Invariants topologiques des Espaces non commutatifs

Abstract

In this thesis, we give a definition of topological K-theory of Kontsevich's noncommutative spaces (ie dg-categories) defined over the complex. The main motivation comes from noncommutative Hodge structures in the sense of Kontsevich--Katzarkov--Pantev on the periodic cyclic homology of smooth and proper dg-algebras. The essential ingredient in the definition is the topological realization functor from presheaves of spectra on the site of complex affine schemes to spectra. Topological K-theory is defined as the Bott inverted topological realization of nonconnective algebraic K-theory. Using a non-abelian generalization of Deligne's proper cohomological descent, we show that the topological realization of the preaheave given by nonconnective K-theory is the spectrum bu of connective topological K-theory. The other main result deals with the relation between topological K-theory of a dg-category T and the moduli stack of perfect modules over T^op. Moreover, topological K-theory can be endowed with a Chern character map to periodic cyclic homology which factorizes Cisinski--Tabuada Chern map. We give a comparison result for smooth schemes of finite type over the complex and for finite dimensional associative algebras.