Global classification of Morse-Smale gradient systems and adapted energy functions: the open core of the program announced in Smale's On Gradient Dynamical Systems (1961)

Smale's 1961 paper On Gradient Dynamical Systems proved that a gradient flow of a Morse function can be $C^1$-approximated by a field with no recurrence and pairwise transverse (normal) intersections of stable and unstable manifolds, that such gradient-like fields are realized by global Morse functions whose critical values equal the corresponding Morse indices with prescribed values on the two parts of the boundary, and deduced the existence of self-indexing Morse functions on compact manifolds with two-part boundary. The paper closed by announcing consequences of these theorems for ordinary differential equations (dynamical systems) and for topology. This record reconstructs the fate of that announcement: the topological branch was completed by 1962-1965 (high-dimensional Poincare conjecture, h-cobordism theorem, textbook self-indexing), and the dynamical branch grew into Morse-Smale theory with Peixoto's surface classification and the Palis-Smale structural stability theorem. The surviving open core is the global classification program the announcement named: complete topological classification of Morse-Smale gradient and gradient-like systems in dimension at least 3 beyond the solved subclasses, together with necessary and sufficient conditions for such systems to admit the adapted, self-indexing-type energy functions the 1961 construction provides.

Background

Let $M^n$ be a compact smooth ($C^\infty$) manifold, possibly with nonempty boundary. A smooth function $f:M\to\mathbb{R}$ is a Morse function if every critical point (a point where the differential $df$ vanishes) is nondegenerate, meaning the matrix of second derivatives at that point is invertible; the Morse index of such a point is the number of negative eigenvalues of that matrix. After choosing a Riemannian metric (a smoothly varying inner product on tangent spaces), the gradient $\operatorname{grad}f$ is the vector field metrically dual to $df$, and the ordinary differential equation $\dot{x}=\operatorname{grad}f(x)$ defines the gradient flow: $f$ strictly increases along nonconstant orbits, so the flow has no recurrence and every orbit tends to critical points forward and backward in time. The stable manifold of a fixed point is the set of points whose forward orbits tend to it; the unstable manifold is defined with time reversed. Smale's 1961 Annals paper On Gradient Dynamical Systems studied these flows on manifolds with boundary and proved three results. Theorem A: if $f$ is a Morse function with distinct critical values whose gradient is transverse to (nowhere tangent to) the boundary, then every $C^1$-neighborhood of $\operatorname{grad}f$ contains a vector field that agrees with $\operatorname{grad}f$ near the critical points, is transverse to the boundary, has all forward and backward limit sets contained in its singularities, and has every unstable manifold meeting every stable manifold with normal intersection, that is, transversally (tangent spaces at a common point span the tangent space of $M$, equivalently $\dim W^u_i+\dim W^s_j-n$ equals the local dimension of the intersection). Theorem B: conversely, a vector field with these properties is realized, up to the choice of metric, by a global Morse function whose critical points are exactly its singularities, which is transverse to the level sets off the critical points, whose critical value at each critical point equals its Morse index, and which takes the prescribed constant values $-\tfrac12$ and $n+\tfrac12$ on the two closed parts $V_1,V_2$ of the boundary. Theorem C: consequently every compact manifold whose boundary is the disjoint union of two closed parts admits a self-indexing Morse function, one with $f(\beta)=\operatorname{ind}(\beta)$ at each critical point and the boundary values above. Sublevel sets of a self-indexing Morse function yield a handle decomposition (handlebody decomposition) of the manifold, attaching one handle, a thickened disk, of dimension equal to the index per critical point; rearranging such decompositions is the technical core of the h-cobordism theorem, which gives a criterion for a cobordism, a compact manifold whose boundary splits into two parts, to be a product, and of Smale's proof of the generalized Poincare conjecture in dimensions greater than four. A dynamical system is called Morse-Smale when it has finitely many hyperbolic fixed or periodic points (hyperbolic meaning that the eigenvalues of the linearization at a fixed point all have nonzero real part, and, for a periodic point, that the eigenvalues of the return map around the orbit all have modulus different from one), no other nonwandering points (so no recurrent behavior beyond the listed periodic points and no cycles among the connecting orbits), and transversal intersections of all stable and unstable manifolds; a cascade is the discrete-time system generated by iterating one diffeomorphism. A Morse-Smale system is gradient-like when it has no periodic orbits at all, so that its nonwandering set is exactly a finite set of fixed points; an orbit whose forward limit is one fixed point and whose backward limit is a different one is a heteroclinic orbit; the stable and unstable manifolds of saddle fixed points are traditionally called separatrices; and in dimension at least 3 these invariant manifolds can be wildly embedded, meaning not locally flat (no neighborhood of a point makes the pair of submanifold and ambient manifold equivalent to a pair of flat disks), so that the ambient embedding, not only the abstract orbit data, carries dynamical information. Two systems are topologically equivalent (flows) or topologically conjugate (cascades) if a homeomorphism of the ambient manifold maps orbits to orbits preserving time orientation or the discrete time; the global dynamical classification problem for a class of systems asks for a computable invariant complete for this equivalence relation, together with the characterization of which manifolds admit systems with given data. An energy function (Lyapunov function) for a system is a function decreasing along all nonconstant orbits whose critical set is exactly the fixed or periodic (nonwandering) set; for a gradient system it is the defining function itself, while for a general Morse-Smale cascade its existence is a nontrivial topological question. The paper closed by announcing that its theorems have consequences for both the theory of ordinary differential equations (dynamical systems) and topology, to be developed in subsequent work. The topological branch was delivered within a few years by Smale himself and codified by Milnor; the dynamical branch became Morse-Smale theory through Peixoto's surface theory, Smale's 1967 survey naming the class, the Palis-Smale structural stability theorem (structural stability means all sufficiently $C^1$-small perturbations of the system are topologically equivalent to it), and the Morse-homology developments following Witten. What survives from the announcement is the classification and energy-function program stated below.

