The sphere (surface of a ball) is a two-dimensional manifold since it can be represented by a collection of two-dimensional maps.

In mathematics (specifically in geometry and topology), a manifold is a mathematical object that on a small enough scale resembles Euclidean space. For example, seen from far away, the surface of the planet Earth is not flat and Euclidean, but on a smaller scale, one may describe each region via a geographic map, a projection of the surface onto the Euclidean plane.

A precise mathematical definition of a manifold is given below.

Lines and circles (but not figure eights) are one-dimensional manifolds (1-manifolds). The plane and the sphere are examples of 2-manifolds. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. A manifold can be endowed with a differentiable structure that allows one to do calculus on manifolds and a Riemannian metric that allows one to measure distances and angles. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

Motivational examples[link]

Circle[link]

Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top half of the unit circle, x² + y² = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this semicircle can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper semicircle to the open interval (−1,1):

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_{\mathrm{top}}(x,y) = x . \,

Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle.

The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χ_top and χ_right each map this part into the interval (0, 1). Thus a function T from (0, 1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and then follows the right chart back to the interval. Let a be any number in (0, 1), then:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} T(a) &= \chi_{\mathrm{right}}\left(\chi_{\mathrm{top}}^{-1}\left[a\right]\right) \\ &= \chi_{\mathrm{right}}\left(a, \sqrt{1-a^2}\right) \\ &= \sqrt{1-a^2} \end{align}

Such a function is called a transition map.

Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.

The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_{\mathrm{minus}}(x,y) = s = \frac{y}{1+x}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi_{\mathrm{plus}}(x,y) = t = \frac{y}{1-x}{}

Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1, 0); t is the mirror image, with pivot point (+1, 0). The inverse mapping from s to (x, y) is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} x &= \frac{1-s^2}{1+s^2} \\ y &= \frac{2s}{1+s^2} \end{align}

It can easily be confirmed that x² + y² = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with

Failed to parse (Missing texvc executable; please see math/README to configure.): t = \frac{1}{s}

Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.

Other curves[link]

Four manifolds from algebraic curves:
■ circles, ■ parabola, ■ hyperbola, ■ cubic.

Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. In this example we see that a manifold need not have any well-defined notion of distance, for there is no way to define the distance between points that don't lie in the same piece.

Manifolds need not be closed; thus a line segment without its end points is a manifold. And they are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola (two open, infinite pieces) and the locus of points on a cubic curve y² = x³−x (a closed loop piece and an open, infinite piece).

However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line (a + is not homeomorphic to a closed interval (line segment) since deleting the center point from the + gives a space with four components (i.e., pieces) whereas deleting a point from a closed interval gives a space with at most two pieces; topological operations always preserve the number of pieces).

Enriched circle[link]

Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable. The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transition functions have these properties as well.

Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold.

History[link]

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology.

Early development[link]

Before the modern concept of a manifold there were several important results.

Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai, and Riemann developed them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively.

Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.

Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces,

Failed to parse (Missing texvc executable; please see math/README to configure.): V - E + F = 2.\

The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topological map with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.^[1] Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V = 1 vertex, E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.

Synthesis[link]

Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables.

Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Simeon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology.

Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann.

Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Notably, the Whitney embedding theorem^[2] showed that the intrinsic definition in terms of charts was equivalent to the extrinsic definition in terms of subsets of Euclidean space.

Topology of manifolds: highlights[link]

Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). Bill Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang-Mills theory), where they serve as a substitute for ordinary 'flat' space-time. Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory.

Mathematical definition[link]

Informally, a manifold is a space that is "modeled on" Euclidean space.

There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. In terms of constructing manifolds via patching, a manifold has an additional structure if the transition maps between different patches satisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole.

Formally, a topological manifold^[3] is a second countable Hausdorff space that is locally homeomorphic to Euclidean space.

Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalized manifolds are discussed in non-Hausdorff manifolds).

Locally homeomorphic to Euclidean space means^[4] that every point has a neighborhood homeomorphic to an open Euclidean n-ball,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B}^n = \{ (x_1, x_2, \dots, x_n)\in\mathbb{R}^n \mid x_1^2 + x_2^2 + \cdots + x_n^2 < 1 \}.

Generally manifolds are taken to have a fixed dimension (the space must be locally homeomorphic to a fixed n-ball), and such a space is called an n-manifold; however, some authors admit manifolds where different points can have different dimensions. If a manifold has a fixed dimension, it is called a pure manifold. For example, the sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected component has a fixed dimension.

Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry.

Broad definition[link]

The broadest common definition of manifold is a topological space locally homeomorphic to a topological vector space over the reals. This omits the point-set axioms, allowing higher cardinalities and non-Hausdorff manifolds; and it omits finite dimension, allowing structures such as Hilbert manifolds to be modeled on Hilbert spaces, Banach manifolds to be modeled on Banach spaces, and Fréchet manifolds to be modeled on Fréchet spaces. Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis.

Charts, atlases, and transition maps[link]

For more exact definitions, see Differentiable manifold.

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure.

Charts[link]

A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.^[5] For a topological manifold, the simple space is some Euclidean space Rⁿ and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.

In the case of a differentiable manifold, a set of charts called an atlas allows us to do calculus on manifolds. Polar coordinates, for example, form a chart for the plane R² minus the positive x-axis and the origin. Another example of a chart is the map χ_top mentioned in the section above, a chart for the circle.

Atlases[link]

The description of most manifolds requires more than one chart (a single chart is adequate for only the simplest manifolds). A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered multiple ways using different combinations of charts. Two atlases are said to be C^k equivalent if their union is also a C^k atlas.

The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas (under the already defined equivalence relation given in the previous paragraph)). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though it is useful for definitions, it is an abstract object and not used directly (e.g. in calculations).

Transition maps[link]

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Asia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in Rⁿ to the manifold and then back to another (or perhaps the same) open ball in Rⁿ. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map.

Additional structure[link]

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all the transition maps are compatible with this structure, the structure transfers to the manifold.

This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of Rⁿ (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms.

