In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, Morse theory enables one to analyze the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
by studying
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s on that manifold. According to the basic insights of
Marston Morse, a typical differentiable function on a manifold will 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
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problem ...
and
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
had developed some of the ideas of Morse theory in the context of
topography
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary sc ...
. Morse originally applied his theory to
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s (
critical points of the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
on the space of paths). These techniques were used in
Raoul Bott's proof of his
periodicity theorem.
The analogue of Morse theory for complex manifolds is
Picard–Lefschetz theory.
Basic concepts
To illustrate, consider a mountainous landscape surface
(more generally, a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
). If
is the
function giving the elevation of each point, then the
inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of a point in
is a
contour line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional gr ...
(more generally, a
level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ...
). Each
connected component of a contour line is either a point, a
simple closed curve
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exteri ...
, or a closed curve with a
double point
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic cur ...
. 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, where the surrounding landscape curves up in one direction and down in the other.
Imagine flooding this landscape with water. When the water reaches elevation
, the underwater surface is
, the points with elevation
or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when
passes the height of a
Critical point (mathematics), critical point, where the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of
is
(more generally, the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
acting as a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
between
tangent spaces
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to ...
does not have maximal
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
). In other words, the topology of
does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a
mountain pass
A mountain pass is a navigable route through a mountain range or over a ridge. Since many of the world's mountain ranges have presented formidable barriers to travel, passes have played a key role in trade, war, and both human and animal migr ...
), or (3) submerges a peak.
To these three types of
critical pointsbasins, passes, and peaks (i.e. minima, saddles, and maxima)one associates a number called the index, the number of
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
directions in which
decreases from the point. More precisely, the index of a non-degenerate critical point
of
is the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of the largest subspace of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to
at
on which the
Hessian of
is negative definite. The indices of basins, passes, and peaks are
and
respectively.
Considering a more general surface, let
be a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
oriented as in the picture, with
again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface
changes as the water level
rises.
Starting from the bottom of the torus, let
and
be the four critical points of index
and
corresponding to the basin, two saddles, and peak, respectively. When
is less than
then
is the empty set. After
passes the level of
when
then
is a
disk, which is
homotopy equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a point (a 0-cell) which has been "attached" to the empty set. Next, when
exceeds the level of
and
then
is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once
passes the level of
and
then
is a torus with a disk removed, which is homotopy equivalent to a
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an ...
with a 1-cell attached (image at right). Finally, when
is greater than the critical level of
is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached.
This illustrates the following rule: the topology of
does not change except when
passes the height of a critical point; at this point, a
-cell is attached to
, where
is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of
In the case of a landscape or a manifold
embedded in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, this perturbation might simply be tilting slightly, rotating the coordinate system.
One must take care to make the critical points non-degenerate. To see what can pose a problem, let
and let
Then
is a critical point of
but the topology of
does not change when
passes
The problem is that the second derivative is
that is, the
Hessian of
vanishes and the critical point is degenerate. This situation is unstable, since by slightly deforming
to
, the degenerate critical point is either removed (
) or breaks up into two non-degenerate critical points (
).
Formal development
For a real-valued
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
on a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
the points where the
differential of
vanishes are called
critical points of
and their images under
are called
critical values. If at a critical point
the matrix of second partial derivatives (the
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
) is non-singular, then
is called a ; if the Hessian is singular then
is a .
For the functions
from
to
has a critical point at the origin if
which is non-degenerate if
(that is,
is of the form
) and degenerate if
(that is,
is of the form
). A less trivial example of a degenerate critical point is the origin of the
monkey saddle.
The
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
of a non-degenerate critical point
of
is the dimension of the largest subspace of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to
at
on which the Hessian is
negative definite. This corresponds to the intuitive notion that the index is the number of directions in which
decreases. The degeneracy and index of a critical point are independent of the choice of the local coordinate system used, as shown by
Sylvester's Law.
Morse lemma
Let
be a non-degenerate critical point of
Then there exists a
chart
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent ...
in a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
such that
for all
and
throughout
Here
is equal to the index of
at
. As a corollary of the Morse lemma, one sees that non-degenerate critical points are
isolated
Isolation is the near or complete lack of social contact by an individual.
Isolation or isolated may also refer to:
Sociology and psychology
*Isolation (health care), various measures taken to prevent contagious diseases from being spread
**Is ...
. (Regarding an extension to the complex domain see
Complex Morse Lemma. For a generalization, see
Morse–Palais lemma In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable c ...
).
Fundamental theorems
A smooth real-valued function on a manifold
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
in the
topology. This is sometimes expressed as "a typical function is Morse" or "a
generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
function is Morse".
As indicated before, we are interested in the question of when the topology of
changes as
varies. Half of the answer to this question is given by the following theorem.
:Theorem. Suppose
is a smooth real-valued function on