Problem Statement

Carry out the dynamical branch of the program announced at the end of Smale's 1961 paper On gradient dynamical systems, namely the systematic application of its existence and approximation theorems to global dynamical classification problems. In the surviving, precise form of that question: (1) classify, up to topological equivalence (for flows) or topological conjugacy (for cascades), the gradient and gradient-like Morse-Smale systems to which the approximation and realization Theorems A and B apply, on manifolds of dimension at least 3: construct a complete invariant, one that is necessary and sufficient for equivalence or conjugacy, that covers systems with heteroclinic intersections of positive dimension and wildly embedded separatrices, beyond the subclasses already classified (all surface systems; gradient-like cascades and cascades without heteroclinic curves on 3-manifolds; orientation-preserving Morse-Smale diffeomorphisms on closed orientable 3-manifolds; Morse-Smale diffeomorphisms of $S^n$, $n>3$, whose saddle invariant manifolds are pairwise disjoint), and derive from the classification the resulting topological characterization of, and restrictions on, the underlying manifolds; and (2) establish necessary and sufficient conditions under which a given Morse-Smale system admits the dynamically adapted Morse-theoretic data that Theorems B and C construct in the gradient case, an energy function whose critical set is exactly the fixed or periodic set, with the self-indexing or dynamically ordered arrangement of critical values, settling in particular the cases of wild invariant-manifold embeddings and of dimension at least 4. These two strands share one objective: completing the dictionary, opened by the 1961 theorems, between the dynamics of the simplest structurally stable systems and the topology of the underlying manifold.

