Smooth structures on pseudomanifolds with isolated conical singularities
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SMOOTH STRUCTURES ON PSEUDOMANIFOLDS WITH ISOLATED CONICAL SINGULARITIES Hông Vân Lê · Petr Somberg · Jiˇrí Vanžura
Received: 11 October 2012 / Published online: 20 February 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C ∞ -rings of smooth functions on M. For a finitely generated smooth structure C ∞ (M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the notion of characteristic classes of M. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on M. We introduce the notion of a conical symplectic form on M and show that it is smooth with respect to a Euclidean smooth structure on M. If a conical symplectic structure is also smooth with respect to a compatible Poisson smooth structure C ∞ (M), we show that its Brylinski–Poisson homology groups coincide with the de Rham homology groups of M. We show nontrivial examples of these smooth conical symplectic-Poisson pseudomanifolds. Keywords C ∞ -ring · Conical pseudomanifold · Symplectic form · Poisson structure Mathematics Subject Classification 51H25 · 53D05 · 53D17
1 Introduction Since the second half of the last century the theory of smooth manifolds has been extended from various points of view to a large class of topological spaces admitting singularities, see e.g. [8, 10–12, 22, 23, 25, 26]. Roughly speaking, a C k -structure, 1 ≤ k ≤ ∞, on a topological space M is defined by a choice of a subalgebra C k (M) of the R-algebra C 0 (X) of
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H.V. Lê ( ) Institute of Mathematics of ASCR, Zitna 25, 11567 Praha 1, Czech Republic e-mail: [email protected] P. Somberg Mathematical Institute, Charles University, Sokolovska 83, 180 00 Praha 8, Czech Republic J. Vanžura Institute of Mathematics of ASCR, Zizkova 22, 61662 Brno, Czech Republic
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all continuous R-valued functions on M, which satisfies certain axioms varying in different approaches. Most of the efforts have been spent on construction of a convenient category of smooth spaces, which should satisfy good formal properties, see [1] for a survey. Notably, the theory of de Rham cohomology has been extended to a large class of singular spaces, see [23, 26]. In this note we develop the theory of smooth structures on singular spaces in a different direction. We pick a class of topological spaces and ask, if we can provide these spaces with a family of reasonable smooth structures and what is the best smooth structure on a singular space. This question is motivated by the question of finding the best compactification of an open smooth manifold. We are looking not only for an extension of classical theorems on smooth manifolds, but we are also looking for new phenomena on these manifolds, which are caused by presence of nontrivial singularities. We study in this note pseudomanifolds w
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