The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible.

These notions are made precise in general through the use of pseudogroups.

Construction[link]

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.

Charts[link]

The chart maps the part of the sphere with positive z coordinate to a disc.

Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R² is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:

Sphere with charts[link]

A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R³:

Failed to parse (Missing texvc executable; please see math/README to configure.): S = \{ (x,y,z) \in \mathbf{R}^3 | x^2 + y^2 + z^2 = 1 \}.

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R². Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): \chi(x,y,z) = (x,y),\

maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere.

This can be easily generalized to higher-dimensional spheres.

Patchwork[link]

A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold.

The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold.

This can be illustrated with the transition map t = ¹⁄_s from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point s = ¹⁄_t on the first copy (the points t = 0 and s = 0 are not identified with any point on the first and second copy, respectively). This gives a circle.

Intrinsic and extrinsic view[link]

The first construction and this construction are very similar, but they represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point.

The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle.

[edit] n-Sphere as a patchwork

The n-sphere Sⁿ is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sⁿ can be constructed by gluing together two copies of Rⁿ. The transition map between them is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}^n \setminus \{0\} \to \mathbf{R}^n \setminus \{0\}: x \mapsto x/\|x\|^2.

This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier.

Identifying points of a manifold[link]

It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold is the real projective space identified as a quotient space of the corresponding sphere.

One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifold and G is the group, the resulting quotient space is denoted by M / G (or G \ M).

Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively).

Manifold with boundary[link]

A manifold with boundary is a manifold with an edge. For example a sheet of paper with rounded corners is a 2-manifold with a 1-dimensional boundary. The boundary of an n-manifold with boundary is an (n − 1)-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (See also Boundary (topology)).

In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open n-ball {(x₁, x₂, …, x_n) | Σ x_i² < 1}. Every boundary point has a neighborhood homeomorphic to the "half" n-ball {(x₁, x₂, …, x_n) | Σ x_i² < 1 and x₁ ≥ 0}. The homeomorphism must send the boundary point to a point with x₁ = 0.

If M is a smooth n-manifold, a compact, embedded n-dimensional submanifold with boundary D ⊂ M is called a regular domain in M.^[6]

Gluing along boundaries[link]

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.

Formally, the gluing is defined by a bijection between the two boundaries^{[dubious – discuss]}. Two points are identified when they are mapped onto each other. For a topological manifold this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly for a differentiable manifold it has to be a diffeomorphism. For other manifolds other structures should be preserved.

A finite cylinder may be constructed as a manifold by starting with a strip [0, 1] × [0, 1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries.

Cartesian products[link]

The Cartesian product of manifolds is also a manifold.

The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite cylinders, for example, as S¹ × S¹ and S¹ × [0, 1], respectively.

A finite cylinder is a manifold with boundary.

Manifolds with additional structure[link]

Topological manifolds[link]

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rⁿ. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rⁿ. These homeomorphisms are the charts of the manifold.

It is to be noted that a topological manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular and consistent choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.

Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable.

The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number n in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. Other authors allow disjoint unions of topological manifolds with differing dimensions to be called manifolds.

Differentiable manifolds[link]

For most applications a special kind of topological manifold, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point.

Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series, which are essentially polynomials of infinite degree). The sphere can be given analytic structure, as can most familiar curves and surfaces.

There are also topological manifolds, i.e., locally Euclidean spaces, which possess no differentiable structures at all.^[7]

A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

Riemannian manifolds[link]

To measure distances and angles on manifolds, the manifold must be Riemannian. A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product 〈⋅,⋅〉 in a manner which varies smoothly from point to point. Given two tangent vectors u and v, the inner product〈u,v〉gives a real number. The dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, angles, areas (or volumes), curvature, gradients of functions and divergence of vector fields.

All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it.

Finsler manifolds[link]

A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a norm, ||·||, in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product.

Any Riemannian manifold is a Finsler manifold.

Lie groups[link]

Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps.

A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as U(1), can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation. Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of n by n matrices with non-zero determinant. If the matrix entries are real numbers, this will be an n²-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are n(n-1)/2 dimensional manifolds, where n-1 is the dimension of the sphere. Further examples can be found in the table of Lie groups.

Other types of manifolds[link]

• A complex manifold is a manifold modeled on Cⁿ with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. Note that an n-dimensional complex manifold has dimension 2n as a real differentiable manifold.
• A CR manifold is a manifold modeled on boundaries of domains in Cⁿ.
• Infinite dimensional manifolds: to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces.
• A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold.
• A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
• A digital manifold is a special kind of combinatorial manifold which is defined in digital space. See digital topology

Classification and invariants[link]

Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds.

The classification of smooth closed manifolds is well-understood in principle, except in dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the Solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions.

Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants.

This is much harder in higher dimensions: higher dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object.

However, one can determine if two manifolds are different if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are invariant under different descriptions.

Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no program exists that can decide whether two manifolds are diffeomorphic.

Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant).

Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between no local invariants and local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global.

Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the simply connected property and orientability (see below). Indeed several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds.

Examples of surfaces[link]

Orientability[link]

In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to Rⁿ. Given an ordered basis for Rⁿ, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable.

Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in 3-space, and (3) the real projective plane, which arises naturally in geometry.

Möbius strip

Möbius strip[link]

Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a single side.

Klein bottle[link]

The Klein bottle immersed in three-dimensional space.

Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. Note that in three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space.

Real projective plane[link]

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes. Although there is no way to do so physically, it is possible to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane".

Genus and the Euler characteristic[link]

For two dimensional manifolds a key invariant property is the genus, or the "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology and cohomology.

Maps of manifolds[link]

A Morin surface, an immersion used in sphere eversion.

Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuous functions and smooth functions generally, there are maps with special properties. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem.

In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.

Scalar-valued functions[link]

3D color plot of the spherical harmonics of degree Failed to parse (Missing texvc executable; please see math/README to configure.): n=5

A basic example of maps between manifolds are scalar-valued functions on a manifold,

Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon M \to \mathbf{R}

or Failed to parse (Missing texvc executable; please see math/README to configure.): f\colon M \to \mathbf{C},

sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold.

