Topology Glossary
Mainly extracted from (a) UC Davis Math:Profile Glossary
( http://www.math.ucdavis.edu/profiles/glossary.html ) by Greg Kuperberg
( http://www.math.ucdavis.edu/profiles/kuperberg.html ), and
(b) Topology Atlas Glossary
( http://www.achilles.net/~mtalbot/TopoGloss.html ).
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absolute retract
A space A is an absolute retract if whenever a normal space X has a
closed subspace B homeomorphic to A, then B is a retract of X. The fact
that I and R1 are absolute retracts follows from the Tietze Extension
Theorem.
accumulation point
Given a subset A of a topological space X, the point p in X is called an
accumulation point of A if each neighborhood of p contains infinitely many
distinct points of A.
algebraic topology
The branch of topology concerned with homology and other algebraic models
of topological spaces.
algebraic variety
A space which is locally the solution locus to a set of polynomial
equations. Algebraic varieties are for algebraic geometers what
manifolds are for manifold topologists. Indeed, many algebraic varieties
are (complex) manifolds. However, algebraic varieties may also have
complicated singular sets and may be parametrized with rings other than
the complex numbers. (For the technical reason that the real numbers are
not algebraically closed, one does not consider algebraic varieties over
the real numbers in the straightforward sense.)
arc
Homeomorphic image of a closed line interval.
automorphism
An isomorphism of a group with itself
band-connected sum
A knot formed from two other knots by connecting them along two parallel
segments called a band. Although the original two knots cannot be
entangled with each other (they must be separated by a sphere), the band
can meander among them in a complicated way.
basis
1. In mathematics, usually means basis in the sense of linear algebra; a
minimal set of vectors that spans a vector space.
2. A basis for a topological space X is a collection of open sets of X
that contains "arbitrarily small" neighborhoods of every point of X.
Specifically, for every point x of X, and for every open set U containing
x, the collection must include a neighborhood of x lying within U.
boundary
Boundary of A is the union of points for which every open set around them
intersects both A and its complement OR the image of a differential map in
a complex.
boundary of Manifold
The boundary Bd M of a manifold M is the set of points of M that have
boundary patches. The boundary of an n-manifold M is an (n-1) manifold
>M whose boundary >(>M) is empty {ż}.
calculus of variations
Calculus problems, especially differentiation and maximization, involving
functions on a set of functions of a real variable. For example, finding
the shape of a cable suspended from both ends.
category theory
The study of abstracted collections of mathematical objects, such as the
category of sets or the category of vector spaces, together with
abstracted operations sending one object to another, such as the
collection of functions from one set to another or linear transformations
from one vector space to another.
characteristic class
A kind of homological model for a decoration or property of a manifold or
other topological space. The simplest characteristic class describes how
a manifold fails to be orientable, that is, in which directions a being
can travel in the manifold and reverse its handedness.
closure
The closure of a subset A of a space X, denoted Cl A, is the minimal
closed set of X that contains A.
codimension
In general, if a mathematical object sits inside or is associated to
another object of dimension n, then it is said to have codimension k if
it has dimension n-k.
combinatorial geometry
The visual study of discrete and finite structures, and the study of
discrete and finite possibilities for the arrangement or features of
geometric objects.
compact
A topological space is compact if every collection of open sets that
covers the space has a finite subset that also covers the space. The
compact subspaces of Rn are the closed and bounded sets.
complement
A word with a relatively specific meaning in mathematics. The complement
of a subset X in a set Y is Y-X, the set of all things in Y but not X.
complex manifold
A manifold with complex coordinates; its ordinary or real dimension is
then twice its complex dimension.
conformal
Angle-preserving or angle-defining. The Mercator map is a conformal map
of the Earth because angles are true. A conformal structure on a
manifold defines angles between curves segments on the manifold but not
their lengths.
connected
A topological space is connected if it cannot be partitioned into two
disjoint, nonempty open sets.
connected sum
1. A manifold formed from two others by removing balls and gluing along
the resulting spherical boundary.
2. The analogous operation for knots; a band-connected sum in which the
band connects the knots in the simplest possible way (by piercing the
separating sphere only once).
contractible
A space is contractible if it can be shrunk to a point within itself.
