Abstract
In this thesis, we give a definition of a topological K-theory of Kontsevich's non-commutative spaces (i.e. of dg-categories) defined over complex numbers. The introduction of this invariant initiates the quest for topological invariants of non-commutative spaces, which are considered as "simplifications" of algebraic ones like algebraic K-theory, cyclic homology, periodic homology as studied by Tsygan, Keller. The main motivation comes from non-commutative Hodge theory in the sense of Katzarkov--Kontsevich--Pantev. In algebraic geometry, the rational part of the Hodge structure is given by rational Betti cohomology, which is the rational cohomology of the underlying space of complex points. The existence of a space associated to a dg-category admits a first answer given by the stack (defined by Toën--Vaquié) classifying perfect dg-modules over this dg-category. The essential ingredient in the definition of the topological K-theory is the topological realization functor of spectral presheaves on the site of complex schemes of finite type. The connective part of the semi-topological K-theory can then be definied as the topological realization of the stack of perfect dg-modules over the space, together with its commutative monoid structure up to homotopy. But to deal with negative K-groups, we realize the presehaf given by non-connective algebraic K-theory. One of our main results relies the two previous definition in the connective case. We show that the topological realization of the presheaf of connective algebraic K-theory for the unit dg-category is equivalent to the usual topological K-theory spectrum. We show this is also true in the non-connective case, using a property of restriction to smooth schemes. This last property leads us to show a generalization of Deligne's proper cohomological descent to the homotopical non-abelian setting. This enables us to define topological K-theory by inverting the Bott element. We point out that the process of the definition involves non-trivial results. We then show that the Chern character from algebraic K-theory to periodic homology factorizes through topological K-theory, giving a natural candidate for the rational part of a non-commutative Hodge structure on the periodic homology of a smooth and proper dg-category. This last claim is written in the form of a conjecture : the lattice conjecture. Our first comparison result deals with the case of a smooth scheme of finite type over complex numbers -- we show the lattice conjecture holds for dg-categories of perfect complexes. We also show this conjecture is true in the case of finite dimensional associative algebras.