In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem.

Generalizations of manifolds[link]

• Orbifolds: An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense.

• Manifold with corners

• Algebraic varieties and schemes: Non-singular algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using sheaves instead of atlases.

Because of singular points, a variety is in general not a manifold, though linguistically the French variété, German Mannigfaltigkeit and English manifold are largely synonymous. In French an algebraic variety is called une variété algébrique (an algebraic variety), while a smooth manifold is called une variété différentielle (a differential variety).

• CW-complexes: A CW complex is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, and hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory, as they are easy to compute with and singularities are not a concern.

See also[link]

• List of manifolds
• Submanifold
• Directional statistics: statistics on manifolds
• Affine Geodesic: paths on manifolds
• Mathematics of general relativity

By dimension[link]

• Curve (1-manifold)
• Surface (2-manifold)
• 3-manifold
• 4-manifold
• 5-manifold

Notes[link]

References[link]

• Freedman, Michael H., and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3.
• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. Inspired by Milnor and commonly used in undergraduate courses.
• Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1.
• Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. The most complete account, with historical insights and excellent, but difficult, problems. The standard reference for those wishing to have a deep understanding of the subject.
• Kirby, Robion C. and Siebenmann, Laurence C. (1977) Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton University Press. ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
• Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2.
• Lee, John M. (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3.
• Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6.
• Milnor, John (1997) Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 0-691-04833-9.
• Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers. ISBN 0-8053-9021-9. The standard graduate text.
• Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.
• Neuwirth, L. P., ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox. Princeton University Press. ISBN 978-0-691-08170-0.
• Riemann, Bernhard, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN 3-253-03059-8.
  • Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears.
  • Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. The 1854 Göttingen inaugural lecture (Habilitationsschrift).
• Lee, John M. (2002), Introduction to Smooth Manifolds, Springer, ISBN 978-0-387-95448-6 

External links[link]

• Dimensions-math.org (A film explaining and visualizing manifolds up to fourth dimension.)
• The manifold atlas project of the Hausdorff Institute for Mathematics in Bonn

William Thurston
William Thurston in 1991
Born	(1946-10-30) October 30, 1946 (age 65)
Nationality	American
Fields	Mathematics
Institutions	Cornell University University of California, Davis University of California, Berkeley Princeton University
Alma mater	New College of Florida University of California, Berkeley
Doctoral advisor	Morris Hirsch
Doctoral students	Richard Canary Benson Farb David Gabai William Goldman Steven Kerckhoff Yair Minsky Oded Schramm Danny Calegari
Notable awards	Fields Medal (1982) Oswald Veblen Prize in Geometry (1976) National Academy of Sciences (1983)

William Paul Thurston (born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

Mathematical contributions[link]

Foliations[link]

His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. His more significant results include:

• The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one).

• The construction of a continuous family of smooth, codimension one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.

• With John Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology.

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ^[1]).

This section requires expansion.

The geometrization conjecture[link]

His later work, starting around the late 1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. This was the first example of a hyperbolic knot.

Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure eight knot complement. He showed that the figure eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next revolutionary theorem.

Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem.

To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.

The geometrization theorem has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. A proof of that conjecture follows from the recent work of Grigori Perelman (2002–2003).

Orbifold theorem[link]

In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Hamilton's work on the Ricci flow.

Education and career[link]

Thurston was born in Washington, D.C. to a homemaker and an aeronautical engineer. He received his bachelors degree from New College (now New College of Florida) in 1967. For his undergraduate thesis he developed an intuitionist foundation for topology. Following this, he earned a doctorate in mathematics from the University of California, Berkeley, in 1972. His Ph.D. advisor was Morris W. Hirsch and his dissertation was on Foliations of Three-Manifolds which are Circle Bundles.

After completing his Ph.D., he spent a year at the Institute for Advanced Study, then another year at MIT as Assistant Professor. In 1974, he was appointed Professor of Mathematics at Princeton University. In 1991, he returned to UC-Berkeley as Professor of Mathematics and in 1993 became Director of the Mathematical Sciences Research Institute. In 1996, he moved to University of California, Davis. In 2003, he moved again to become Professor of Mathematics at Cornell University.

His Ph.D. students include Richard Canary, Renaud Dreyer, David Gabai, William Goldman, Benson Farb, Detlef Hardorp, Craig Hodgson, Richard Kenyon, Steven Kerckhoff, Robert Meyerhoff, Yair Minsky, Lee Mosher, Igor Rivin, Oded Schramm, Richard Schwartz, Martin Bridgeman, William Floyd and Jeffrey Weeks. His son Dylan Thurston is an assistant professor of mathematics at Barnard College, Columbia University.

Thurston has turned his attention in recent years to mathematical education and bringing mathematics to the general public. He has served as mathematics editor for Quantum Magazine, a youth science magazine, and as head of The Geometry Center. As director of Mathematical Sciences Research Institute from 1992 to 1997, he initiated a number of programs designed to increase awareness of mathematics among the public.

In 2005 Thurston won the first AMS Book Prize, for Three-dimensional Geometry and Topology. The prize recognizes an outstanding research book that makes a seminal contribution to the research literature.^[2]

Thurston has an Erdős number of 2.

Selected works[link]

• William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978–1981).
• William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5
• William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246.
• William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.
• William Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431
• Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. xii+330 pp. ISBN 0-86720-244-0
• Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, RI, 1998. x+66 pp. ISBN 0-8218-0776-5

See also[link]

References[link]

External links[link]

• William Thurston at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "William Thurston", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Thurston.html .
• Thurston's page at Cornell

This article relies largely or entirely upon a single source. Please help improve this article by introducing citations to additional sources. Discussion about the problems with the sole source used may be found on the talk page. (November 2009)

The Fast Show, known as Brilliant in the US, was a BBC comedy sketch show programme that ran for three series from 1994 to 1997 with a special Last Fast Show Ever in 2000. The show's central performers were Paul Whitehouse, Charlie Higson, Simon Day, Mark Williams, John Thomson, Arabella Weir and Caroline Aherne. Other significant cast members included Paul Shearer, Felix Dexter, Rhys Thomas, Jeff Harding, Maria McErlane, Eryl Maynard, Colin McFarlane and Donna Ewin.