The homotopy that does this is called a 'contraction'. Contractible
spaces are simply connected.
covering
A collection of sets whose union is the whole space.
covering space
The domain of a covering map; also called a 'cover'. A space that looks
locally like the space it covers, but whose parts may be connected
together differently.
covering transformation
Also called 'deck transformation'. A covering transformation is a
homeomorphism of a covering space with itself that preserves the covering
map. For any two liftings of a connected object, there is a covering
transformation that carries one to the other, provided that the covering
space is connected and locally path-connected.
convex
A region in the plane, in Euclidean space, or in some other geometry with
lines such as hyperbolic space, is convex if it always contains the line
segment connecting two points if it contains the two points themselves.
A convex body is, technically, a closed and bounded convex set with
non-zero volume.
convex geometry
The study of convex shapes, usually in Euclidean geometry.
convolution
A convolution of two planar regions is the set of all vector sums of a
point in one region with a point in the other.
correlation
1. The relationship between any two random variables which may or may not
be independent. It may be expressed in terms of conditional
probabilities or the mutual probability distribution of the random
variables.
2. The quadratic term in the relationship between two real-valued random
variables; the expectation of the product minus the product of the
expectations, suitably normalized.
curvature
In Riemannian geometry, usually means the intrinsic curvature of a
manifold with a Riemannian metric. The curvature at a point is positive
(negative) if the sum of the angles of a small approximate triangle at
that point is greater than (less than) 180 degrees.
cusp
A trumpet-shaped salient of a hyperbolic structure or Riemannian metric
which is typically infinitely long. The neighborhood of an ideal vertex
is a kind of cusp.
deformation retract
A subspace A of a space X is a deformation retract if X can be shrunk
down to A without moving any point of A. The homotopy that does the
shrinking is called a 'deformation retraction'.
Dehn filling
Another view of Dehn surgery due to Thurston and motivated by hyperbolic
geometry. If the complement of a knot in a 3-manifold has a family of
hyperbolic structures, then typically many of them can be completed or
filled in to realize different Dehn surgeries along the knot.
Dehn surgery
A modification of a 3-manifold in which a solid torus around a knot is
removed and replaced by a solid torus in a different position.
Dehn twist
On a surface of genus g > 0, cut apart one of the handles along a circle,
give one handle a 360ˇ twist, and glue the handles back together.
deRham cohomology
The nth deRham cohomology is the the space of n-forms w with dw = 0,
modulo those of the form du where u is an (n-1)-form. Its dimension is
called the nth Betti number of the space.
diffeomorphism
A bijection between two manifolds that preserves all smooth structure.
differential geometry
The general study of smooth manifolds decorated by continuous structures
such as foliations, Riemannian metrics, and symplectic structures.
Riemannian geometry is a disproportionate part of differential geometry.
embedding
A mapping into a space whose image is homeomorphic to the domain. The
parametrization of a submanifold by means of a standard model. A knotted
sphere in 4-space is an embedding of the familiar round sphere.
Whitney's theorem says that an n-dimensional manifold is guaranteed to
have an embedding in Euclidean n-space.
enumeration
The goal in combinatorics of counting the number of some type of
mathematical object. Example: There are n! permutations of n distinct
letters. An enumeration is considered obscure unless the formula for the
number of some object is much simpler than the set of objects itself.
enumerative combinatorics
The branch of mathematics concerned with finite counting problems, for
example counting arrangements of non-attacking queens on a chessboard.
essential
In low-dimensional topology, a non-trivial kind of curve or surface in a
manifold that only sometimes exists. An essential sphere in a
3-manifold, for example, is a sphere that does not bound a ball.
Euclidean space
A finite-dimensional vector space, such as ordinary 3-dimensional space,
together with a metric that satisfies the Pythagorean theorem.
