such that $\omega_{\Delta}\in \Omega^{p}(\cA^{!!}_{\Delta};\Delta)$ for each Δ∈K, satisfying the compatibility condition concerning the restrictions of ωΔ to the faces of Δ, that is, if Δ′ is a face of Δ, the restriction of the form ωΔ to the simplex Δ′ coincides with the form ωΔ′. The set $\Omega^{\ast}(\cA;K)$ of all piecewise smooth forms on $\cA$ is a cochain algebra. One has a natural morphism \Omega^{\ast}(\cA;M)\rightarrow \Omega^{\ast}(\cA;K) of cochain algebras given by restriction of a smooth form defined on $\cA$ to a smooth form defined on $\cA^{!!}_{\Delta}$, for all simplices Δ of K. In this paper, we prove that, for triangulated compact manifolds, the cohomology of this construction is isomorphic to the Lie algebroid cohomology of $\cA$, in which the isomorphism is induced by the restriction map.
Добавил в систему: Мищенко Александр Сергеевич

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