Properly positive scalar curvature

An interesting (at least to me) research theme in the geometry of manifolds is the question about the existence of positive scalar curvature metrics on closed manifolds. Since I also like to do coarse geometry, I therefore also consider the corresponding question on non-compact manifolds. But what is the ‘corresponding’ question on non-compact manifolds? Currently, … Continue reading "Properly positive scalar curvature"

An implication of the Farrell-Jones conjecture

A ‘well-known’ implication of the Farrell-Jones conjecture (for a given group G) is that the map \[\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}\] in reduced algebraic K-theory is rationally trivial. What at first might seem as a technical statement about algebraic K-theory turns out to have an interesting geometric consequence. It implies the Bass conjecture, which is equivalent to … Continue reading "An implication of the Farrell-Jones conjecture"

Topological CAT(0)-manifolds

It is an interesting and important fact that a contractible manifold (without boundary) is not necessarily homeomorphic to Euclidean space. This makes the classical Cartan-Hadamard theorem, stating that a contractible manifold equipped with a Riemannian metric of non-positive sectional curvature is diffeomorphic to Euclidean space, even more powerful. One can ask now whether one can … Continue reading "Topological CAT(0)-manifolds"

PSC obstructions via infinite width and index theory

In a recent preprint (arXiv:2108.08506), Yosuke Kubota proved an intriguing new result on the relation of largeness properties of spin manifolds and index-theoretic obstructions to positive scalar curvature (psc): Let \(M\) be a closed spin \(n\)-manifold. If \(M\) has infinite \(\mathcal{KO}\)-width, then its Rosenberg index \(\alpha(M) \in \mathrm{KO}_n(\mathrm{C}^\ast_{\max} \pi_1 M)\) does not vanish. Let us … Continue reading "PSC obstructions via infinite width and index theory"