Euler characteristic
An integer associated to a manifold or other topological space which is
particularly easy to compute. Example: The Euler characteristic of a
surface is given by the number of faces minus edges plus vertices.
fiber bundle
A topological space that divides into locally parallel fibers that may
twist globally. The set of fibers is itself a topological space called
the base space. For example, a Mobius strip is a fiber bundle over a
circle with line segments as fibers.
fixed-point free
A flow or map is fixed-point free if it does not send any point to
itself; any such point would be a fixed point.
flow
A vector field in Euclidean space or on a manifold which gives the
velocity of a particle moving through the manifold as a function of its
position. More technically, an autonomous ordinary differential
equation.
foliation
A decoration of a manifold in which the manifold is partitioned into
sheets of some lower dimension, and the sheets are locally parallel.
(More technically, the foliated manifold is locally homeomorphic to a
vector space decorated by cosets of a subspace).
Fredholm determinant
A complex analytic function which generalizes the characteristic
polynomial of a matrix. It is defined for those operators which have
continuous kernels, i.e., kernels in the sense of analysis.
frontier
The frontier of a subset A in a space X, denoted Fr A, is Cl A - Int A:
the set of points that lie in the closure of A but not in the interior
of A.
function
A mapping from a domain to a range such that an element has only one
image
functor
A correspondence from one category to another mapping objects to objects
and preserving morphisms.
functional analysis
The study of differentiation, integration, estimates, and asymptotics of
functions of real numbers. The modern research version of calculus.
fundamental group
1. A group, often non-abelian, that rather fully describes the
periodicity or the 1-dimensional holes in a topological space.
2. The group of homotopy classes of loops at a base point of a space
is a topological invariant. It measures the holes in a space.
Gelfand-Fuchs cohomology
An ad hoc modification of certain cohomology groups in various
infinite-dimensional settings that is more relevant and easier to compute
than ordinary cohomology.
genus
1. In geometric topology, the number of holes of a surface. Usually this
means the maximum number of disjoint circles that can be drawn on the
surface such that the complement is connected. If there are no such
circles, the surface is planar, and the genus then sometimes means the
maximum number of disjoint arcs with the same property.
2. A measure of the twistedness of a fiber bundle defined to A. Schwarz.
Technically, if X is a topological space with a fiber bundle F, it is the
minimum number of open sets that cover X such that F is trivial over each
open set.
3. Some other number which generalizes or is analogous to the topological
genus, such as the genus of an algebraic curve.
geodesic
On a Riemannian manifold or some other metric space, a curve which is the
shortest path between any two points on it that are sufficiently close
together.
Geometrization Conjecture
The conjecture of Thurston that, after cutting along essential spheres
and tori, every compact 3-manifold admits a special Riemannian structure
known as a geometry, usually hyperbolic geometry. The conjecture
subsumes the Poincaré conjecture and many other standard conjectures
about 3-manifolds, and constitutes a classification of 3-manifolds.
global analysis
The study of partial differential equations (and other structures from
analysis) on manifolds.
graph
In research mathematics, usually means set of points with another set of
edges connecting them.
graph theory
The study of graphs, either for their own sake, or as models of such
diverse things as groups (in pure mathematics) or computer networks.
Gromov norm
An invariant associated with the homology of a topological space that
measures how many simplices are needed to represent a given homology
class.
group
A mathematical system consisting of elements from a set G and a binary
operation * such that
1.x*y is a member of G whenever x and y are
2.(x*y)*z=x*(y*z) for all x, y, and z
3.there is an identity element e such that e*x=x*e=e for all x
4.each member x in G has an inverse element y such that x*y=y*x=e
Haefliger structure
A relatively complicated structure on an n-manifold which consists of a
k-plane field plus auxiliary information derived from the diffeomorphism
group of n-k-dimensional Euclidean space. A foliation has a parallel
Haefliger structure just as it has a parallel plane field, but a
Haefliger structure may have a singular locus where it is not parallel to
a foliation.
Haken
A noted mathematician with the privilege of having an adjective named
after him. A 3-manifold is Haken if it is closed, orientable, and has no
essential sphere, but has an incompressible surface.
Hamiltonian
An expression that represents the equations of motion of a classical
physical system via Hamilton's equations. The equations are related to
the Hamiltonian via phase space or symplectic geometry. The solutions
conserve the Hamiltonian if it is time-independent, and the value of the
Hamiltonian can be interpreted as the energy of the system.
