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"

(Non-)Vanishing results for Lp-cohomology of semisimple Lie groups

For a locally compact, second countable group \(G\) one can define the continuous \(L^p\)-cohomology \(H^*_{ct}(G,L^p(G))\) of \(G\) and the reduced version \(\overline{H}^*_{ct}(G,L^p(G))\) for all \(p > 1\). In his influential paper “Asymptotic invariants of infinite groups” Gromov asked if \[H^j(G,L^p(G)) = 0 \] when \(G\) is a connected semisimple Lie group and \(j < \mathrm{rk}_{\mathbb{R}}(G)\). … Continue reading "(Non-)Vanishing results for Lp-cohomology of semisimple Lie groups"

Rigidity implication of the Novikov conjecture

Part of my own research is related to the Novikov conjecture, and hence I am always interested to see applications of it. When reading the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892) I saw another one of these applications (Theorem D therein). I do not … Continue reading "Rigidity implication of the Novikov conjecture"

Space of psc-metrics on non-spin manifolds

Recently I was having a look at the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892). In Theorem C therein they construct a closed, totally non-spin manifold (i.e., one whose universal cover is even not spin) whose space of Riemannian metrics of positive scalar curvature has … Continue reading "Space of psc-metrics on non-spin manifolds"

A question about the first \(L^2\)-Betti number

In a recent arXiv preprint (arxiv:2106.15750) J. A. Hillman discusses a new homological approach towards some old results on 3-manifold groups due to Elkalla. In his article he runs into an interesting question concerning the first \(L^2\)-Betti number of finitely generated groups: Assume that a finitely generated group G has infinite subgroups \(N\leq U\) such … Continue reading "A question about the first \(L^2\)-Betti number"

Computer assisted verification of contemporary mathematics

Half a year ago Steffen Kionke wrote a blog post on condensed mathematics wherein he mentioned that Peter Scholze put up a challenge to formally verify a key fundamental result of a joint paper with Clausen: a result Scholze terms his most important theorem to date. Today I want to inform you that the mentioned … Continue reading "Computer assisted verification of contemporary mathematics"