Kervaire semi-characteristics in KK-theory and an Atiyah type vanishing theorem
Hao Zhuang
TL;DR
The paper extends the analytic and topological interpretation of the Kervaire semi-characteristic to noncompact manifolds with proper cocompact group actions by defining the topological invariant $k(M,G)$ via modular-character–twisted invariant cohomology and establishing an Atiyah-type vanishing theorem when two $G$-invariant vector fields exist. It develops a generalized mod 2 index ind$_2^G$ using KK-theory and the Baum–Connes assembly map, and proves its equality with $k(M,G)$ through a proper cocompact Hodge theory that links kernels to twisted cohomology. A skew-adjoint Fredholm picture is constructed to represent ind$_2^G([\mathscr{D}_{\text{sig}}])$, with stable perturbations showing the analytic and topological viewpoints coincide. The work also analyzes the role of the modular character and invariant cohomology, highlighting when invariant cohomology provides a suitable substitute and presenting explicit nonunimodular versus unimodular examples to illustrate the limitations and applicability of the framework.
Abstract
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.