Hausdorff separation axiom
For any two elements p and q of a topological space X there exists
disjoint open subsets P and Q in X such that p is in P and q is in Q.
Hausdorff-Besicovitch dimension
DH(S) is the value of d for which the Hausdorff measure of d-dimensional
Volume of a set S. namely hd(S), changes from infinity to zero, it is a
space filling characterisation.
Hausdorff space
A Hausdorff space is defined by the property that every two distinct
points have disjoint neighborhoods
Heegaard splitting
A division of a 3-manifold into two handlebodies. An early result in
topology states that every closed 3-manifold (closed meaning that the
manifold is finite and connected but has no boundary) has a Heegaard
splitting and a resulting description in terms of a Heegaard diagram,
which describes how the two handlebodies are glued together. The surface
lying between the two handlebodies of the splitting is a Heegaard
surface.
homeomorphism
A bijection between two topological spaces that preserves all continuous
structure; the basic notion of equivalence in topology.
homomorphism
A function that preserve the operators associated with the specified
structure.
homological algebra
The algebraic study of the homology and cohomology of manifolds and other
mathematical objects. Homological algebra is a grand generalization of
linear algebra.
homology/cohomology
Homology and cohomology are algebraic objects associated to a manifold or
other mathematical object which give one measure of the number of holes
of the object. The homology of a topological space has a relatively
technical definition, but it is relatively easy to compute and study with
tools from linear algebra.
Hopf algebra
An abstract algebraic object, generalizing a group or a Lie algebra, with
enough structure to have a representation theory.
horocycle/horosphere
A generalized (round) circle or sphere in the hyperbolic plane or
hyperbolic space. A horosphere has infinite radius and meets the sphere
at infinity at one point, and its geometric center is the same point.
hyperbolic geometry
In modern terms, hyperbolic geometry is the study of manifolds with
Riemannian metrics with constant negative curvature. The hyperbolic
plane is a particular hyperbolic manifold which is in a sense universal
among hyperbolic surfaces; similarly there is also hyperbolic n-space.
Escher's circle limit prints are excellent illustrations of the
hyperbolic plane.
hyperbolic PDE
A partial differential equation that resembles a wave equation with one
time coordinate, one or several space coordinates, and a finite speed of
propogation for features of solutions.
hyperplane/hypersurface
A high-dimensional plane or submanifold in a vector space or manifold
with codimension 1.
ideal vertex
A vertex at infinity of a hyperbolic polyhedron. Geometrically, the
faces of the polyhedron taper together at an exponential rate as they
extend towards the ideal vertex.
immersion
A locally (but not globally) smoothly invertible mapping of one
manifold into another. The image may have self-intersections; the
figure-8 is an immersion of the circle in 2D.
incompressible
1. In physics, a fluid that cannot change volume or a flow that is
possible for such a fluid.
2. In 3-dimensional topology, a surface in a 3-manifold with the property
that no essential circle in the surface bounds a disk in the manifold.
incompressible fluid
A fluid that does not change its volume except under extreme conditions,
or a mathematically abstracted fluid that strictly conserves its volume.
infinite dimensions
In mathematics, the concept of an infinite-dimensional space considered
literally. It is a vector space with an infinite basis or a space with
infinitely many coordinates.
injection
A function which is one-to-one, i.e. if f(x)=f(y), then x=y.
int
A symbol used to denote the interior operator, interior of set A can
also be denoted by Aˇ or Ao.
interior
Interior of A is the union of points which have an open set containing
them which is completely contained inside A. Also the maximal open set
contained in A.
invariant
In topology, a number, polynomial, or other quantity associated to a
topological object such as a knot or 3-manifold which depends only on the
underlying object and not on its specific description or presentation.
isometry
A mapping of metric spaces which preserves the metric.
isotopy
A homotopy of an object produced by a deformation of the ambient space,
so therefore the object cannot develop new self-intersections. The
deformation of the teapot to a torus is an isotopy, but the deformation
to a point is not.
Jacobian variety
A space associated to a Riemann surface defined most succinctly as the
complex cohomology (as a vector space) divided by the integral cohomology
(as a lattice). It is simultaneously a complex manifold, an algebraic
variety, and a Lie group.
