Higher localised  genera for proper actions and applications

Abstract

For a finitely generated discrete group Γacting 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 BΓfor proper actions. In the cocompact case, this leads to a natural generalisation of Gromov-Lawson’s notion of higher Â-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∗(BΓ, 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 Â-genera.