### Lie groups acting on countable sets

Does every connected Lie group act faithfully on a countable set? In other words: is every Lie group a subgroup of $$\mathrm{Sym}(\mathbb{N})$$? This question is sometimes called Ulam’s problem and there is recent progress in a paper of Nicolas Monod. Monod proves that every nilpotent connected Lie group acts faithfully on a countable set. It … Continue reading "Lie groups acting on countable sets"

### 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"

### (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"

### New book about Freedman’s proof

Today I learnt from an article in the QuantaMagazine (link to article) that there is finally a new book trying to explain Freedman’s proof of the 4-dimensional Poincaré conjecture (link to book). The article is fun to read since it contains statements of the involved people about how the whole ‘situation’ about the non-understandable write-up … Continue reading "New book about Freedman’s proof"

### Loewner’s “forgotten” theorem

Yesterday I discovered an instructive paper on “Loewner’s forgotten theorem” by Peter Albers and Serge Tabachnikov on the arXiv. I don’t know in how far the result has really been forgotten, it was published in the Annals in 1948. Fair enough, it has only 4 citations on MathSciNet. So what is Loewner’s theorem? It says … Continue reading "Loewner’s “forgotten” theorem"

### 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"