Jones polynomial
A famous invariant of knots and links discovered by Vaughan Jones. It
has several extremely elementary definitions and at the same time
involves deep mathematics.
kernel
1. In algebra, the set of vectors annihilated (sent to zero) by a matrix,
linear operator, homomorphism, or any similar function.
2. In analysis, a continuous analogue of a matrix. Given a vector space
of functions of a parameter or functions on a manifold, an operator may
have a kernel or matrix whose rows and columns are indexed by the
parameter or by points on the manifold.
knot/link
A link is a collection of disjoint circles lying in a 3-manifold, often
but not always Euclidean 3-space or the 3-sphere. A knot is link which
happens to be a single circle. Manifold topologists usually study tame
knots and links, which can be represented by smooth or polygonal curves,
but there are also wild links which are infinitely knotted. Ordinary
knots and links were topologically classified (in a certain sense) by
Haken and Thurston.
lamination
A decoration of a manifold in which some subset of the manifold is
partitioned into sheets of some lower dimension, and the sheets are
locally parallel. It may or may not be possible to fill the gaps in a
lamination to make a foliation.
lattice
A periodic arrangement of points such as the vertices of a tiling of
space by parallelepipedons or the positions of chemical atoms in a
monoatomic crystal. More technically, a discrete abelian subgroup of an
n-dimensional vector space which is not contained in an n-1 dimensional
vector space. Lattices play a central role in the theory of Lie groups,
in number theory, in error-correcting codes, and many other areas of
mathematics.
least-area surface
A surface (manifold) lying in space or in a manifold which minimizes area
(surface volume) among a class of similar surfaces. Example: The round
sphere has least area among surfaces in Euclidean space that enclose a
fixed volume.
leopard spot
In the proof of the Geometrization Theorem, when the skinning map is
applied to a conformal structure A to produce a new one B, B is made up
of many (unrolled) copies of A that resemble leopard spots.
Lie algebra
An algebraic structure on a vector space which describes multiplication
of elements of a Lie group which are very close to the identity
(infinitesimal transformations). Lie algebras are almost as important as
their comrades-at-arms, Lie groups.
Lie group
A group (in the sense of abstract algebra) which is at the same time a
manifold. Example: The group of rotations in n dimensions is a Lie
group of dimension n(n-1)/2. Lie groups are fundamental objects in
mathematics and physics, especially quantum field theory.
lift
Also called 'lifting'. With respect to a covering map p:M->X, a lift of
a map a:C->X is any map a':C->M such that p Ľ a' = a. The covering map p
is the covering of a sheet by its blanket.
lifting
The process of converting maps into a base space to maps into its
covering space.
local
A property of topological spaces is usually said to hold locally in a
space X if it holds within arbitrarily small neighborhoods of every point
of X. (For properties that open sets do not normally have, such as
compactness, the definition has to be modified somewhat.) For example,
a space is locally path-connected if it has a basis of path-connected
sets.
loop
A map of a circle into a space OR a map of the unit interval [0,1] into
a space with the same beginning and end points, i.e. f(0)=f(1).
loop space
Given a topological space X, its loop space is the topological space of
all continuous functions from a circle to X. Loop spaces are important
examples of new topological spaces formed from old ones, as well as
examples of infinite-dimensional spaces in mathematics.
Loop and sphere theorems
The two most important foundational theorems in 3-manifold topology,
proved by Papakyriakopoulos in the 1950's. They generalize Dehn's lemma,
which asserts that if a knot in three dimensions bounds a disk which
intersects itself in its interior, the disk can be simplified to remove
all self-intersection and the knot must be trivial.
lozenge
A rhombus with a 60 degree angle.
manifold
A fundamental mathematical object which locally resembles a line, a
plane, or space (is locally homeomorphic to a vector space). For
example, the surface of a sphere or a doughnut, the playing field of the
video game Asteroids, and (in the theory of General Relativity) physical
space are all manifolds. The term n-manifold means a manifold which
locally resembles n-dimensional space, not one which might lie in
n-dimensional space.
metric
A distance function on some space or set; an assignment of distance to
every unordered pair of points that satisfies the triangle inequality.
