Equivalence of Morse functions and Morse-Smale flows on 3-manifolds
by
Alexander O. Prishlyak
Kiev University, Ukraine

Let M is a closed smooth n-manifold

Recall that a function f is a Morse function if all its critical points are isolated and nondegenerate. Two functions f, g are topologically (smoothly) equivalent if there exist a homeomorphisms (diffeomorphisms) h: M --> M, h':R --> R such that fh=h'g.

A gradient like Morse-Smale flow on M is a flow, which satisfy next conditions: 1) flow has a finite number of fixed points and all of them are hyperbolic; 2)alpha- and omega- limit sets of trajectories are fixed points; 3)the stable and unstable manifolds have transversal intersections. Two flows are topologically equivalent if there exist homeomorphism of manifolds, which maps trajectories onto trajectories preserving it orientations. The gradient like Morse-Smale flow on M is called ordered if the order of fixed points is given. Such flows are equivalent if there exist topological equivalence between them that preserve the order of fixed points.

A handle of index k (k-handle) h^k = D^k ×D^n-k attached to manifold M with boundary is the union M \cup _f h^k along the boundary of M according to an embedding f: S^k-1 ×D^n-k --> \partialM A handles decompositions is a sequence of imbeddings M₀ subset M₁ subset ... subset M_N = M such that M₀ is a union of n-disks (0-handles), M_i+1 is obtained from M_i by gluing handle.

An order on the set X is the map onto X --> {1, 2, ...., N}. For Morse function f using h':R --> R we can obtain function with set {1, 2, ...., N} as critical value set, so there is natural order on the set of critical points of Morse function. For almost all Riemann metrics on M the gradient flow grad f is a gradient like Morse-Smale flow. We order the fixed points of this flow in the order of critical value of correspondent critical points of Morse function. Conversely, having gradient like Morse-Smale flows one can construct Morse function for which this flow is gradient flow [Smale].

Proposition. Two Morse functions are topologically equivalent iff there exists Reamann metrics in which its gradient flow is equivalent as ordered gradient like MS flows.

Let us construct such handle decomposition that core and cocore of a handle lie in stable and unstable manifold of correspondent fixed points of flow. The function induce the order of handles such that if the handle h₁ was attached before the handle h₂ then the order of h₁ is no more than the order of h₂.

Lemma. Two Morse function are topological equivalent iff in the handles decomposition which associated with it correspondent handles are gluing on isotopic imbeddings.

Using isotopies of attaching spheres one can made handles decomposition such that attaching and belt spheres has transversal intersections and such that each handle attach to the union of less dimensional handles. We call such handles decomposition simple.

The handles decomposition is called ordered if the map of handles set to set {1, 2, ..., N} is given. Each Morse function assign order in the corresponding handles decomposition: we can choose such h': R¹ --> R¹ that critical values set maps on the set of numbers {1, 2, ..., N} we put in correspondence to each handles the number of correspondent critical points number.

Ordered simple handles decompositions (OSHD) are isomorphic if there exist homeomorphism of manifolds which maps handles on handles, cores on cores, cocores on cocores and preserve order of handles. Denote by M^k the union of handles which indexes are no more then k and L^k = \partial M^k.

Theorem. Two Morse function are topological equivalent iff from first function OSHD we can obtain second function OSHD using 1) attaching spheres isotopies in L^k with support in the boundary of handles with less numbers; 2) replacements handle H_i by H_i#H_j, if the number of H_i is more than the number of H_j (handles have the same index). The new handle H_i#H_j has the same number as the handle H_i.

Let N₀ \cup N₁=M be a Heegaard decomposition of the 3-manifold M, F=\partial N₀ = \partial N₁ be the common surface of N and N'. The set u={u₁, u₂, ..., u_n} of non intersected closed curves on surface F are called generalized meridian system for N₀, if it is the boundary of disks D_i subset H and if we cut H along disks D_i we obtain disconnected union of 3-disks. Let v={v₁, v₂, ..., v_m} be the generalized meridian system for N₁.

The triple (F, u, v) is called the generalized Heegaard diagram of manifold M. Diagrams (F, u, v) and (F', u', v') are called homeomorphic if there exist such homeomorphism h: F --> F, that h(u)=u', h(v)=v'. Diagrams (F, u, v) and (F', u', v') are called semiisotopic, if there exist such isotopies j_t, \psi_t :F --> F, that j₀=\psi₀ = 1, j₁(u)=u', \psi₁(v)=v'.

Let us define the operation of meridian addition: the sum u₁# u₂ of two meridian u₁ and u₂ along simple curve \alpha, which connect u₁ and u₂ is a such component of the union neighborhood boundary \partialU(u₁ \cup u₂ \cup \alpha) which don't isotopic neither u₁ not u₂.

Denote by U₁, U₂, ..., U_k that domains which are obtained after cutting surface F along meridians u₁, u₂, ..., u_n and by V₁, V₂, ..., V_l - correspondent domains for meridians v₁, v₂, ..., v_m. Diagram is called ordered if the map \sigma of set {U₁, U₂, ..., U_k, u₁, u₂, ..., u_n, v₁, v₂, ..., v_m, V₁, V₂, ..., V_l} on set {1, 2, ..., N} is given.

Ordered generalized Heegaard diagrams (OGHD) are called equivalent if one from another can be obtained using homeomorphisms, semiisotopies of diagrams (finger moves between u_i and v_j, if \sigma(u_i) > \sigma(v_j)), replacement meridians u_i on u_i# u_j if \sigma(u_i) < \sigma(u_j) and replacement meridians v_i on v_i# v_j if \sigma(v_i) > \sigma(v_j). If we replace u_i on u_i# u_j then \sigma(u_i# u_j) = \sigma(u_i).

Theorem. Two Morse functions on 3-manifolds are topological equivalent iff associated ordered generalized Heegaard diagrams are equivalent.

Theorem. Two gradient like Morse-Smale flows on 3-manifolds are topologically equivalent iff associated generalized Heegaard diagrams are homeomorphic.

Theorem. Two Morse functions f and f' on 3-manifolds can be connected by path in the space of Morse functions iff surfaces F and F' are isotopic in M.

References:

1. S.Smale. On gradient dynamical systems // Ann.Math., v.74, N.1, 1961, - P. 199-206.

2. A.O.Prishlyak. Classification of 3-dimensional gradient like Morse-Smale dynamical systems // Some Problems of Modern Mathematics. Proc. of Inst. of Math. Ukr.Nat.Ac.Sc., v.25, Kiev, 1998, - P.305-313. [In Russian, Engl.Transl: math.DS/9901002]

3. A.O.Prishlyak. Conjugation of Morse function // Some Problems of Modern Mathematics. Proc. of Inst. of Math. Ukr.Nat.Ac.Sc., v.25, Kiev, 1998, - P.319-325. [In Russian, Engl.Transl: math.GT/9812150]

4. V.V.Sharko. Functions on manifolds (algebraic and topological aspects).- Kiev: Naukova dumka, 1990.-196p.

Paper reference: arXiv:math.DS/9901002

Date received: June 24, 1999