Singular levels and topological invariants of Morse--Bott foliations on non orientable surfaces

José Martínez-Alfaro, Ingrid S. Meza-Sarmiento, Regilene D. S. Oliveira



We investigate the classification of closed curves and eight curves of saddle points defined on non-orientable closed surfaces, up to an ambient homeomorphism. The classification obtained here is applied to Morse-Bott foliations on non-orientable closed surfaces in order to define a complete topological invariant.


Morse-Bott functions; topological invariants; foliations; non-orientable surfaces

Full Text:



F.D. Ancel and C.R. Guilbault, An extension of Rourke’s proof that Ω3 = 0 to nonorientable manifolds, Proc. Amer. Math. Soc. 115 (1992), 283–291.

V.I. Arnold, Topological classification of Morse functions and generalizations of Hilbert’s 16-th problem, Math. Phys. Anal. Geom. 10 (2007), 227–236.

A. Banyaga and D. Hurtubise, A proof of the Morse–Bott lemma, Expo. Math. 22 (2004), 365–373.

A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, Boca Raton, Fla., Chapman & Hall/CRC, 2004.

R. Bott, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 331–358.

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, 2012.

R. Ghrist, Barcodes: The persistent topology of data, Bull. Bull. Amer. Math. Soc. 45 (2008), 67–75.

P. Giblin, Graphs, Surfaces and Homology, Cambridge University Press, New York, 2010.

D.P. Lychak and A.O. Prishlyak, Morse functions and flows on nonorientable surfaces, Methods Funct. Anal. Topology 15 (2009), 251–258.

T. Machon and G.P. Alexander, Knots and nonorientable surfaces in chiral nematics, Proc. Natl. Acad. Sci. USA 110 (2013), 14174–14179.

S.I. Maksymenko, Functions with isolated singularities on surfaces, geometry and topology of functions on manifolds, Proc. Inst. Math. Ukrainian National Academy of Sciences 7 (2010), 7–66.

S.I. Maksymenko, Singular levels and topological invariants of Morse–Bott integrable systems on surfaces, J. Differential Equations 260 (2016), 688–707.

S.I. Maksymenko, Topological classification of simple Morse–Bott functions on surfaces, Real and Complex Singularities. Contemp. Math., vol. 675, Amer. Math. Soc., Providence, RI, 2016, 165–179.

J. Matous̆ek, E. Sedgwick, M. Tancer and U. Wagner, Untangling two systems of noncrossing curves, Israel J. Math. 212 (2016), 37–79.

N.D. Mermin, The topological theory of defects in ordered media, Rev. Modern Phys. 51 (1979), 591–648.

J.R. Munkres, Elements of Algebraic Topology, Perseus Books Publishing, L.L.C., Boulder, 1984.

D. Neumann and T. O’Brien, Global structure of continuous flows on 2-manifolds, J. Differential Equations 22 (1976), 89–110.

L.I. Nicolaescu, An Invitation to Morse Theory, Springer, New York, London, 2007.

A.A. Oshemkov and V.V Sharko, Classification of Morse–Smale flows on two-dimensional manifolds, Sb. Math. 189 (1998), 1205–1250.

M.M. Peixoto, On the classification of flows on two-manifolds, Dynamical Systems, (M.M. Peixoto, ed.), Academic Press, New York, 1973, 389–419.

G. Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C.R. Acad. Sci. Paris Sér. 222 (1946), 847–849.

O. Saeki, Topology of Singular Fibers of Differentiable Maps, Lecture Notes in Mathematics, vol. 1854, Springer, Berlin, 2004.

V.V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukrainian Math. J. 55 (2003), 832–846.

A. Shima, A Klein bottle whose singular set consists of three disjoint simple closed curves, Knots in Hellas’98 24 (2000), 411–435.

J.W.T. Youngs, The extension of a homeomorphism defined on the boundary of a 2manifold, Bull. Amer. Math. Soc. 54 (1948), 805–808.


  • There are currently no refbacks.

Partnerzy platformy czasopism