The first Betti numbers of orbits of Morse functions on surfaces

Keywords: Wreath products, Homology groups, Morse functions

Abstract

UDC 515.1

Let $M$ be a connected compact orientable surface and let $P$ be the real line $\mathbb{R}$ or circle $S^1.$
The group $\mathcal{D}$ of diffeomorphisms on $M$ acts in the space of smooth mappings $C^{\infty} (M,P)$ by the rule $(f,h)\longmapsto f\circ h,$ where $h \in \mathcal{D},$ $f\in C^\infty (M,P).$
For $f\in C^{\infty}(M,P),$ let $\mathcal{O}(f)$ denote the orbit of $f$ relative to the specified action.
By $\mathcal{M}(M,P)$ we denote the set of isomorphism classes of the fundamental groups $\pi_1\mathcal{O}(f)$ of orbits of all Morse mappings $f\colon M\to P.$

S. I. Maksymenko and B. G. Feshchenko studied the sets of isomorphism classes $\mathcal{B}$ and $\mathcal{T}$ of groups generated by direct products and certain wreath products.
In this case, they succeeded to prove the inclusions $\mathcal{M}(M,P) \subset \mathcal{B}$ under the condition that $M$ is distinct from the 2-sphere $S^2$ and 2-torus $T^2$ and $\mathcal{M} (T^2, \mathbb{R})\subset \mathcal{T}.$
In the present paper, we show that these inclusions are equalities and describe some subclasses from $\mathcal{M} (M,P)$ under certain restrictions on the behavior of functions on the boundary $\partial M.$

We also prove that for any group $G \in \mathcal{B}$ $(G \in \mathcal{T})$, the center $Z(G)$ and the quotient group by the commutator subgroup $G/[G,G]$ are free Abelian groups of the same rank easily calculated by using the geometric properties of a Morse mapping $f$ such that $\pi_1\mathcal{O}(f)\simeq G.$
In particular, this rank is the first Betti number of the orbit $\mathcal{O}(f)$ of $f.$

References

B. G. Feshchenko, Deformation of smooth functions on 2-torus whose Kronrod – Reeb graphs is a tree, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 12 204 – 219 (2015).

S. I. Maksymenko, Homotopy types of right stabilizers and orbits of smooth functions on surfaces, Ukr. Math. J., 64, № 9, 1186 – 1203 (2012), https://doi.org/10.1007/s11253-013-0721-x DOI: https://doi.org/10.1007/s11253-013-0721-x

S. I. Maksymenko, B. G. Feshchenko, Smooth functions on 2-torus whose Kronrod – Reeb graph contains a cycle, Methods Funct. Anal. and Topology, 21, № 1, 22 – 40 (2015).

Sergiy Maksymenko, Deformations of functions on surfaces by isotopic to the identity diffeomorphisms, Topology and Appl., 282, 107312, 48 (2020), https://doi.org/10.1016/j.topol.2020.107312 DOI: https://doi.org/10.1016/j.topol.2020.107312

S. I. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. and Geom., 29, № 3, 241 – 285 (2006), https://doi.org/10.1007/s10455-005-9012-6 DOI: https://doi.org/10.1007/s10455-005-9012-6

Bohdan Feshchenko, Actions of finite groups and smooth functions on surfaces, Methods Funct. Anal. and Topology, 22, № 3, 210 – 219 (2016),

E. A. Kudryavtseva, Special framed Morse functions on surfaces, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 67, № 4, 14 – 20 (2012), https://doi.org/10.3103/S0027132212040031 DOI: https://doi.org/10.3103/S0027132212040031

E. A. Kudryavtseva, The topology of spaces of Morse functions on surfaces, Math. Notes, 92, № 1-2, 219 – 236 (2012), https://doi.org/10.1134/S0001434612070243 DOI: https://doi.org/10.1134/S0001434612070243

E. A. Kudryavtseva, On the homotopy type of spaces of Morse functions on surfaces, Sb. Math., 204, № 1, 75 – 113 (2013), https://doi.org/10.1070/SM2013v204n01ABEH004292 DOI: https://doi.org/10.1070/SM2013v204n01ABEH004292

E. A. Kudryavtseva, Topology of spaces of functions with prescribed singularities on the surfaces, Dokl. Akad. Nauk, 93, № 3, 264 – 266 (2016), https://doi.org/10.1134/s1064562416030066 DOI: https://doi.org/10.1134/S1064562416030066

B. Feshchenko, Deformations of smooth functions on 2-torus, Proc. Int. Geom. Cent., 12, № 3, 30 – 50 (2019), https://doi.org/10.15673/tmgc.v12i3.1528 DOI: https://doi.org/10.15673/tmgc.v12i3.1528

S. I. Maksymenko, B. G. Feshchenko, Orbits of smooth functions on 2-torus and their homotopy types, Mat. Stud., 44, № 1, 67 – 84 (2015), https://doi.org/10.15330/ms.44.1.67-83 DOI: https://doi.org/10.15330/ms.44.1.67-83

Allen Hatcher, Algebraic topology, Cambridge Univ. Press, Cambridge (2002).

S. I. Maksymenko, B. G. Feshchenko, Homotopy properties of spaces of smooth functions on 2-torus, Ukr. Math. J., 66, № 9, 1205 – 1212 (2014), https://doi.org/10.1007/s11253-015-1014-3 DOI: https://doi.org/10.1007/s11253-015-1014-3

J. D. P. Meldrum, Wreath products of groups and semigroups, Pitman Monogr. and Surv. Pure and Appl. Math., vol. 74, Longman, Harlow (1995).

Published
22.02.2021
How to Cite
Kuznietsova, I. V., and Y. Y. Soroka. “The First Betti Numbers of Orbits of Morse Functions on Surfaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 179 -00, doi:10.37863/umzh.v73i2.2383.
Section
Research articles