The show produced two national tours, the first in 1998 with the cast of the BBC surrealist comedy quiz show Shooting Stars and the second being their 'Farewell Tour' in 2002. The Fast Show was loosely structured and relied on character comedy, recurring running gags, and many catchphrases. Its fast-paced "blackout" style set it apart from traditional sketch series because of the number and relative brevity of its sketches; a typical half-hour TV sketch comedy of the period might have consisted of nine or ten major items, with contrived situations and extended setups, whereas the premiere episode of The Fast Show featured twenty-seven sketches in thirty minutes,^[1] with some items lasting less than ten seconds and none running longer than three minutes. Its innovative style and presentation influenced many later series such as The Catherine Tate Show and Little Britain.

It was one of the most popular sketch shows of the 1990s. The show has been released on VHS, DVD and audio CD. Some of its characters, Ron Manager, Ted and Ralph, Swiss Toni and Billy Bleach have had their own spin-off programmes.

Charlie Higson announced on 5 September 2011 that The Fast Show would return for a new online only series starting 14 November.^[2] The premiere date was changed later to 10 November.^[3]

Style and content[link]

The Fast Show was the brainchild of Paul Whitehouse and his writing partner and friend, Charlie Higson (who had previously enjoyed some success in the UK as a musician with the band The Higsons). After meeting through a mutual friend (Whitehouse's longtime flatmate, guitarist and writer David Cummings) comedian Harry Enfield invited Whitehouse to write for him and Whitehouse in turn asked Higson to help him out; soon after, Enfield got his break into TV with the series Harry Enfield and Chums and became nationally famous in the UK.^[1]

In the early 1990s Higson and Whitehouse worked extensively with Vic Reeves and Bob Mortimer, writing for and performing in the series The Smell of Reeves and Mortimer and Bang Bang, It's Reeves and Mortimer (both of which Higson produced). These series also featured occasional appearances by future Fast Show cast members Caroline Aherne, Simon Day and Mark Williams. Higson made many appearances in minor roles, while Williams and Whitehouse had recurring roles (with Vic and Bob) in The Smell of Reeves and Mortimer, parodying the members of rock group Slade in the popular "Slade in Residence" and "Slade on Holiday" sketches.

Inspired by a press preview tape of Enfield's show, compiled by producer friend Geoffrey Perkins and consisting of fast-cut highlights of Enfield's sketches, the pair began stockpiling material and developing the idea of a rapid-fire 'MTV generation' format based wholly on quick cuts and soundbites/catchphrases. After unsuccessfully trying to sell the series to ITV through an independent production company, Higson and Whitehouse approached the new controller of BBC2, Michael Jackson; fortunately, he was then looking for new shows to replace several high-profile series that had been recently lost to BBC1, and their show was picked up by BBC2.^[1]

Whitehouse and Higson, the co-producers and main writers, then assembled the original team of writers and performers, which included David Cummings, Mark Williams, Caroline Aherne, Paul Shearer, Simon Day, Arabella Weir, John Thomson, Graham Linehan and Arthur Mathews (of Father Ted fame), Vic Reeves, Bob Mortimer and Craig Cash (who went on to write and perform with Aherne in The Royle Family). Musical director Philip Pope was also an established comedy actor with extensive experience in TV and radio comedy, and had previously appeared in series such as Who Dares Wins and KYTV; he also enjoyed success as a comedy recording artist as part of the Bee Gees parody group The HeeBeeGeebees.

The Fast Show was a working title disliked by both Whitehouse and Higson but it went unchanged through production and eventually remained as the final title.^{[citation needed]}

The first series introduced many signature characters and sketches including Ted and Ralph, Unlucky Alf, The Fat Sweaty Coppers, Ron Manager, Roy and Renée, Ken and Kenneth (The Suit You Tailors), Arthur Atkinson, Bob Fleming, Brilliant Kid, Insecure Woman, Janine Carr, Denzil Dexter, Carl Hooper, Ed Winchester, the Patagonian buskers, "Jazz Club" and the popular parody "Chanel 9".

Many characters were never given any 'official' name, with their sketches being written to give their catchphrase as the punchline of each sketch. Examples include "Anyone fancy a pint?" (played by Whitehouse), "You ain't seen me, right!" (a mysterious gangster-like character played by Mark Williams), "I'll get me coat" (Williams) and "Ha!", a sarcastic elderly woman played by Weir.

Other long-standing running jokes in the programme included the fictitious snack food "Cheesy Peas" in various forms, shapes and flavours, in satirical adverts presented by a twangy, Northern lad (Paul Whitehouse) who claims, "They're good for your teas!"and has since become a reality thanks to UK TV chef Jamie Oliver [1]. The dire earnestness of the born-again Christian was parodied in another popular group of sketches where various characters responded to any comment or question by extolling the virtues of "Our Lord Jesus" and ended the sketch with the exclamation "He died for all our sins, didn't he?" or something similar; and most controversially, "We're from the Isle of Man", featuring a stereotype of weird, surreal, townsfolk in a setting portrayed as an abjectly impoverished and desolate cultural wasteland.

Some of the characters resembled parodies of well-known personalities: for example, Louis Balfour, host of "Jazz Club" was reminiscent of Bob Harris of The Old Grey Whistle Test^{[citation needed]} and Ron Manager of football pundits Trevor Brooking and Graham Taylor. However, the parodic intent of this character is broader, and portrays how often football pundits have little to say of any real substance and sometimes waffle.^{[citation needed]} Paul Whitehouse said that Ron Manager was based on ex-Luton Town & Fulham manager Alec Stock [2]. Arthur Atkinson is a composite of Arthur Askey and Max Miller.