metric space
A space with a "distance" function, called a metric, defined above.
min cut, max flow principle
The principle that, given a flow of some substance through a network of
pipes with a maximum allowed flow in each pipe, the maximum possible
total flow equals the capacity of the most constricted set of pipes
which, if cut, would separate the source from the drain.
minimal
A widely used, context-dependent word in mathematics that generally means
atomic or unsimplifiable. For example, prime numbers are minimal with
respect to factorization of whole numbers.
minimal surface
1. A surface (or a manifold) which locally minimizes surface area (or
surface volume), which means that one cannot replace small patches of the
surface and decrease the area. See also least-area surface.
2. A surface that locally has the smallest area given a particular
topological shape for it, and possibly, constrained by a fixed boundary
(soap-films) or prescribed behavior at infinity.
moduli space
Given a topological object, usually low-dimensional, with a continuous
family of possible geometric structures, the moduli space is the set of
these structures considered itself as a geometric object, usually
high-dimensional. Two structures that differ by a topological
automorphism of the low-dimensional object are represented by the same
point. In particular, the space of conformal structures on a surface.
normal
Said of topological spaces. In a normal space, every two disjoint closed
sets have disjoint neighborhoods. All metric spaces are normal.
non-abelian
Non-commutative or order-dependent. For example, the group of
manipulations of the Rubik's cube is non-commutative because the state of
the cube depends greatly on the order that moves are performed on it.
orbifold
A topological object defined by Thurston which is locally modelled by
Euclidean spaces divided by finite groups of symmetries. Orbifolds are
manifolds with singularities such as reflection surfaces, where they
resemble manifolds with boundary, and cone lines, where they are modelled
(in the direction perpendicular to the cone line) by a cone with an angle
of 360/n degrees for some n.
path
A continuous function with domain I=[0,1]
path component
The path components of a topological space are its maximal path connected
subsets. Two points lie in the same path component of a space X iff there
is a path in X from one point to the other.
path-connected
A topological space is path-connected if every pair of its points can be
connected by a path. Every path-connected space is connected, but not
vice versa. If a space is connected and locally path-connected,
however, then it is path connected.
Path homotopy
A homotopy between paths that fixes their endpoints; or the relation of
being path-homotopic. Two paths are path-homotopic if there is a path
homotopy between them.
PL flow
A "piecewise linear" motion on a space or a manifold, akin to a flow
given by a vector field, in which every particle in a given simplex of
some triangulation moves with constant velocity and in the same
direction, so that the particle trajectories are polygons.
plane field
A decoration of a n-manifold that assigns a tangent k-dimensional plane
to every point in a continuous fashion. The general notion includes line
fields and hyperplane fields; line fields are very similar to vector
fields such as the magnetic field.
plane partition
A stack of unit cubes in a rectangular box or in the positive octant in
space such that to the left, behind, and below every cube lies either
another cube or a wall. A plane partition in a box is equivalent to a
lozenge tiling of a hexagon in the plane.
Platonism
The belief that mathematical objects exist independent of physical
models. It is a useful pretense in mathematics, especially in geometry.
pleated surface
A surface in Euclidean or hyperbolic space which resembles a polyhedron
in the sense that it has flat faces that meet along edges. Unlike a
polyhedron, a pleated surface has no corners, but it may have infinitely
many edges that form a lamination.
Poincare conjecture
The conjecture that a closed, simply-connected 3-manifold must be
(homeomorphic to) the 3-sphere. Many mathematicians, including Poincaré
himself, have presented incomplete or fallacious proofs of the
conjecture.
polytope
The n-dimensional generalization of a polygon or a polyhedron.
projection
Usually refers to composition with the covering map. For example, if
§ is a path in a blanket M, and p:M->S is the covering map, then the
projection of § is the path pĽ§:I->S.
projective plane
A 2-manifold obtained by gluing a Mobius strip and a disk along their
circle boundaries.
quotient space
A space obtained from another by identifying or "gluing" some points to
some others. Formally, Y is a quotient space of X if there is a
surjective map f:X->Y such that the open sets of Y are those subsets U
of Y for which f^(-1)(U) is open in X.
rational map
A map from some field such as the real or complex numbers (plus infinity)
to itself given by a rational polynomial function.
representation/linear representation
A realization of a group, Lie group, or Lie algebra by matrices or linear
transformations. More technically, a homomorphism from a group to a
group of matrices.
representation theory
The study of linear representations of groups, Lie groups and Lie
algebras.
retract
A subspace A of a space X is a retract if there is a map f:X->A that
fixes every point of A. The map f is called a 'retraction'.
