Morse-Bott functions with two critical values on a surface

Irina Gelbukh

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 3, page 865-880
  • ISSN: 0011-4642

Abstract

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We study Morse-Bott functions with two critical values (equivalently, nonconstant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.

How to cite

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Gelbukh, Irina. "Morse-Bott functions with two critical values on a surface." Czechoslovak Mathematical Journal 71.3 (2021): 865-880. <http://eudml.org/doc/297750>.

@article{Gelbukh2021,
abstract = {We study Morse-Bott functions with two critical values (equivalently, nonconstant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.},
author = {Gelbukh, Irina},
journal = {Czechoslovak Mathematical Journal},
keywords = {Morse-Bott function; height function; surface; critical value; Reeb graph},
language = {eng},
number = {3},
pages = {865-880},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Morse-Bott functions with two critical values on a surface},
url = {http://eudml.org/doc/297750},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Gelbukh, Irina
TI - Morse-Bott functions with two critical values on a surface
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 3
SP - 865
EP - 880
AB - We study Morse-Bott functions with two critical values (equivalently, nonconstant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.
LA - eng
KW - Morse-Bott function; height function; surface; critical value; Reeb graph
UR - http://eudml.org/doc/297750
ER -

References

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  1. Banyaga, A., Hurtubise, D. E., 10.1016/S0723-0869(04)80014-8, Expo. Math. 22 (2004), 365-373. (2004) Zbl1078.57031MR2075744DOI10.1016/S0723-0869(04)80014-8
  2. Banyaga, A., Hurtubise, D. E., 10.1017/S0143385708000928, Ergodic Theory Dyn. Syst. 29 (2009), 1693-1703. (2009) Zbl1186.37038MR2563088DOI10.1017/S0143385708000928
  3. Duan, H., Rees, E. G., Functions whose critical set consists of two connected manifolds, Bol. Soc. Mat. Mex., II. Ser. 37 (1992), 139-149. (1992) Zbl0867.57025MR1317568
  4. Gelbukh, I., 10.1007/s10587-013-0034-0, Czech. Math. J. 63 (2013), 515-528. (2013) Zbl1289.57009MR3073975DOI10.1007/s10587-013-0034-0
  5. Gelbukh, I., 10.1515/ms-2016-0298, Math. Slovaca 67 (2017), 645-656. (2017) Zbl1424.14003MR3660746DOI10.1515/ms-2016-0298
  6. Gelbukh, I., 10.2298/FIL1907031G, Filomat 33 (2019), 2031-2049. (2019) MR4036359DOI10.2298/FIL1907031G
  7. Hurtubise, D. E., Three approaches to Morse-Bott homology, Afr. Diaspora J. Math. 14 (2012), 145-177. (2012) Zbl1311.57043MR3093241
  8. Jiang, M.-Y., 10.12775/TMNA.1999.007, Topol. Methods Nonlinear Anal. 13 (1999), 147-161. (1999) Zbl0940.57034MR1716589DOI10.12775/TMNA.1999.007
  9. Kudryavtseva, E. A., 10.1070/SM1999v190n03ABEH000392, Sb. Math. 190 (1999), 349-405. (1999) Zbl0941.57026MR1700994DOI10.1070/SM1999v190n03ABEH000392
  10. Leininger, C. J., Reid, A. W., 10.2140/agt.2002.2.37, Algebr. Geom. Topol. 2 (2002), 37-50. (2002) Zbl0983.57001MR1885215DOI10.2140/agt.2002.2.37
  11. Martínez-Alfaro, J., Meza-Sarmiento, I. S., Oliveira, R. D. S., 10.1090/conm/675/13590, Real and Complex Singularities Contemporary Mathematics 675. American Mathematical Society, Providence (2016), 165-179. (2016) Zbl1362.37078MR3578724DOI10.1090/conm/675/13590
  12. Martínez-Alfaro, J., Meza-Sarmiento, I. S., Oliveira, R. D. S., 10.12775/TMNA.2017.051, Topol. Methods Nonlinear Anal. 51 (2018), 183-213. (2018) Zbl1393.37057MR3784742DOI10.12775/TMNA.2017.051
  13. Milnor, J. W., 10.1515/9781400881802, Annals of Mathematics Studies 51. Princeton University Press, Princeton (1963). (1963) Zbl0108.10401MR0163331DOI10.1515/9781400881802
  14. Nicolaescu, L. I., 10.1007/978-1-4614-1105-5, Universitext. Springer, Berlin (2011). (2011) Zbl1238.57001MR2883440DOI10.1007/978-1-4614-1105-5
  15. Panov, D., Immersion in R 3 of a Klein bottle with Morse-Bott height function without centers, MathOverflow Available at https://mathoverflow.net/q/343792 2019. 
  16. Prishlyak, A. O., 10.1016/S0166-8641(01)00077-3, Topology Appl. 119 (2002), 257-267. (2002) Zbl1042.57021MR1888671DOI10.1016/S0166-8641(01)00077-3
  17. Rot, T. O., 10.1016/j.crma.2016.08.003, C. R., Math., Acad. Sci. Paris 354 (2016), 1026-1028. (2016) Zbl1350.57035MR3553908DOI10.1016/j.crma.2016.08.003
  18. Saeki, O., 10.1093/imrn/rnaa301, (to appear) in Int. Math. Res. Not. DOI10.1093/imrn/rnaa301

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