Surface (topology)

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world.

In general

In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R³, such as spheres. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself, while, in topology and differential geometry, it may not.
A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it.
The concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface.

Definitions and first examples

A surface is a topological space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E². Such a neighborhood, together with the corresponding homeomorphism, is known as a chart. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.
In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second-countable, and Hausdorff. It is also often assumed that the surfaces under consideration are connected.
The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second-countable, and connected.
More generally, a surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half-plane H² in C. These homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis. A point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of such points is known as the boundary of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the x-axis is an interior point. The collection of interior points is the interior of the surface which is always non-empty. The closed disk is a simple example of a surface with boundary. The boundary of the disc is a circle.
The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces.
The Möbius strip is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not.
In differential and algebraic geometry, extra structure is added upon the topology of the surface. This added structure can be a smoothness structure, a Riemannian metric, a complex structure, or an algebraic structure.

Extrinsically defined surfaces and embeddings

Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the locus of zeros of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger space, and as such was termed extrinsic.
In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean. This topological space is not considered a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is intrinsic.
A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space. It may seem possible for some surfaces defined intrinsically to not be surfaces in the extrinsic sense. However, the Whitney embedding theorem asserts every surface can in fact be embedded homeomorphically into Euclidean space, in fact into E⁴: The extrinsic and intrinsic approaches turn out to be equivalent.
In fact, any compact surface that is either orientable or has a boundary can be embedded in E³; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E³. Steiner surfaces, including Boy's surface, the Roman surface and the cross-cap, are models of the real projective plane in E³, but only the Boy surface is an immersed surface. All these models are singular at points where they intersect themselves.
The Alexander horned sphere is a well-known pathological embedding of the two-sphere into the three-sphere.
Image:KnottedTorus.svg|right|thumb|A knotted torus.
The chosen embedding of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into E³ in the "standard" manner or in a knotted manner. The two embedded tori are homeomorphic, but not isotopic: They are topologically equivalent, but their embeddings are not.
The image of a continuous, injective function from R² to higher-dimensional Rⁿ is said to be a parametric surface. Such an image is so-called because the x- and y- directions of the domain R² are 2 variables that parametrize the image. A parametric surface need not be a topological surface. A surface of revolution can be viewed as a special kind of parametric surface.
If f is a smooth function from R³ to R whose gradient is nowhere zero, then the locus of zeros of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities.

Construction from polygons

Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels, so that the arrows point in the same direction, yields the indicated surface.
Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex. During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield

sphere:
real projective plane:
torus:
Klein bottle:.

Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon.
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem.
Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum.

Connected sums

The connected sum of two surfaces M and N, denoted M # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The Euler characteristic of is the sum of the Euler characteristics of the summands, minus two:
The sphere S is an identity element for the connected sum, meaning that. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing.
Connected summation with the torus T is also described as attaching a "handle" to the other summand M. If M is orientable, then so is. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.
The connected sum of two real projective planes,, is the Klein bottle K. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus; in a formula,. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.