The show ended in 2000, with a three-part "Last Ever" show, in the first episode of which Fast Show fan Johnny Depp had a guest-starring role as a customer of The Suit You Tailors, after three series and a Christmas special.

The theme tune was "Release Me", a song which had been a hit for pop singer Engelbert Humperdinck. In the first series it was performed over the opening credits by Whitehouse in the guise of abnormally transfiguring singer Kenny Valentine. In subsequent series, the tune only appeared in the closing credits, played on the saxophone.

Major characters[link]

The show featured many characters and sketches. Some of the more prominent recurring characters/sketches are:

• 'Unlucky' Alf, the lonely old pensioner living somewhere in northern England for whom nothing ever goes right. His hook is his resigned "Oh bugger!" as something terrible happens. He often predicts a bad event that is quite obvious, only to find something else occurs as he tries to avoid the first problem. (Paul Whitehouse, all series)
• Anyone fancy a pint? A man (Whitehouse) who find himself in boring or bizarre situations, such as a dinner party where a woman is talking about how she was abandoned as a child and crying about everyone letting her down. Whitehouse then interrupts at the most insensitive moment asking "anyone fancy a pint?", before he and most of the men in the room leave. One early sketch featured Higson portraying an earnest claymation animator (a parody of Nick Park) who describes the animation process in excruciatingly tedious detail by moving each feature "just a tiny amount" until Whitehouse's character sneaks away, whispering the punchline. According to the commentary in The Ultimate Fast Show, Nick Park loved the sketch and sent copies of it to friends and family that year as a video Xmas card.
• Archie the pub bore. Talks to people in the pub, and when they mention their profession, no matter what it is and however unlikely, he always claims to have had the same profession ("I used to be a single mother myself"), saying that it is the 'hardest game in the world. Thirty years, man and boy!' He has an obsession with Frank Sinatra, almost invariably steering the conversation towards the singer and weakly singing the title line of "High Hopes", after mentioning how he and his friend Stan fared on a recent fishing trip. (Whitehouse, series 3)
• Arthur Atkinson, parody of 1940s music hall entertainers such as Max Miller and Arthur Askey, played by Paul Whitehouse, introduced by Tommy Cockles (Simon Day), himself a parody of presenters of TV history, especially Denis Norden (Whitehouse, all series)
• Billy Bleach, tousle-mopped, interfering pub know-it-all who gets it all wrong, usually ending up with others losing money. His catchphrases include "Hold the bells" and "Someone's sitting there, mate". This character starred in his own series, Grass which was shown on BBC Three, later shown on BBC Two. (Day, all series)
• Bob Fleming, the ageing, incompetent West Country host of Country Matters, who has an extremely bad cough. The character's surname is a pun on 'phlegm-ing.' (Higson, all series) 'Country matters' is also an in-joke for those who know it is a Shakespearean euphemism for cunnilingus.
• Brilliant Kid. In the first draft of the script for the pilot, the character was called Eric and was described as "a young Yorkshire man"^[4] but in the series he is never named. He delivers an edited monologue listing everyday things, all of which he declares to be 'brilliant!' or 'fantastic!' as he walks through a series of random backgrounds (filmed in widely varied locations ranging from the Tees Valley to Iceland) during which the quality and format of the images also randomly changes (e.g. from colour to b/w). In one episode it is shown that he isn't sure about whether everything really is 'brilliant' or not, and as he's walking through one solid background, an abandoned funfair, he debates with himself half heartedly ("Everything is brilliant...right? I mean...it might not be...nah, it is!") (Whitehouse, all series)
• Carl Hooper, Australian presenter of That's Amazing, a spoof of pop-science shows. Normally the person on his show was trying to pass off an everyday animal or object as something magical. The one occasion where a guest had a truly amazing story to tell was unbroadcastable due to the guest's inability to refrain from swearing excitedly while relating the tale (Day, all series)
• Chanel 9, a low-budget television channel from a country known only as "Republicca", ruled by "El Presidente" who resembled a stereotypical Latin American dictator. Chanel 9 parodied the sort of programmes that British people end up watching on holiday around the Mediterranean. The stars, usually Paul Whitehouse, Paul Shearer and Caroline Aherne, speak a concocted language loosely based on Italian, Greek, Spanish and Portuguese, mashed together with nonsensical phrases (e.g. "sminky pinky") and incongruous English names and words (e.g. footballer Chris Waddle). Early segments featured the Chanel 9 news, read by anchormen Poutremos Poutra-Poutremos (Whitehouse) and Kolothos Apollonia (Shearer), followed by the weather forecast with meteorologist Poula Fisch (Aherne), invariably reporting a temperature for all locations of 45 °C (113 °F) while exclaiming "scorchio!" with apparent surprise. Sporting news was presented by Antonios Gubba - whose name is based on the commentator Tony Gubba - (Simon Day), seated at a much lower desk and talking with a low voice. Later Chanel 9 sketches consisted of several short segments, usually interspersed with parodic advertisements for the ubiquitous "Gizmo" a bizarre mechanical product which could be used with equal efficacy in the kitchen, garden or bedroom. Other Chanel 9 programmes, including a cartoon, a current affairs discussion, a quiz show, a soap opera ("El Amora Y El Passionna") and a variety series, were added in later sketches. Key catchphrases included the Republiccan greeting and farewell, "Buono estente" and "Boutros Boutros-Ghali" (the name of the then UN Secretary-General).
• Chip Cobb, the deaf stuntman, a TV and film stuntman who, because of his hearing problems, always mishears his instructions and proceeds to carry them out wrongly before anyone can stop him, much to the despair of the film crew. (Thomson, series 3)
