Mathematics: Morse theory

According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Morse originally applied his theory to geodesics (critical points of the energy functional on paths). These techniques were used in Raoul Bott's proof of his celebrated periodicity theorem.

a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction (like a horse saddle or a mountain pass).

We therefore appear to have the following rule: the topology of M^α does not change except when α passes the height of a critical point, and when α passes the height of a critical point of index γ, a γ-cell is attached to M^α. This does not address the question of what happens when two critical points are at the same height. That situation can be resolved by a slight perturbation of f. In the case of a landscape (or a manifold embedded in Euclidean space), this perturbation might simply be tilting the landscape slightly, or rotating the coordinate system.
This rule, however, is false as stated. To see this, let M equal R and let f(x)=x³. Then 0 is a critical point of f, but the topology of M^α does not change when α passes 0. In fact, the concept of index does not make sense. The problem is that the second derivative is also 0 at 0. This kind of situation is called a degenerate critical point. Note that this situation is unstable: by rotating the coordinate system under the graph, the degenerate critical point either is removed or breaks up into two non-degenerate critical points.

The index of a non-degenerate critical point b of f is the dimension of the largest subspace of the tangent space to M at b on which the Hessian is negative definite. This corresponds to the intuitive notion that the index is the number of directions in which f decreases. The degeneracy and index of a critical point are independent of the choice the local coordinate system used, as shown by Sylvester's Law.

The Morse lemma

Let b be a non-degenerate critical point of f : M → R. Then there exists a chart (x₁, x₂, ..., x_n) in a neighborhood U of b such that x_i(b)=0 for all i and

f(x) = f(b) − (x₁)² − ... − (x_α)² + (x_α+1)² + ... + (x_n)²

throughout U. Here α is equal to the index of f at b. As a corollary of the Morse lemma we see that non-degenerate critical points are isolated. (For a generalization, see Morse-Palais lemma).
For functions from R² to R with a critical point at the origin, the Morse lemma implies that after rotation of coordinates f will be of the form

f (x, y) = a + (A x 2 + B y 2) / 2 + higher order terms,

which will be degenerate if A = 0 or B = 0.
A smooth real-valued function on a manifold M is a Morse function if it has no degenerate critical points. A basic result of Morse theory says that almost all functions are Morse functions. Technically, the Morse functions form an open, dense subset of all smooth functions M → R in the C² topology. This is sometimes expressed as "a typical function is Morse." or "a generic function is Morse".
As indicated before, we are interested in the question of when the topology of M^a = f⁻¹(-∞, a] changes as a varies. Half of the answer to this question is given by the following theorem.

Theorem. Suppose f is a smooth real-valued function on M, a < b, f⁻¹[a, b] is compact, and there are no critical values between a and b. Then M^a is diffeomorphic to M^b, and M^b deformation retracts onto M^a.

It is also of interest to know how the topology of M^a changes when a passes a critical point. The following theorem answers that question.

Theorem. Suppose f is a smooth real-valued function on M and p is a non-degenerate critical point of f of index γ, and that f(p) = q. Suppose f⁻¹[q − ε, q + ε] is compact and contains no critical points besides p. Then M^{q + ε} is homotopy equivalent to M^{q − ε} with a γ-cell attached.

These results generalize and formalize the 'rule' stated in the previous section. As was mentioned, the rule as stated is incorrect; these theorems correct it.
Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an n-cell for each critical point of index n. To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level.

[edit] The Morse inequalities

Morse theory can be used to prove some strong results on the homology of manifolds. The number of critical points of index γ of f: M → R is equal to the number of γ cells in the CW structure on M obtained from "climbing" f. Using the fact that the alternating sum of the ranks of the homology groups of a topological space is equal to the alternating sum of the ranks of the chain groups from which the homology is computed, then by using the cellular chain groups (see cellular homology) it is clear that the Euler characteristic is equal to the sum

$\sum(-1)^{\gamma}C^{\gamma}\,$

where C^γ is the number of critical points of index γ. Also by cellular homology, the rank of the n^th homology group of a CW complex M is less than or equal to the number of n-cells in M. Therefore the rank of the γ^th homology group is less than or equal to the number of critical points of index γ of a Morse function on M. These facts can be strengthened to obtain the Morse inequalities:

$C^\gamma -C^{\gamma -1}+-\cdots \pm C^0 \ge {\rm{Rank}}[H_\gamma (M)]-{\rm{Rank}}[H_{\gamma -1}(M)]+- \cdots \pm {\rm{Rank}}[H_0 (M)].$

[edit] Morse homology

Morse homology is a particularly perspicuous approach to the homology of smooth manifolds. It is defined using a generic choice of Morse function and Riemannian metric. The basic theorem is that the resulting homology is an invariant of the manifold (i.e. independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities. An infinite dimensional analog of Morse homology is known as Floer homology.
Ed Witten developed another related approach to Morse theory in 1982 using harmonic functions.

[edit] Morse–Bott theory

The notion of a Morse function can be generalized to consider functions that have degenerate critical points.

[edit] Definition

A Morse–Bott function is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the Hessian at critical points is non-degenerate in every direction, i.e., has no kernel).
The index is most naturally thought of as a pair

$(i_-, i_+),\$

where i₋ is the dimension of the unstable manifold at a given point of the critical manifold, and i₊ is i₋ plus the dimension of the critical manifold. If the Morse-Bott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between i₋ and i₊).
Morse-Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. Raoul Bott used Morse-Bott theory in his original proof of the Bott periodicity theorem.
Round functions are examples of Morse-Bott functions, where the critical sets are (disjoint unions of) circles.
Morse homology can also be formulated for Morse-Bott functions; the differential in Morse-Bott homology is computed by a spectral sequence. Frederic Bourgeois developed a neat approach in the course of his work on a Morse-Bott version of symplectic field theory.

[edit] See also

[edit] References

Bott, Raoul (1988). Morse Theory Indomitable. Publications Mathématiques de l'IHÉS. 68, 99–114.
Bott, Raoul (1982). Lectures on Morse theory, old and new., Bull. Amer. Math. Soc. (N.S.) 7, no. 2, 331–358.
Cayley, Arthur (1859). On Contour and Slope Line. The Philosophical Magazine 18 (120), 264-268.
Guest, Martin (15 April 2001), Morse Theory in the 1990's, http://www.comp.metro-u.ac.jp/~martin/RESEARCH/mg.ps, survey article; arXiv abstract.
Matsumoto, Yukio (2002). An Introduction to Morse Theory
Maxwell, James Clerk (1870). On Hills and Dales. The Philosophical Magazine 40 (269), 421–427.
Milnor, John (1963). Morse Theory. Princeton University Press. ISBN 0-691-08008-9. A classic advanced reference in mathematics and mathematical physics.
Milnor, John (1965). Lectures on the h-Cobordism theorem - scans available here
Morse, Marston (1934). "The Calculus of Variations in the Large", American Mathematical Society Colloquim Publication 18; New York.
Matthias Schwarz: Morse Homology, Birkhäuser, 1993.
Seifert, Herbert & Threlfall, William (1938). Variationsrechnung im Grossen
Witten, Edward (1982). Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661–692.

Saturday, October 3, 2009

Morse theory