NationMaster - Encyclopedia: Morse theory

A Morse function is also an expression for an anharmonic oscillator Anharmonicity is the deviation of a system from being a harmonic oscillator. ...

In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. 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. Before Morse, Arthur Cayley and James Clerk Maxwell developed some of the ideas of Morse theory in the context of topography. 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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Harald Calvin Marston Morse (24 March 1892 - 22 June 1977) was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... The handle decomposition of an n-manifold M is a representation of that manifold as an exhaustion where each is obtained from by attaching a -handle. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematical physicist, born in Edinburgh. ... Surface of the Earth Topography, a term in geography, has come to refer to the lay of the land, or the physiogeographic characteristics of land in terms of elevation, slope, and orientation. ... Raoul Bott (Harvard University News Office) Raoul Bott, FRS (born September 24, 1923, died December 20, 2005) was a mathematician known for numerous basic contributions to geometry in its broad sense. ... In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory. ...

1 Basic concepts
2 Formal development
3 See also
4 Further reading

Basic concepts

A saddle point

Consider, for purposes of illustration, a mountainous landscape M. If f is the function M → R sending each point to its elevation, then the inverse image of a point in R (a level set) is simply a contour line. Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... Partial plot of a function f. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... Elevation contour map A contour line (also level set, isopleth, isogram or isarithm) for a function of two variables is a curve connecting points where the function has a same particular value. ...

Contour lines around a saddle point

Imagine flooding this landscape with water. Then, assuming the ground is porous, the region covered by water when the water reaches an elevation of a is f⁻¹ (-∞, a], or the points with elevation less than or equal to a. Consider how the topology of this region changes as the water rises. It appears, intuitively, that it does not change except when a passes the height of a critical point; that is, a point where the gradient of f is 0. In other words, it does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak. Image File history File links By submitter. ... Image File history File links By submitter. ... In mathematics, a critical point (or critical number) is a point on the domain of a function where the derivative is equal to zero. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... In a range of hills, or especially of mountains, a pass (also gap, notch, col, saddle, bwlch or bealach) is a lower point that allows easier access through the range. ...

The torus

To each of these three types of critical points - basins, passes, and peaks (also called minima, saddles, and maxima) - one associates a number called the index. Intuitively speaking, the index of a critical point b is the number of independent directions around b in which f decreases. Therefore, the indices of basins, passes, and peaks are 0, 1, and 2, respectively. Image File history File links Created by me. ... Image File history File links Created by me. ...

Define M^a as f⁻¹(-∞, a]. Leaving the context of topography, one can make a similar analysis of how the topology of M^a changes as a increases when M is a torus oriented as in the image and f is projection on a vertical axis, taking a point to its height above the plane.

These figures are homotopy equivalent

When a is less than 0, M^a is the empty set. After a passes the level of p (a critical point of index 0), when 0<a<f(q), then M^a is a disk, which is homotopy equivalent to a point, (a 0-cell) which has been "attached" to the empty set. Next, when a exceeds the level of q (a critical point of index 1), and f(q) <a<f(r), then M^a is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once a passes the level of r (a critical point of index 1), and f(r)<a<f(s), then M^a is a torus with a disk removed, which is homotopy equivalent to a cylinder with a 1-cell attached (image at right). Finally, when a is greater that the critical level of s (a critical point of index 2) M^a is a torus. A torus, of course, is the same as a torus with a disk removed with a disk (a 2-cell) attached. Image File history File links By submitter. ... Image File history File links By submitter. ... Image File history File links By submitter. ... Image File history File links By submitter. ...

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. In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

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.

Formal development

For a smooth function f from a differentiable manifold M to the reals the points where the gradient of f in a local coordinate system is 0 are called critical points and their images under f are called critical values. If at a critical point b the matrix of second partial derivatives (the Hessian matrix) is non-singular, then b is called a non-degenerate critical point; if the Hessian is singular then b is a degenerate critical point. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Chemistry In chemistry, a critical point is the conditions ( temperature, pressure) at which the liquid state of the matter ceases to exist. ... In differential topology, a critical value of a differentiable map between differentiable manifolds is the image of a critical point. ... In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...

For the functions

f (x) = a + b x + c x 2 + d x 3 + ...

from R to R, f has a critical point at the origin if b=0, which is non-degenerate if c≠0,(f is of the form a+cx²+...) and degenerate if c=0,(f is of the form a+dx³+...). A less trivial example of a degenerate critical point is the origin of the monkey saddle. In mathematics, the monkey saddle is the surface defined by the equation z = x3 âˆ’ 3xy2. ...

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. It is easy to see that this corresponds to the intuitive notion that the index is the number of directions in which f decreases. In mathematics, an index is a superscript or subscript to a symbol. ...

The Morse Lemma

Let b be a non-degenerate critical point of f. 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 In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

f=f(b)−(x₁)²− ... −(x_α)²+(x_α+1)²+ ... +(x_n)²+ higher order terms

throughout U and 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 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". Look up Generic in Wiktionary, the free dictionary. ...

As indicated before, we are interested in the question of when the topology of M^α changes as α 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^α changes when α 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 for ε sufficiently small 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.

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 In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

∑ (-1)^γC^γ,

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:

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

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. A Morse function on a manifold gives rise to a homology theory isomorphic to singular homology. ... Homology is an important concept in several disciplines: Homology (anthropology) in archaeology and anthropology. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, Floer homology refers to a family of homology theories which share similar characteristics and are believed by experts to be closely related. ...

Ed Witten developed another related approach to Morse theory in 1982 using harmonic functions. Edward Witten at Harvard University Edward Witten (born August 26, 1951) is a professor at the Institute for Advanced Study. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

Morse-Bott Theory

The notion of a Morse function can be generalized to consider functions that have degenerate critical manifolds; a Morse function is the special case where the critical manifolds are zero-dimensional. 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.

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. Raoul Bott (Harvard University News Office) Raoul Bott, FRS (born September 24, 1923, died December 20, 2005) was a mathematician known for numerous basic contributions to geometry in its broad sense. ... In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory. ...

See Round function, for an instance. In topology and in calculus, a round function is a scalar function , over a manifold , whose critical points form one or several connected components, each homeomorphic to the circle , also called critical loops. ...

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. A Morse function on a manifold gives rise to a homology theory isomorphic to singular homology. ... In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (En,dn) such that En+1 = H(En) = ker dn / im dn is the homology of En. ...

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