We study the classification of group actions on ${\text{C}}^{\ast }$-algebras up to equivariant KK-equivalence. We show that any group action is equivariantly KK-equivalent to an action on a simple, purely infinite ${\text{C}}^{\ast }$-algebra. We show that a conjecture of Izumi is equivalent to an equivalence between cocycle conjugacy and equivariant KK-equivalence for actions of torsion-free amenable groups on Kirchberg algebras. Let $G$ be a cyclic group of prime order. We describe its actions up to equivariant KK-equivalence, based on previous work by Manuel Köhler. In particular, we classify actions of $G$ on stabilised Cuntz algebras in the equivariant bootstrap class up to equivariant KK-equivalence.

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the $\mathit{I}$-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow–Witt ring. For both maps, we establish compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant $\text{Spin}$-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M∕G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

We prove a $K$-homology index theorem for Toeplitz operators obtained from the multishifts of Bergman spaces on several classes of egg-like domains. This generalizes our earlier work with Douglas and Yu for the unit ball.