Higher localised [formula omitted]-genera for proper actions and applications

Abstract

For a finitely generated discrete group Γ acting properly on a spin manifold M, we formulate new topological obstructions to Γ-invariant metrics of positive scalar curvature on M that take into account the cohomology of the classifying space B_Γ for proper actions.In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher Aˆ-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of Γ-invariant positive scalar curvature on M. For classes arising from the subring of H⁎(B_Γ,R) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted L2-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of Wang-Wang [24] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of Mathai [17], which provided a partial answer to a conjecture of Gromov-Lawson on higher Aˆ-genera.If M is non-cocompact, we obtain obstructions to M being a partitioning hypersurface inside a non-cocompact Γ-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [12], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.