RT Dissertation/Thesis
T1 Homological properties of transitive Lie algebroids via Sullivan models
A1 Ribeiro Oliveira, Jose Manuel
K1 Lie algebroids
K1 Sullivan models
K1 Homological properties
K1 Rham theorem
AB The aim of the present work is to prove Mishchenko's conjecture. For this purpose,we have used a structure called a complex of Lie algebroids. This structure commencesby  xing a smooth triangulation of the base of a transitive Lie algebroid by a simplicialcomplex and taking the restriction of the Lie algebroid to all simplices of the triangulation.Since the Lie algebroid is transitive, the restriction of the Lie algebroid to each simplexalways exists. When a complex of Lie algebroids is given, we de ne the notion of piecewisesmooth form in a similar way to Whitney forms on a simplicial complex and the set ofall piecewise smooth forms de ned on a complex of Lie algebroids is naturally equippedwith a di erential, yielding a commutative di erential graded algebra. Its cohomology is,by de nition, the piecewise smooth cohomology of the Lie algebroid. Each smooth formde ned on the Lie algebroid gives a piecewise smooth form de ned on the correspondingcomplex of Lie algebroids by taking the restriction of the form to each simplex. Thiscorrespondence is a natural map from the usual algebra of the smooth forms of the Liealgebroid to the algebra of the piecewise smooth forms of the corresponding complex of Liealgebroids. Based on three crucial results, namely the triviality of a transitive Lie algebroidover a contractible smooth manifold (Mackenzie, Weinstein), the K unneth theorem for Liealgebroids (Kubarski) and the de Rham-Sullivan theorem for smooth manifolds, we showthat this map is an isomorphism in cohomology.
YR 2013
FD 2013-10-23
LK http://hdl.handle.net/10347/9272
UL http://hdl.handle.net/10347/9272
LA eng
DS Minerva
RD 27 abr 2026