Coarse topology, enlargeability, and essentialness

Hanke, Bernhard; Kotschick, Dieter; Roe, John; Schick, Thomas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 473-495.

En utilisant des méthodes de topologie à grande échelle, on prouve que les classes fondamentales des variétés agrandissables ne s’annulent pas, ni dans l’homologie rationnelle de leurs groupes fondamentaux, ni dans la $K$-théorie des $C^{*}$-algèbres réduites correspondantes. Nos résultats ne dépendent pas de la conjecture de Baum-Connes, et confirment de façon indépendante certaines conséquences de cette conjecture.

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the $K$-theory of the corresponding reduced $C^{*}$-algebras. Our proofs do not depend on the Baum-Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.

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