• Chris the Crafty Cockney, claims to be an incurable kleptomaniac ("I'll nick anything, me"). He is left alone with something valuable and invariably steals it. Because of how up-front he is of his thieving nature, other people tend to believe he's joking. In one sketch, he even alludes to being an actual clinical kleptomaniac and involuntarily steals from his friend Dan after he trusts Chris to watch his newspaper stall, after extensively warning him of the risks involved in doing so. (Whitehouse, series 2–3)
• Colin Hunt, unfunny and irritating office joker whose name gives an indication of his personality (Higson, series 2–3)
• Competitive Dad, who is overcritical and demanding of his kids, and always has to get one up on them. (Day, series 2–3). Day explained in an interview that he had based the idea for the character from a man he noticed in a public swimming pool that challenged his two young children to a race. Day thought he'd let them win, but instead he took off and stood on the other end of the pool waiting for his toddler sons to struggle their way across the pool. He (Day) thought of it as "sick".
• Dave Angel, Eco-Warrior, a classic Essex geezer who, despite his get-up and rather lavish lifestyle, is improbably concerned about saving the planet (though this is often undermined by his wife's behaviour), Mike Oldfield records, and swinging. A parody of a late-night magazine programme presented by Mike Reid. "Moonlight Shadow" by Mike Oldfield & Maggie Reilly is used as the theme tune to sketches featuring the character (Day, series 3)
• Professor Denzil Dexter of the University of Southern California and his various bizarre scientific experiments, long-haired and highly laid-back. (Thomson, series 1–2, online series)
• The 13th Duke of Wymbourne, posh, rumpled dinner-jacketed, lecherous cigar smoker, reminisces about finding himself in wholly unsuitable places considering his 'reputation'. Based on the characters of Terry-Thomas. (Whitehouse, series 3)
• Ed Winchester, an American reporter. He beams at the camera and says "Hi! I'm Ed Winchester!" in a very upbeat voice, before the camera cuts to another scene. In his last appearance in Series 2 he appears to have been cut off as he then mentions Jesus straight after introducing himself. (Jeff Harding).
• Even Better Than That!, a slack-jawed, not too bright man who comes back from the shops with something ridiculously unnecessary instead of what his wife sent him for (Williams, series 3)
• The Fat, Sweaty Coppers A squad of police officers who cannot do their job properly as they are extremely overweight due to their constant eating and drinking.(Thomson and Weir included, series 1-2)
• Gideon Soames, white-haired, posh-talking architecture and history professor, possibly a cross between Simon Schama & Brian Sewell, with some elements of Bamber Gascoigne. Despite the serious tone of his speeches, the "history" of them becomes increasingly ridiculous. (Day series 2–3)
• Giggly Woman, a woman who seems very confident until a man she likes enters the room, and then goes all childish, as if she fancies him. The character debuted in series 1 during a small segment in the credits, but only became a recurring character later on. (Weir, series 3)
• The Historian, a jubilant, but emotionally imbalanced man who patrols the corridors of an historic boys academy alone whilst telling tall tales about former traditions both cruel and unreasonable. (Williams, series 3)
• I'll Get Me Coat, a socially inept Brummie, who is unable to make any appropriate contribution to a conversation, and therefore disgraces himself with a faux pas before using the punchline and leaving (Williams, all series)
• I'm not Pissed, a family - mother (Maria McErlane), father (Williams) and son (Day) - who are regularly pointing out that they are not drunk despite the fact they are taking regular swigs from gin bottles, beer cans etc. (series 2)
• Insecure Woman, who appears in a variety of different locations, usually exclaiming, "Does my bum look big in this?" (Weir, all series)
• Invisible Woman, who tries to converse with a group of men but is completely ignored, only for a man to repeat what she said and receive congratulations from the others for such a good idea. (Weir)
• Jesse, a taciturn country bumpkin who exclaims his strange diets, fashion tastes and experiments, in a single sentence "This week, I 'ave been mostly..." - except for one sketch, where he said "This week, I 'aven't been 'ungry." (Williams, series 2–3)
• John Actor, who plays Inspector Monkfish, the tough uncompromising cop who often exclaims to the nearest woman, "Put your knickers on and make me a cup of tea!" (Day, series 2–3, online series). Loosely based on the BBC series Dangerfield. There were variations on the show's format, two examples being Monkfish as a tough, uncompromising doctor in Monkfish M.D. and Monkfish as a tough, uncompromising vet in All Monkfish Great and Small. One Monkfish sketch even crossed over onto Chanel 9. Sometime between the end of series 3 and the last episode, John Actor died yet the series is apparently continuing in the manner of Taggart after the death of the lead actor.
• Johnny Nice Painter, who is painting a scene and describing all the colours. Whenever he mentions the colour "black", however, he becomes more and more depressed, eventually going somewhat insane and shouting wildly about the despair of mankind, despite the best efforts of his wife Katie (Weir) to prevent him from doing so. His appearance is based on TV painter Alwyn Crawshaw (Higson, series 3, online series)
• Ken and Kenneth, two tailors in a men's formal wear shop, who bombard potential customers with sexually explicit innuendo about their private life, frequently interjecting the catchphrase "Ooh! Suit you sir!", much to the discomfort of the customer. They become confused and even frightened in two episodes; one when they get a customer who is gay, and another with a customer (played by Day) who is as willing to talk about sexual deviance as they are. Due to Williams absence from the online series, his character Kenneth will be written out and replaced by Kenton, played by Charlie Higson. (Whitehouse and Williams: 1,2,3, and Whitehouse and Higson: online specials)
