Codimension two index obstructions to positive scalar curvature

Hanke, Bernhard; Pape, Daniel; Schick, Thomas

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2681-2710.

Nous dérivons une obstruction générale à l’existence d’une métrique à courbure scalaire positive sur une variété compacte spin, qui est basée sur des sous-variétés de codimension deux. La preuve utilise la théorie d’indice grossier (synonymement “indice a grande échelle”) pour l’opérateur de Dirac tordu par un fibré de $C^{*}$-modules Hilbertiens.

En cours de route nous donnons une preuve complète et indépendant du fait que la clôture minimale d’un opérateur de type Dirac sur une variété complète, tordu par un fibré de $C^{*}$-modules Hilbertiens, est régulière et auto-adjointe comme operateur non-borné sur le $C^{*}$-module Hilbertien des sections $L^2$-intégrables de ce fibré.

En outre, nous donnons une preuve nouvelle du théorème de Roe affirmant que l’indice grossier de l’opérateur de Dirac est nul pour une variété Riemannienne complète non-compacte avec courbure scalaire uniformement positive en dehors d’un sous-ensemble compact. Notre preuve se généralise immédiatement aux operateurs de Dirac tordu par un fibré plat de $C^{*}$-modules Hilbertiens.

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert $C^{*}$-module bundles.

Along the way we give a complete and self-contained proof that the minimal closure of a Dirac type operator twisted with a Hilbert $C^{*}$-module bundle on a complete Riemannian manifold is a regular and self-adjoint operator on the Hilbert $C^{*}$-module of $L^2$-sections of this bundle.

Moreover, we give a new proof of Roe’s vanishing theorem for the coarse index of the Dirac operator on a complete non-compact Riemannian manifold whose scalar curvature is uniformly positive outside of a compact subset. This proof immediately generalizes to Dirac operators twisted with flat Hilbert $C^{*}$-module bundles.

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