Reuleaux polytope
A convex body in the plane or in higher dimensions which, like a Reuleaux
triangle, consists of pieces of round spheres, each centered at one of
the corners of the convex body.
Reuleaux triangle
A three-cornered curve consisting of three equal arcs of circles, each
centered at the opposite corner.
Riemann matrix
In the cohomology of a Riemann surface, the integral cohomology has a 2g
x 2g generating matrix. After a standard basis manipulation using a
natural inner product on the cohomology and its structure as a complex
vector space, one obtains a g x g matrix called the Riemann matrix. It
determines the Riemann surface and is unique up to transformations coming
from homeomorphisms of the surface.
Riemann-Schottky problem
The problem of determining whether a given complex matrix is the Riemann
matrix of some Riemann surface.
Riemann sphere
A topological sphere consisting of the complex plane and the point at
infinity; an example of a Riemann surface.
Riemann surface/complex curve
A surface with a conformal structure; a complex manifold with one complex
dimension.
Riemannian geometry
The study of curvature and other properties of Riemannian metrics on
manifolds. A Riemannian metric is a metric on a manifold which is
locally like ordinary distance in Euclidean space. Here ``locally'' is
not meant in the usual sense that every point has a region around it that
is identical to a region in Euclidean space; rather, a Riemannian metric
agrees with Euclidean distance to first order in the sense of calculus.
Sometimes also called differential geometry.
round
In topology, the terms circle and sphere refer to topological objects and
not geometric ones, so that the surface of an egg shape is a sphere. A
round sphere, then, is a sphere with constant curvature; a sphere in the
sense of geometry.
saddle function
A function F(x,y) of two vectors x and y (which typically lie in
different vector spaces) is a saddle function if it is concave up in x
and concave down in y.
Schrodinger equation
The partial differential equation i hbar psi_t = H psi, where H is some
spatial linear differential operator called a Hamiltonian. It gives the
evolution over time of a probability cloud in quantum mechanics.
separable
A separable space is one that has a countable dense subset, that is a
countable subset whose closure is the whole space.
simplex
The n-dimensional generalization of the triangle and the tetrahedron; a
polytope in n dimensions with n+1 vertices.
simplicial
Made up of simplices. E.g. a simplicial polytope has simplices as faces
and a simplicial complex is a collection of simplices pasted together in
any reasonable vertex-to-vertex and edge-to-edge arrangement.
simply connected
A topological space is simply connected if
1. it is path connected,
2. every loop in that space can be continuously shrunk to a point.
skein theory
An inductive definition of an invariant of knots or links which
postulates a linear relation between the invariant of a given link and
the invariant of the same link with a crossing switched or otherwise
simplified. Sometimes skein theories also involve tangled graphs.
skinning map
An iterative map on conformal structures on a surface that appears in
Thurston's proof of the Geometrization Theorem. In the proof, the
hyperbolic structure of a cut-open manifold must be varied until it fits
with itself when the manifold is glued back together. Such a hyperbolic
structure is determined by a conformal structure at the wounds of the cut
(ends). Given one such structure A, the skinning map produces a new one
B chosen to match A; structure B also comes closer to matching itself.
smooth
1. Infinitely differentiable; possessing infinitely many derivatives.
For example, sin(x) is a smooth function, while |x|^3 is not. More
complicated mathematical objects such as manifolds are called smooth if
they are defined or described by smooth functions.
2. Continuously differentiable; possessing a continuous tangent or
derivative.
space
A term with many mathematical meanings. When a mathematical object is
called a space, there is a Platonist connotation that some kind of being
could exist in it and examine its properties as if it were a jail or the
entire universe.
spectrum
1. In quantum physics, the set of allowed energy levels of a particle or
system. It is directly related to bright or dark lines in a spectrum of
light produced by a prism.