References

1. Stephen Smale, On Gradient Dynamical Systems, Annals of Mathematics, Second Series 74 (1961) 199-206, DOI 10.2307/1970311, https://doi.org/10.2307/1970311 (zbMATH review record Zbl 0136.43702).
2. Stephen Smale, Morse inequalities for a dynamical system, Bulletin of the American Mathematical Society 66 (1960) 43-49, DOI 10.1090/S0002-9904-1960-10386-2, https://doi.org/10.1090/S0002-9904-1960-10386-2.
3. Stephen Smale, Generalized Poincare's Conjecture in Dimensions Greater Than Four, Annals of Mathematics, Second Series 74 (1961) 391-406, DOI 10.2307/1970239, https://doi.org/10.2307/1970239.
4. S. Smale, On the Structure of Manifolds, American Journal of Mathematics 84 (1962) 387-399, DOI 10.2307/2372978, https://doi.org/10.2307/2372978.
5. John Milnor, Morse Theory (AM-51), Annals of Mathematics Studies 51, Princeton University Press, 1963, DOI 10.1515/9781400881802, https://doi.org/10.1515/9781400881802.
6. John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow), Princeton University Press, 1965, DOI 10.1515/9781400878055, https://doi.org/10.1515/9781400878055.
7. M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962) 101-120, DOI 10.1016/0040-9383(65)90018-2, https://doi.org/10.1016/0040-9383(65)90018-2.
8. S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society 73 (1967) 747-817, DOI 10.1090/S0002-9904-1967-11798-1, https://doi.org/10.1090/S0002-9904-1967-11798-1.
9. J. Palis and S. Smale, Structural stability theorems, in Global Analysis (Proc. Sympos. Pure Math. XIV, Berkeley, 1968), American Mathematical Society, 1970, 223-231, DOI 10.1090/pspum/014/0267603, https://doi.org/10.1090/pspum/014/0267603.
10. Edward Witten, Supersymmetry and Morse theory, Journal of Differential Geometry 17 (1982) 661-692, DOI 10.4310/jdg/1214437492, https://doi.org/10.4310/jdg/1214437492.
11. A. O. Prishlyak, The Classification of Three-Dimensional Gradient Like Morse-Smale Dynamic Systems, arXiv:math/9901002 (1999), https://arxiv.org/abs/math/9901002.
12. V. Grines, F. Laudenbach, O. Pochinka, Self-indexing Energy Function for Morse-Smale Diffeomorphisms on 3-Manifolds, Moscow Mathematical Journal 9 (2009) 801-821, DOI 10.17323/1609-4514-2009-9-4-801-821, arXiv:0804.0699, https://arxiv.org/abs/0804.0699.
13. Viatcheslav Grines, Francois Laudenbach, Olga Pochinka, Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds, arXiv:1101.6036 (2011), https://arxiv.org/abs/1101.6036.
14. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, Classification of Morse-Smale systems and topological structure of the underlying manifolds, Russian Mathematical Surveys 74 (2019) 37-110, DOI 10.1070/RM9855, https://doi.org/10.1070/RM9855.
15. C. Bonatti, V. Grines, O. Pochinka, Topological classification of Morse-Smale diffeomorphisms on 3-manifolds, Duke Mathematical Journal 168 (2019) 2507-2558, DOI 10.1215/00127094-2019-0019, arXiv:1710.08292, https://arxiv.org/abs/1710.08292.
16. V. Grines, E. Gurevich, O. Pochinka, D. Malyshev, On topological classification of Morse-Smale diffeomorphisms on the sphere $S^n$ ($n>3$), Nonlinearity 33 (2020) 7088-7113, DOI 10.1088/1361-6544/abaf60, arXiv:1911.10234, https://arxiv.org/abs/1911.10234.
17. Vladislav Kibkalo, Tomoo Yokoyama, Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows, Discrete and Continuous Dynamical Systems 42 (2022) 4787-4822, DOI 10.3934/dcds.2022072, https://doi.org/10.3934/dcds.2022072.
18. M. K. Barinova, V. Z. Grines, O. V. Pochinka, Criterion for the Existence of an Energy Function for a Regular Homeomorphism of the 3-Sphere, Proceedings of the Steklov Institute of Mathematics 321 (2023) 37-53, DOI 10.1134/S0081543823020037, https://doi.org/10.1134/S0081543823020037.
19. Olga V. Pochinka, Elena A. Talanova, Morse-Smale diffeomorphisms with non-wandering points of pairwise different Morse indices on 3-manifolds, Russian Mathematical Surveys 79 (2024) 127-171, DOI 10.4213/rm10141e, https://doi.org/10.4213/rm10141e.
20. E. Ya. Gurevich, On topological classification of flows with heteroclinic curves on four-dimensional manifolds, Mathematics and Theoretical Computer Science 3 (2025) 20-42, DOI 10.26907/2949-3919.2025.3.20-42, https://doi.org/10.26907/2949-3919.2025.3.20-42.
21. Elena Ya. Gurevich, Ilya A. Saraev, Topological Classification of Polar Flows on Four-Dimensional Manifolds, Regular and Chaotic Dynamics 30 (2025) 254-278, DOI 10.1134/S1560354725020054, https://doi.org/10.1134/S1560354725020054.
22. E. Ya. Gurevich, I. A. Saraev, Classification of gradient-like flows without heteroclinic intersections on four-dimensional manifolds, Izvestiya: Mathematics 90 (2026) 480-514, DOI 10.4213/im9689e, https://doi.org/10.4213/im9689e.