arXiv subjects
Publications and source records attributed to Jose R. Oliveira.
Cohomology of a compatible family of Lie algebroids defined on a family of transverse manifolds is defined. A sheaf of differential forms on a compatible family of Lie algebroids defined over regular open subsets of a simplicial complex is constructed. It is proved that sheaf is fine.
Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $Δ$ of $K$, the inverse images of $Δ$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of $G$ does not depend on the triangulation of the base.
It is shown that the Mayer-Vietoris sequence holds for the cohomology of complexes of Lie algebroids which are defined on simplicial complexes and satisfy the compatibility condition concerning restrictions to the faces of each simplex. The Mayer-Vietoris sequence will be obtained as a consequence of the extension lemma for piecewise smooth forms defined on complexes of Lie algebroids.
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 $\cA^{!!}_Δ$. A piecewise smooth form of degree $p$ on $\cA$ is a family $ω=(ω_Δ)_{Δ\in K}$ such that $ω_Δ\in Ω^{p}(\cA^{!!}_Δ;Δ)$ for each $Δ\in K$, satisfying the compatibility condition concerning the restrictions of $ω_Δ$ to the faces of $Δ$, that is, if $Δ'$ is a face of $Δ$, the restriction of the form $ω_Δ$ to the simplex $Δ'$ coincides with the form $ω_{Δ'}$. The set $Ω^{\ast}(\cA;K)$ of all piecewise smooth forms on $\cA$ is a cochain algebra. One has a natural morphism $$Ω^{\ast}(\cA;M)\rightarrow Ω^{\ast}(\cA;K)$$ of cochain algebras given by restriction of a smooth form defined on $\cA$ to a smooth form defined on $\cA^{!!}_Δ$, 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.