• Louis Balfour, pretentious and ultra laid-back presenter of Jazz Club (a parody of The Old Grey Whistle Test), based on a blend of Bob Harris and Roger Moore. He also has a remarkable similarity to Geoffrey Smith, presenter of Jazz Record Requests on Radio 3). Seemingly having done his 'research', he introduces his guests by comparing them to avant-garde jazz musicians or describing their style/technique by using complex musical phraseology. These guests usually turn out to be utterly talentless 'experimentalists', much to his bemusement. His catchphrase "Nice!" was delivered by turning to a different camera for that word only. Later he delivered other words in a similar manner. (Thomson, series 2–3, online series)
• Luvvie an over-the-top thespian describes his character and mentions that he needs to spend hours being made up when he is actually done in a few seconds (Thomson, series 3)
• Monster Monster, a vampire who creeps up on a slumbering woman and gives her betting advice (Whitehouse, series 3). The set-up is a parody of a scene from the 1922 German classic horror film Nosferatu, while the vampire's voice and catchphrase of "Monster, monster" are based on Eric Hall.
• No Offence, a rude, orange-faced South African department store cosmetics saleswoman who has no qualms about informing women of their physical imperfections, seemingly oblivious to the fact that she looks like a dried-out old orange herself. (Weir, series 3)
• "Our" Janine Carr, teenage mum with a unique world outlook. She refuses to reveal who the father of her baby is because "it's not fair to grass on your headmaster" (Aherne, series 1–2, online series)
• The Offroaders, Simon and Lindsey, despite their unusually high confidence and esteem, are useless at their hobby ("sorted!", "gripped!"). (Higson and Whitehouse, all series)
• Patrick Nice, a man who tells far-fetched, sometimes odd stories followed by him calmingly saying his catchphrase, "Which was nice." (Williams, series 2–3)
• Prawn Sandwich Man, a man who seems to talk a lot about football as if a true Arsenal supporter, but makes it glaringly obvious he knows nothing about the game (Thomson, series 3)
• Ron Manager, a football commentator who speaks in incoherent sentence fragments on randomly divergent trains of thought. He usually appears with interviewer (Day) and fellow commentator Tommy (Williams), and whenever a question is posed to Tommy, Ron Manager often begins one of his nonsensical train-of-thoughts based on one of the words or names in the question. Based on former football manager from the 60s and 70s, Alec Stock.^[5] (Whitehouse: all TV and online series)
• Rowley Birkin QC, a retired barrister, tells mostly unintelligible stories at the fireside. Occasionally, his speech becomes coherent for a short while, containing strange phrases such as "The whole thing was made completely out of matchsticks" or "Snake! Snake!". Almost always ends his stories with a sly "I'm afraid I was very, very drunk!" In the final episode of series 2, his rambling anecdote appeared to involve a woman for whom he had great affection and ended with a close-up of faint tears on his cheeks, while the usual "very drunk" line was delivered in an unexpectedly moving, sorrowful voice. . The character is reprised as a working barrister in the spin-off feature Ted and Ralph. Whitehouse revealed on the UK chatshow Parkinson that the character was based on Andrew Rollo whom he met on a fishing trip to Iceland; Rollo appeared in the Fast Show documentary, which revealed how closely Rowley's speech resembled that of his real-life inspiration. At the end of a Christmas episode the caption revealed that he died. "Rowley Birkin QC 1918 - 2000", however despite this on-screen demise he will appear in the 2011 online specials.(Whitehouse, TV series 2–3, and online series)
• Roy & Renée, endless chattering from Renée and her quiet, submissive husband Roy, whom she expects to meekly agree with everything she says. Roy always embarrasses her at the end of every sketch, after which he gets a stinging reprimand from his wife. She makes her last appearance in the show during the 1996 Christmas Special, when Roy's mother finally gives in to holding back the resentment towards Renée's smug attitude. (Thomson and Aherne, series 1–2)
• Rubbish Dad, the father and opposite of Brilliant Kid who proclaims everything to be "rubbish". He is usually only seen in an industrial scrapheap area. The one thing he does like is Des Lynam. (Thomson, all series)
• The Shags, a child-less couple who are seen in the midst of very graphic sexual intercourse, much to the discomfort of their neighbours. Among other things they were seen at it in a tent in a sports shop, in a tree in the park and even on a bed being carried by removal men as they moved into the neighbourhood.
• Swiss Toni, a car salesman who compares everything to seducing and making love to a beautiful woman, usually in the presence of his bemused trainee Paul (Rhys Thomas). This was also the title of a short-lived spin-off sitcom, featuring Toni in the car dealership which he worked. Swiss is the only non-original character in the show. He had previously appeared in the second series of The Smell of Reeves and Mortimer in 1995, which was produced by Higson and featured cameos from many members of The Fast Show. (Higson: TV series 3 and online)
• Ted & Ralph – country squire Lord Ralph Mayhew attempts to strike-up an intimate relationship with his introverted Irish estate worker Ted, by way of subtle romantic/erotic subtexts in his conversations with him (Whitehouse and Higson, all series). This was also the title of a one-off, hour-long spin-off feature, reprising the characters, with cameos from a few other characters as well. (Higson and Whitehouse: all TV and online series)
• "You ain't seen me: right?" - An unknown traveller who is probably a criminal who says "You ain't seen me: right?" to some minor characters in the show and sometimes the viewer of the show. He comes up in the show in various locations and is at one point on Chanel 9 Neus sitting in the sports reporter's seat. He is also seen in the background when the "Brilliant" Kid walks past talking about the usual things.(Williams, all series)
• Inept Zookeeper - a zookeeper who is frightened and/or disgusted by virtually every aspect of his job (cleaning up elephant dung, feeding the penguins, etc.) and is thus rendered unable to perform his tasks properly. (Williams, series 3)

