An equivariant higher index theory and nonpositively curved manifolds

Abstract

In this paper, we define an equivariant higher index map from K ∗ Γ ( X ) to K ∗ ( C ∗ ( X ) Γ ) if a torsion-free discrete group Γ acts on a manifold X properly, where C ∗ ( X ) Γ is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum–Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1–20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C ∗ -algebras, in: C ∗ -Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291]. When Γ is trivial, this equivariant higher index map is the coarse Baum–Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective.