Oka principle for Levi flat manifolds

The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in terms of regularity or bidegree of the bundle, and partial results

Bordism, rho-invariants and the Baum-Connes conjecture

Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-free and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^*-algebra. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof re-establishes this result and also extends it to the delocalized eta-invariant of Lott. Our method also gives some information about the eta-invariant itself (a much more saddle object than the rho-invariant).Comment: LaTeX2e, 60 pages; the gap pointed out by Nigel Higson and John Roe is now closed and all statements of the first version of the paper are proved (with some small refinements

Homotopically trivializing the circle in the framed little disks

This paper confirms the following suggestion of Kontsevich. In the appropriate derived sense, an action of the framed little disks operad and a trivialization of the circle action is the same information as an action of the Deligne-Mumford-Knudsen operad. This improves an earlier result of the author and Bruno Vallette.Comment: 36 pages. This version accepted for publication by the Journal of Topolog

Hilbert's Theorem 90 and algebraic spaces

In modern form, Hilbert's Theorem 90 tells us that R^1f_*(G_m)=0, where f is the canonical map between the etale site and the Zariski site of a scheme X. I construct examples showing that the corresponding statement for algebraic spaces does not hold. The first example is a nonseparated smooth 1-dimensional bug-eyed cover in Kollar's sense. The second example is a nonnormal proper algebraic space obtained by identifying points on suitable nonprojective smooth proper schemes.Comment: 6 pages, to appear in J. Pure Appl. Algebr

The surgery exact sequence, K-theory and the signature operator

The main result of this paper is a new and direct proof of the natural transformation from the surgery exact sequence in topology to the analytic K-theory sequence of Higson and Roe. Our approach makes crucial use of analytic properties and new index theorems for the signature operator on Galois coverings with boundary. These are of independent interest and form the second main theme of the paper. The main technical novelty is the use of large scale index theory for Dirac type operators that are perturbed by lower order operators.Comment: 29 pages, AMS-LaTeX; v2: small corrections and (hopefully) improved exposition, as suggested by the referee. Final version, to appear in Annals of K-Theor