Surface (topology)
In the part of mathematics referred to as
Topological surfaces are sometimes equipped with additional information, such as a
In general
In
A surface is a
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 (topological) surface is a
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 (topological) surface with boundary is a
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 (because the real projective plane with one point removed is homeomorphic to the open Möbius strip).
In
Extrinsically defined surfaces and embeddings
Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the
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 (see Gramain).
The Alexander horned sphere is a well-known pathological embedding of the two-sphere into the three-sphere.
The chosen embedding (if any) 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 (which looks like a bagel) or in a knotted manner (see figure). The two embedded tori are homeomorphic, but not isotopic: They are topologically equivalent, but their embeddings are not.
The
If f is a smooth function from R3 to R whose
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 (A with A, B with B), 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 (square).
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a
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 M # N 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 S # M = M. 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 T # M. 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, P # P, 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, P # K = P # T. 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.
Closed surfaces
A closed surface is a surface that is
A surface embedded in three-dimensional space is closed if and only if it is the boundary of a solid. As with any closed manifold, a surface embedded in Euclidean space that is closed with respect to the inherited Euclidean topology is not necessarily a closed surface; for example, a disk embedded in that contains its boundary is a surface that is topologically closed but not a closed surface.
Classification of closed surfaces
The classification theorem of closed surfaces states that any
- the sphere,
- the connected sum of g tori for g ≥ 1,
- the connected sum of k real projective planes for k ≥ 1.
The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of g tori is 2 − 2g.
The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of k of them is 2 − k.
It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism.
Closed surfaces with multiple
Monoid structure
Relating this classification to connected sums, the closed surfaces up to homeomorphism form a
Geometrically, connect-sum with a torus (# T) adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle (# K) adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane (# P), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation.
Proof
The classification of closed surfaces has been known since the 1860s,[1] and today a number of proofs exist.
Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a
A geometric proof, which yields a stronger geometric result, is the uniformization theorem. This was originally proven only for Riemann surfaces in the 1880s and 1900s by Felix Klein, Paul Koebe, and Henri Poincaré.
Surfaces with boundary
This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing k open discs yields a compact surface with k disjoint circles for boundary components. The precise locations of the holes are irrelevant, because the
Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the cone) yields a closed surface.
The unique compact orientable surface of genus g and with k boundary components is often denoted for example in the study of the mapping class group.
Non-compact surfaces
Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold. On the other hand, any open subset of a compact surface is itself a non-compact surface; consider, for example, the complement of a
A non-compact surface M has a non-empty space of ends E(M), which informally speaking describes the ways that the surface "goes off to infinity". The space E(M) is always topologically equivalent to a closed subspace of the Cantor set. M may have a finite or countably infinite number Nh of handles, as well as a finite or countably infinite number Np of projective planes. If both Nh and Np are finite, then these two numbers, and the topological type of space of ends, classify the surface M up to topological equivalence. If either or both of Nh and Np is infinite, then the topological type of M depends not only on these two numbers but also on how the infinite one(s) approach the space of ends. In general the topological type of M is determined by the four subspaces of E(M) that are limit points of infinitely many handles and infinitely many projective planes, limit points of only handles, limit points of only projective planes, and limit points of neither.[3]
Assumption of second-countability
If one removes the assumption of second-countability from the definition of a surface, there exist (necessarily non-compact) topological surfaces having no countable base for their topology. Perhaps the simplest example is the Cartesian product of the long line with the space of real numbers.
Another surface having no countable base for its topology, but not requiring the Axiom of Choice to prove its existence, is the
In 1925, Tibor Radó proved that all Riemann surfaces (i.e., one-dimensional
Surfaces in geometry
Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2. This elaboration allows calculus to be applied to surfaces to prove many results.
Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.
Smooth surfaces equipped with
This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds).
Another way in which surfaces arise in geometry is by passing into the complex domain. A complex one-manifold is a smooth oriented surface, also called a Riemann surface. Any complex nonsingular algebraic curve viewed as a complex manifold is a Riemann surface. In fact, every compact orientable surface is realizable as a Riemann surface. Thus compact Riemann surfaces are characterized topologically by their genus: 0, 1, 2, .... On the other hand, the genus does not characterize the complex structure. For example, there are uncountably many non-isomorphic compact Riemann surfaces of genus 1 (the elliptic curves).
Complex structures on a closed oriented surface correspond to
A complex surface is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves defined over fields other than the complex numbers, nor are algebraic surfaces defined over fields other than the real numbers.
See also
- Boundary (topology)
- Volume form, for volumes of surfaces in En
- Poincaré metric, for metric properties of Riemann surfaces
- Roman surface
- Boy's surface
- Tetrahemihexahedron
- Crumpled surface, a non-differentiable surface obtained by deforming (crumpling) a differentiable surface
Notes
References
- S2CID 118123073
Simplicial proofs of classification up to homeomorphism
- Seifert, Herbert; Threlfall, William (1980), A textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press, ISBN 0126348502, English translation of 1934 classic German textbook
- Ahlfors, Lars V.; Sario, Leo (1960), Riemann surfaces, Princeton Mathematical Series, vol. 26, Princeton University Press, Chapter I
- Maunder, C. R. F. (1996), Algebraic topology, Dover Publications, ISBN 0486691314, Cambridge undergraduate course
- Massey, William S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag. ISBN 0-387-97430-X.
- ISBN 0-387-97926-3.
- Jost, Jürgen (2006), Compact Riemann surfaces: an introduction to contemporary mathematics (3rd ed.), Springer, ISBN 3540330658, for closed oriented Riemannian manifolds
Morse theoretic proofs of classification up to diffeomorphism
- Hirsch, M. (1994), Differential topology (2nd ed.), Springer
- Gauld, David B. (1982), Differential topology: an introduction, Monographs and Textbooks in Pure and Applied Mathematics, vol. 72, Marcel Dekker, ISBN 0824717090
- Shastri, Anant R. (2011), Elements of differential topology, CRC Press, ISBN 9781439831601, careful proof aimed at undergraduates
- Gramain, André (1984). Topology of Surfaces. BCS Associates.
- A. Champanerkar; et al., Classification of surfaces via Morse Theory (PDF), an exposition of Gramain's notes
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Other proofs
- Lawson, Terry (2003), Topology: a geometric approach, Oxford University Press, ISBN 0-19-851597-9, similar to Morse theoretic proof using sliding of attached handles
- Francis, George K.; Weeks, Jeffrey R. (May 1999), "Conway's ZIP Proof" (PDF), JSTOR 2589143; page discussing the paper: On Conway's ZIP Proof
- Thomassen, Carsten (1992), "The Jordan-Schönflies theorem and the classification of surfaces", Amer. Math. Monthly, 99 (2): 116–13, JSTOR 2324180, short elementary proof using spanning graphs
- Prasolov, V.V. (2006), Elements of combinatorial and differential topology, Graduate Studies in Mathematics, vol. 74, American Mathematical Society, ISBN 0821838091, contains short account of Thomassen's proof
External links
- Classification of Compact Surfaces in Mathifold Project
- The Classification of Surfaces and the Jordan Curve Theorem in Home page of Andrew Ranicki
- Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
- Math Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewing
- The Classification of Surfaces Lecture Notes by Z.Fiedorowicz
- History and Art of Surfaces and their Mathematical Models
- 2-manifolds at the Manifold Atlas