Kervaire semi-characteristics in KK-theory and an Atiyah type vanishing theorem

On (4n + 1)-dimensional (noncompact) manifolds admitting proper cocompact Lie group actions, we explore the analytic and topological sides of Kervaire semi-characteristics. The analytic side puts together two interpretations, one via assembly maps, and the other via dimensions of kernels. The topological side is ensured by the proper cocompact version of the Hodge theorem. The two sides coincide and admit an Atiyah type vanishing theorem.