This project is concerned with non-compact Riemannian manifolds which are so-called “large”. Concretely, we consider macroscopic dimension as a measure of largeness, and being hypereuclidean as a strong form of having full macroscopic dimension. The goals are to investigate the behaviour of these two notions in geometric situations where we have a suitable large submanifold, to relate them to macroscopic positive scalar curvature, and to explore them in the world of simplicial non-positive curvature.
For a countable group G we construct a small, idempotent complete, symmetric monoidal, stable ∞-category KK^G_sep whose homotopy category recovers the triangulated equivariant Kasparov category of separable G-C*-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable ∞-category KK^G which receives a symmetric monoidal functor kk^G from possibly non-separable G-C*-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying G. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite K-homology theory on proper and locally compact G-topological spaces, allowing for coefficients in arbitrary G-C*-algebras. Finally, we extend the functor kk^G from G-C*-algebras to G-C*-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.
45Macroscopic invariants of manifolds