2. In mathematics, the set of eigenvalues of a linear transformation. By
historical coincidence, it is equivalent to the notion of a spectrum in
quantum mechanics.
sphere
In topology, any manifold equivalent (homeomorphic) to the usual round
hollow shell in some dimension. An sphere in n+1-dimensional is called
an n-sphere, because that is its dimension as a manifold.
steepest descent method
A particular way of guiding an isotopy of an embedded surface to one
which minimizes a function that measures its shape. Moving down the
gradient of the area function often terminates at a minimal surface.
straight
A path in a flat m-manifold is straight if its projection to Rm is linear
and nonconstant.
SU(2)
The 3-dimensional Lie group of 2 x 2 unitary matrices; the most common
Lie group in mathematics and physics after the circle.
submanifold
A subset of a manifold that is itself a manifold.
subpath
A subpath of a path µ is any path of the form µs:t for s,t in I. Defined:
µs:t(x)=µ((1-x)s+xt).
subspace
A subset A of a topological space X with the inherited topology: the open
set in A are the intersections of the open sets of X with A.
support
A straight path § in R2 supports a piecewise linear path w at s in (0,1)
if §(1)=w(s) and w turns towards §(0) at s. If wr:s and ws:t are segments
of w, we also say that § supports these segments.
surjection
A function where every element in the range has a preimage in the domain.
tangled graph
A graph in 3-dimensional space; equivalently, a graph drawn in the plane
so that when edges cross, one edge goes over the other.
tensor
An object in linear algebra, generalizing a vector, an inner product, and
a linear transformation, with a multi-dimensional matrix.
thin position
A representation of a knot or graph in Euclidean space, or some analogous
structure, with the simplest possible horizontal slices. Between U-turns
where the knot or link has a horizontal tangent (and between vertices of
the graph), one counts the number of intersections with a horizontal
plane. Thin position is achieved when the total of these counts is
minimized.
top
Category of all topological groups.
topological complexity
A lower bound on computational complexity defined by Smale that involves
topology. A task with high computational complexity requires a computer
to make many decisions, sometimes arbitrary decisions, to untangle the
topology of the space of possible inputs or outputs or the space of
possibilities for an intermediate quantity.
topological quantum field theory
A quantum field theory, such as the Chern-Simons field theory in three
dimensions, whose integrals for a manifold produce topological
invariants.
topology
The study of how geometric objects are intrinsically connected to
themselves. Since topologists are not concerned with the geometric
measurements of objects, people often say that they study objects up to
continuous deformation. But usually topologists consider spaces which
have a topology (a qualitative shape or connectivity) but no predefined
(quantitative) geometry. Knots and manifolds are typical examples of
topological objects.
torsion
An R-module has torsion if some non-zero element, a, and some non-zero
scalar, r, imply ra = 0.
torus
The surface of a donut, i.e. a rectangle with the top and bottom
identified to form a cylinder and the the ends of the cylinder
identified to form a "donut". Constructed as the cartesian product
x of two 1-spheres (circles) T 2 = S^(1)xS^(1).
train track
A graph drawn on a surface such that every vertex has degree three, and
such that all three edges meeting at a vertex have a common tangent, two
edges on one side and one on the other. Every lamination of a surface is
carried by (approximately parallel to) a train track.
tree
A tree is a graph with the property that there is a unique path from any
vertex to any other vertex traveling along the edges.
triangulation
A tiling of some object such as a manifold by simplices.
trivalent graph
A graph such that each vertex has three edges.
unimodal
A function is unimodal if it goes up and then it goes down, once. For
example, a bell curve.
valuation
In convex geometry, a function f on convex sets such that f(A) + f(B) =
f(A cap B) + f(A cup B).
variation
In analysis, the amount that a function increases plus the amount that it
decreases.
weak/weakly
In mathematics and especially in analysis, an object is called weak if it
is of a generalized kind with fewer properties, and a property holds
weakly if it holds in a lesser sense. For instance, a weak solution to
an equation might be a discontinuous solution if a straightforward
interpretation implies continuity.