Recent zbMATH articles in MSC 57Shttps://www.zbmath.org/atom/cc/57S2021-04-16T16:22:00+00:00WerkzeugA small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists.https://www.zbmath.org/1456.570152021-04-16T16:22:00+00:00"Omori, Genki"https://www.zbmath.org/authors/?q=ai:omori.genkiLet \(N_{g,n}\) denote the connected closed nonorientable surface of genus \(g \geq 1\) with \(n \geq 0\) boundary components, and let \(N_{g,0} := N_g\). Let \(\mathcal{M}(N_{g,n})\) be the mapping class group of \(N_{g,n}\), and let \(\mathcal{T}(N_{g,n})\) denote the twist subgroup of \(\mathcal{M}(N_{g,n})\) generated by Dehn twists. \textit{W. B. R. Lickorish} [Proc. Camb. Philos. Soc. 59, 307--317 (1963; Zbl 0115.40801); ibid. 61, 61--64 (1965; Zbl 0131.20802)] proved that \(\mathcal{M}(N_{g,n})\) is generated by a finite collection of Dehn twists and a \(Y\)-homeomorphism. \textit{B. Szepietowski} [Geom. Dedicata 117, 1--9 (2006; Zbl 1091.57014)] showed that \(g\) Dehn twists and a \(Y\)-homeomorphism from a generating set derived by \textit{D. R. J. Chillingworth} [Proc. Camb. Philos. Soc. 65, 409--430 (1969; Zbl 0172.48801)], generate \(\mathcal{M}(N_{g,n})\). This generating set was shown by \textit{S. Hirose} [Kodai Math. J. 41, No. 1, 154--159 (2018; Zbl 1402.57019)] to be minimal among generating sets consisting of Dehn twists and \(Y\)-homeomorphisms.
It is known [Chillingworth, loc. cit.] that for \(g >3\), \(\mathcal{T}(N_{g,n})\) is generated by \(\frac{3g-1}{2}\) Dehn twists, when \(g\) is odd, and \(\frac{3g}{2}\) Dehn twists, when \(g\) is even. Moreover, for \(n \in\{ 0,1\}\), \textit{M. Stukow} [Bull. Korean Math. Soc. 53, No. 2, 601--614 (2016; Zbl 1355.57021)] derived a finite presentation for \(\mathcal{T}(N_{g,n})\) that involved \(g+2\) Dehn twists. The main result in this paper asserts that for \(g \geq 4\) and \(n \leq 1\), \(\mathcal{T}(N_{g,n})\) is generated by \(g+1\) Dehn twists, which form a proper subset of the generating set derived by Stukow [loc. cit.]. The author proves this assertion by applying an argument of Hirose [loc. cit.] to establish that when \(g \geq 4\) and \(n \leq 1\), any minimal generating set for \(\mathcal{T}(N_{g,n})\) comprising Dehn twists has cardinality at least \(g\). The paper leaves open the question of whether \(\mathcal{T}(N_{g,n})\) is minimally generated by \(g\) or \(g+1\) Dehn twists.
Reviewer: Kashyap Rajeevsarathy (Bhopal)Homotopy properties of smooth functions on the Möbius band.https://www.zbmath.org/1456.570272021-04-16T16:22:00+00:00"Kuznietsova, Iryna"https://www.zbmath.org/authors/?q=ai:kuznietsova.iryna"Maksymenko, Sergiy"https://www.zbmath.org/authors/?q=ai:maksymenko.sergii-ivanovychLet \(B\) be a Möbius band and \(f:B\rightarrow \mathbb{R}\) be a Morse map taking a constant value on the boundary \(\partial B\), and \(S(f;\partial B)\) be the group of diffeomorphisms \(h\) of \(B\) fixed on \(\partial B\) and preserving \(f\) in the sense that \(f\circ h=f\). In this interesting paper the authors compute, under certain assumptions on \(f\), the group \(\pi_0S(f;\partial B)\) of isotopy classes of such diffeomorphisms (Theorem 1.5).
The paper is organized into eight sections dealing with the following aspects: description of the main result, Kronrod-Reeb graph, proof of Theorem 1.5, diffeomorphisms of non-orientable manifolds, Hamiltonian like flows for \(g \in \mathcal{F}(M,P)\), group \(\Delta (f)\), functions on the annulus, completing the proof of Theorem 1.5.
Other papers by the second author directly connected to this topic are [\textit{S. Maksymenko}, Ann. Global Anal. Geom. 29, No. 3, 241--285 (2006; Zbl 1099.37013); Methods Funct. Anal. Topol. 16, No. 2, 167--182 (2010; Zbl 1224.57017); Zb. Pr. Inst. Mat. NAN Ukr. 7, No. 4, 7--66 (2010; Zbl 1240.57019); Ukr. Math. J. 64, No. 9, 1350--1369 (2013; Zbl 1272.58008); translation from Ukr. Mat. Zh. 64, No. 9, 1186--1203 (2012)]; \textit{S. I. Maksymenko} and \textit{B. G. Feshchenko}, ibid. 66, No. 9, 1346--1353 (2015; Zbl 1354.57036); translation from Ukr. Mat. Zh. 66, No. 9, 1205--1212 (2014); Mat. Stud. 44, No. 1, 67--83 (2015; Zbl 1376.37048); Methods Funct. Anal. Topol. 21, No. 1, 22--40 (2015; Zbl 1340.57009)].
Reviewer: Dorin Andrica (Riyadh)Notes on functions of hyperbolic type.https://www.zbmath.org/1456.430022021-04-16T16:22:00+00:00"Monod, Nicolas"https://www.zbmath.org/authors/?q=ai:monod.nicolasSummary: Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of \(\mathbf{SU}(1,n)\) and of its infinite-dimensional kin \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\). We further classify all the self-representations of \(\text{Is}(\mathbf{H}_{\mathbf{C}}^{\infty})\) that satisfy a compatibility condition for the subgroup \(\text{Is}(\mathbf{H}_{\mathbf{R}}^{\infty})\). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.The fundamental groups of open manifolds with nonnegative Ricci curvature.https://www.zbmath.org/1456.530072021-04-16T16:22:00+00:00"Pan, Jiayin"https://www.zbmath.org/authors/?q=ai:pan.jiayinSummary: We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.