Recent zbMATH articles in MSC 57S https://zbmath.org/atom/cc/57S 2022-06-24T15:10:38.853281Z Werkzeug Minimal model-universal flows for locally compact Polish groups https://zbmath.org/1485.37011 2022-06-24T15:10:38.853281Z "Jahel, Colin" https://zbmath.org/authors/?q=ai:jahel.colin "Zucker, Andy" https://zbmath.org/authors/?q=ai:zucker.andy Let $$G$$ be a locally compact, non-compact Polish group. A continuous action of $$G$$ on a compact space is called a $$G$$-flow. A $$G$$-flow is called minimal if every orbit is dense. By a $$G$$-system, the authors mean a Borel $$G$$-action on a standard Lebesgue space $$(X,\mu)$$ which preserves the measure $$\mu$$. A $$G$$-system $$(X,\mu)$$ is called free if the set $\mathrm{Free}(X)=\{ x\in X\mid gx\not= x \, \forall x\in(G\setminus \{1_G\})\}$ has measure $$1$$. Moreover, a compact metric $$G$$-flow $$Y$$ is called model-universal if for every free $$G$$-system $$(X,\mu)$$, there is a $$G$$-invariant regular Borel probability measure $$\nu$$ on $$Y$$ such that the $$G$$-systems $$(X,\mu)$$ and $$(Y,\nu)$$ are isomorphic. The main result of the paper states that there exists a minimal model-universal flow for $$G$$. The result extends a theorem of \textit{B. Weiss} [Contemp. Math. 567, 249--264 (2012; Zbl 1279.37010)], who proved a similar result for infinite countable groups. Reviewer: Marja Kankaanrinta (Helsinki) The conformal group of a compact simply connected Lorentzian manifold https://zbmath.org/1485.53086 2022-06-24T15:10:38.853281Z "Melnick, Karin" https://zbmath.org/authors/?q=ai:melnick.karin "Pecastaing, Vincent" https://zbmath.org/authors/?q=ai:pecastaing.vincent The main result of this paper is the compactness of the conformal group of closed simply-connected real analytic Lorentzian manifolds. This extends a well-known and difficult result of \textit{G. D'Ambra} [Invent. Math. 92, No. 3, 555--565 (1988; Zbl 0647.53046)], on the compactness of the isometry group of such manifolds, based on methods developed by Gromov. Differently from the Riemannian setting, isometry groups (and hence conformal groups) of closed semi-Riemannian manifolds are often noncompact. The only cases covered by D'Ambra's theorem are those conformal groups that preserve a finite volume, since they are the isometry group of some metric in the same conformal class. For the general (essential) case, the authors carry out a deep analysis of the dynamics of sequences of conformal transformations, and associated holonomy sequences''. As additional motivation for their results, the authors mention the Lorentzian Lichnerowicz Conjecture, which is a Lorentzian counterpart to the theorem of Lelong-Ferrand and Obata that, up to conformal equivalence, the only closed Riemannian manifold with noncompact conformal group is the round sphere. In this direction, some recent progress in dimension 3 was announced by the first named author in joint work with C. Frances. Reviewer: Renato G. Bettiol (New York) On the sign ambiguity in equivariant cohomological rigidity of GKM graphs https://zbmath.org/1485.55011 2022-06-24T15:10:38.853281Z "Yamanaka, Hitoshi" https://zbmath.org/authors/?q=ai:yamanaka.hitoshi \textit{V. Guillemin} and \textit{C. Zara} defined an abstract GKM (Goresky-Kottwitz-MacPherson) graph $${\mathcal{G}}$$ and its graph equivariant cohomology in [Asian J. Math. 3, No. 1, 49--76 (1999; Zbl 0971.58001)]. The graph equivariant cohomology $$H_T^\ast({\mathcal{G}})$$ is a graded algebra over the integral cohomology $$H^\ast(BT)$$, where $$T$$ is a torus and $$BT$$ is the classifying space of $$T$$. In this paper the author introduces the notion of equivariant total Chern class of a GKM graph $${\mathcal{G}}$$. This is an element in $$H_T^\ast({\mathcal{G}})$$. Let $${\mathcal{G}}$$ and $${\mathcal{G}}'$$ be GKM graphs. Assume there is an isomorphism $$\varphi\colon H^\ast_T({\mathcal{G}}')\to H^\ast_T({\mathcal{G}})$$ of graded $$H^\ast(BT)$$-algebras that preserves the equivariant total Chern classes. The main result of the paper states that there is a geometric isomorphism $${\mathcal{G}}\to {\mathcal{G}}'$$ inducing $$\varphi$$. Thus the graph equivariant cohomology and the equivariant total Chern class together completely determine a GKM graph. The paper is a sequel to [\textit{M. Franz} and \textit{H. Yamanaka}, Proc. Japan Acad., Ser. A 95, No. 10, 107--110 (2019; Zbl 1442.55006)]. Reviewer: Marja Kankaanrinta (Helsinki) Cohomology algebra of orbit spaces of free involutions on some Wall manifolds https://zbmath.org/1485.57029 2022-06-24T15:10:38.853281Z "Paiva, Thales Fernando Vilamaior" https://zbmath.org/authors/?q=ai:paiva.thales-fernando-vilamaior "dos Santos, Edivaldo Lopes" https://zbmath.org/authors/?q=ai:dos-santos.edivaldo-lopes Can a topological group $$G$$ act freely on a topological space $$X$$, and if so, what is the cohomology of the orbit space $$X/G$$? For specific groups and spaces, this question has a rich history and led to deep and important results in topology. The authors answer this question in a specific setting. Theorem. Let $$X$$ be a compact Hausdorff space with the mod $$2$$ cohomology of a Wall manifold $$Q(1,n)$$, with $$n \ge 3$$. Suppose that $$G = \mathbb Z_2$$ acts freely on $$X$$, and the induced action on the mod $$2$$ cohomology is trivial. Then $H^*(X/G; \mathbb Z_2) \cong \frac{\mathbb Z_2 [\alpha, \beta, \gamma, \delta]}{(\alpha^3, \beta^3, \gamma^2, \beta^2+\beta\gamma, \delta^{\frac{n+1}{2}})},$ where $$\deg \alpha = \deg \beta = \deg \gamma = 1$$ and $$\deg \delta = 4$$. Reviewer: Karl Heinz Dovermann (Honolulu) Transformation groups of certain flat affine manifolds https://zbmath.org/1485.57030 2022-06-24T15:10:38.853281Z "Saldarriaga, O." https://zbmath.org/authors/?q=ai:saldarriaga.omar "Flórez, A." https://zbmath.org/authors/?q=ai:florez.alvaro-jose Authors' abstract: The paper under review characterizes the group of affine transformations of a flat affine simply connected manifold whose developing map is a diffeomorphism. This is proved by making use of some simple facts about homeomorphisms of $$\mathbb R^n$$ preserving open connected sets. Moreover, there are presented some examples where the characterization is useful. Reviewer: Andrzej Szczepański (Gdańsk)