In popular culture[link]

• Arabella Weir later turned Insecure Woman into Jackie Payne, heroine of her novel Does My Bum Look Big in This?
• In Pirates of the Caribbean: The Curse of the Black Pearl, Jack Sparrow quotes the show; Johnny Depp is a fan and had a cameo role in the farewell special .
• In 2010, chef Jamie Oliver advertised his forthcoming cookery series (examining the cuisine of several European/Mediterranean countries) speaking in the made-up language used by Chanel 9 news.

Filming locations[link]

Unusually for a sketch show, a significant proportion of The Fast Show was shot externally. During the early series much of this filming was done around the Tees Valley, Yorkshire Dales and Newcastle upon Tyne in the North East of England. Locations include:

• Ashington, Northumberland – at least two scenes involving Unlucky Alf and one involving Brilliant Kid were filmed on Station Road.
• Aske Hall – Background in early Ted & Ralph scenes
• Darlington – 'The Running Family' were shown around various locations in the town centre, including The Cornmill Centre. Darlington was the childhood home of Jim Moir (Vic Reeves) whose longterm comedy partner Bob Mortimer was one of the writers. The Cornmill and The High Street with scenes involving Brilliant Kid.
• Durham – The market place with scenes involving Brilliant Kid
• Hartlepool – One Unlucky Alf scene saw him sat in the empty Rink End Stand of Hartlepool United's ground, Victoria Park. Also, one Ed Winchester scene is in front of the Mill House Stand. Some of the Brilliant Kid scenes were also fimed at nearby Seal Sands.
• Keld, North Yorkshire – The campsite used in a Dave Angel scene
• Langley Park – Railway Street is used in Unlucky Alf scenes
• Middlesbrough – docks used in 'hard of hearing stuntman' scenes, scene on Transporter Bridge as well as the Riverside Stadium
• Newcastle upon Tyne – including the 'Shore Leave' sketch, the scene where Chris the Crafty Cockney steals the woman's suitcases (shot in Newcastle Central station), and some of the Sir Geoffrey Norman MP sketches, such as the one where he is pulled over by a policeman for speeding and the one where he refuses to pay the taxi driver after getting out of the car (shot outside the main entrance to Newcastle Central station), one Ed Winchester scene is near Monument station. One scene from the Brilliant Kid showed him in Exhibition Park in Newcastle upon Tyne. Many scenes involving Janine Carr, filmed in greyscale, used the backdrop of the concrete flyover and underpasses of the junction of the A167(M) and the B1318 of the Great North Road in Jesmond. A few scenes were filmed inside the Newcastle City Library (which has now been demolished and a new Library building has replaced it).
• Richmond – The market place in Ted & Ralph's trip to the shops
• Scotch Corner – Garage used in Swiss Toni's early scenes
• Seaton Carew - One "You ain't seen me right?" scene sees the main character sat on a child's ride in one of the seafront amusement arcades.
• The Spanish City, Whitley Bay, Tyne and Wear – a number of scenes involving Brilliant Kid
• Stockton On Tees, Swiss Toni scenes, filmed at a car showroom on Norton Road.

Also for the third series the production extended abroad:

• Iceland – Scenes with Brilliant Kid and Billy Bleach were shot with volcanic landscapes, waterfalls and hot springs in the background.

Down the Line[link]

In 2006, Higson and Whitehouse produced and starred in Down the Line, a spoof phone-in show for BBC Radio 4. The show also featured many of the regular Fast Show cast, including Simon Day, Arabella Weir, Rhys Thomas and Felix Dexter. Further series were broadcast in 2007, 2008 and 2011. A follow-on TV series, Bellamy's People, was broadcast in 2010.

The future[link]

Speaking on the BBC Two show Something for the Weekend on 9 September 2007, Higson mentioned the upcoming DVD boxed set release and that a reunion of some sort to help promote it was being considered. This took place at The Dominion Theatre in London on Sunday 4 November, and was a collection of some new sketches, videos of cast favourites and performances of classic sketches (including the return of Ed Winchester). Higson and Whitehouse stated they were working on a film script which would feature the Fast Show team, but wouldn't have any of the characters from the show. A new online only series was commissioned in a sponsorship deal with Foster's Lager, and aired beginning 14 November 2011; the trailer was released on 9 November on Foster's YouTube Channel^[6][7] New episodes featured the original cast with the exception of Mark Williams, who declined involvement in the project.^[8]

DVDs[link]

This section may require copy-editing.

Numerous Fast Show DVDs are available including :

• The Fast Show : Series 1 (includes cast interviews with Paul Whitehouse, Charlie Higson, Arabella Weir and Mark Williams)
• The Fast Show : Series 2
• The Fast Show : Series 3 and 1996 Christmas Special
• The Fast Show : The Last Fast Show Ever, Part One
• The Fast Show Live
• The Fast Show Farewell Tour
• A 7 DVD box set, The Ultimate Fast Show Collection, was released in the UK on 5 November 2007, which compiled nearly all their material, except the two live DVD releases and their spin-off series/specials.

Wikiquote has a collection of quotations related to: The Fast Show

[edit] You Ain't Seen These, Right!

You Ain't Seen These, Right! was a one-off programme, shown during BBC Two's Fast Show Night, featuring various sketches which were filmed but did not make it onto the final show. Some of these were:

• An ensemble series of sketches made by the whole male team, as members of a golf club, in which Charlie Higson's character was dating a beautiful young woman. The rest of the team are initially dismissive of him as a sad old man, but cannot help gawking over her.
• A chain-smoking car driver played by Mark Williams who rants about anything and everything through his wound-down window. A study of road rage. "Shoe shop?! Shoe Shop?!". He drives around in Harlesden, London.
• A medieval king played by Simon Day, who 'loves being king' because he gets to boss everyone about.
• A middle aged man, played by John Thomson, who always finds an excuse to leave the room as soon as the conversation gets round to "women's things."
• A Paul Whitehouse character who responds to almost every question, accusation and situation with the phrase "Sorry, but I was up all night, shagging."

These sketches are included in the UK edition of the boxed VHS videotape set of Series 3, and also on the 7 disc Ultimate Fast Show DVD box set.

References[link]

1. ^ ^{a b c} Dewhurst, Keith (2007), "The Fast Show - A Personal View" (notes for The Ultimate Fast Show Collection DVD set, BBC)
2. ^ https://twitter.com/#!/monstroso/status/110748812594790400
3. ^ http://fosters.co.uk/tagged/The_Fast_Show
4. ^ liner notes to The Ultimate Fast Show Collection, (BBC, 2007)
5. ^ http://www.wsc.co.uk/content/view/4024/29/}}
6. ^ http://www.youtube.com/fosters
7. ^ http://offlicencenews.co.uk/news/fullstory.php/aid/12379/Fast_Show_returns_in_Foster_s_deal.html
8. ^ http://www.telegraph.co.uk/culture/tvandradio/8744496/The-Fast-Show-returns.html

External links[link]

• The Fast Show at BBC Programmes
• The Fast Show at the Internet Movie Database