Special framed Morse functions on surfaces

Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M and $\mathbb{F}^1$ be the space of framed Morse functions both endowed with the C ^??-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping is a homotopy equivalence. In the case when at least x(M) + 1 critical points of each function of F are marked, the homotopy equivalences and are proved, where is the complex of framed Morse functions, is the universal moduli space of framed Morse functions, is the group of self-diffeomorphisms of M homotopic to the identity.     

Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f ⁻¹(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.     

Uniform Morse lemma and isotopy criterion for Morse functions on surfaces

Let M be a smooth compact (orientable or not) surface with or without a boundary. Let $ \mathcal{D}_0 $ \mathcal{D}_0 ⊂ Diff(M) be the group of diffeomorphisms homotopic to id _M . Two smooth functions f, g: M → ℝ are called isotopic if f = h ₂ ℴ g ℴ h ₁ for some diffeomorphisms h ₁ ∈ $ \mathcal{D}_0 $ \mathcal{D}_0 and h ₂ ∈ Diff⁺(ℝ). Let F be the space of Morse functions on M which are constant on each boundary component and have no critical points on the boundary. A criterion for two Morse functions from F to be isotopic is proved. For each Morse function f ∈ F, a collection of Morse local coordinates in disjoint circular neighborhoods of its critical points is constructed, which continuously and Diff(M)-equivariantly depends on f in C ^∞-topology on F (“uniform Morse lemma”). Applications of these results to the problem of describing the homotopy type of the space F are formulated.     

The number of critical elements of discrete Morse functions on non-compact surfaces

This paper is focused on looking for links between the topology of a connected and non-compact surface with finitely many ends and any proper discrete Morse function which can be defined on it. More precisely, we study the non-compact surfaces which admit a proper discrete Morse function with a given number of critical elements. In particular, given any of these surfaces, we obtain an optimal discrete Morse function on it, that is, with the minimum possible number of critical elements.     

The algebraic construction of the Novikov complex of a circle-valued Morse function

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction of the real-valued Morse function to a fundamental domain of the pullback infinite cyclic cover of M. Received: 23 November 2000 / Revised version: 3 May 2001 / Published online: 28 February 2002     

Morse functions on cobordisms

We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of the exact values of Morse numbers of smooth cobordisms.     

Morse functions statistics

We answer a question of Arnold concerning the growth rate of the number of Morse functions on the two-sphere. This work was supported in part by the NSF (Grant No. DMS-0303601).     

Deformations of smooth toric surfaces

For a complete, smooth toric variety Y, we describe the graded vector space ${T_Y^1}$ . Furthermore, we show that smooth toric surfaces are unobstructed and that a smooth toric surface is rigid if and only if it is Fano. For a given toric surface we then construct homogeneous deformations by means of Minkowski decompositions of polyhedral subdivisions, compute their images under the Kodaira-Spencer map, and show that they span ${T_Y^1}$ .     

Linear independence of Morse functions on manifolds

It is proved that on a smooth n-dimensional manifold there exist n linearly independent Morse functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 835–838, June, 1990.     

Morse inequalities for almost-periodic functions

Translated from Matematicheski Zametki, Vol. 50, No. 2, pp. 146–151, August, 1991.     

Characterizing equivalent discrete Morse functions

We get a characterization theorem for equivalent discrete Morse functions defined on simplicial complexes in terms of their gradient vector field. As a consequence, we also characterize them in the 1-dimensional case by using critical elements. The authors are partially supported by the P.A.I. project (Junta de Andalucia, SPAIN, 2009/FQM-189 and 327) and by the MEC-FEDER grants MTM2007-61284 and MTM2007-65726 (MEC, SPAIN, 2007).     

Deformation of Morse functions. I

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 237–243, February, 1989.     

Minimal Morse functions on a pair of manifolds

The existence theorem for a minimal Morse function on a pair of manifolds (M ⁿ,N ^k), wheren-k 3,k 6, is proved.Translated from Ukrainskii Matematiccheskii Zhurnal, Vol. 45, No. 1, pp. 143–144, January, 1993.     

Morse theory for minimal surfaces in manifolds

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.     

The number of excellent discrete Morse functions on graphs

In Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S² was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S¹, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph.