Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere
| dc.contributor.author | Anna Kravchenko, Sergiy Maksymenko | |
| dc.date.accessioned | 2019-05-07T13:57:12Z | |
| dc.date.available | 2019-05-07T13:57:12Z | |
| dc.date.issued | 2019 | |
| dc.description.abstract | Let $M$ be a compact two-dimensional manifold and, $f in C^{infty}(M, R)$ be a Morse function, and $Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{infty}$, and by $S(f)={hin D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $hin S(f)$ induces an automorphism of the graph $Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere. The present paper is devoted to the case $M = S^2$. In this situation $Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 to R$ whose fixed subtree $Fix(G)$ consists of more than one point. | |
| dc.identifier.issn | 2409-8906 | |
| dc.identifier.uri | https://card-file.ontu.edu.ua/handle/123456789/8167 | |
| dc.identifier.uri | https://doi.org/10.15673/tmgc.v11i4.1306 | |
| dc.source | Proceedings of the International Geometry Center | |
| dc.title | Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere |