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Closed surfaces {{#invoke:Hatnote|hatnote}}

A closed surface is a surface that is compact and without boundary. Examples are spaces like the sphere, the torus and the Klein bottle. Examples of non-closed surfaces are: an open disk, which is a sphere with a puncture; a cylinder, which is a sphere with two punctures; and the Möbius strip.

Classification of closed surfaces

Some examples of orientable closed surfaces (left) and surfaces with boundary (right). Left: Some orientable closed surfaces are the surface of a sphere, the surface of a torus, and the surface of a cube. (The cube and the sphere are topologically equivalent to each other.) Right: Some surfaces with boundary are the disk surface, square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:

  1. the sphere;
  2. the connected sum of g tori, for <math>g \geq 1</math>;
  3. the connected sum of k real projective planes, for <math>k \geq 1</math>.

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 connected components are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.

Monoid structure

Relating this classification to connected sums, the closed surfaces up to homeomorphism form a commutative monoid under the operation of connected sum, as indeed do manifolds of any fixed dimension. The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation P # P # P = P # T, which may also be written P # K = P # T, since K = P # P. This relation is sometimes known as Dyck's theorem after Walther von Dyck, who proved it in , and the triple cross surface P # P # P is accordingly called Dyck's surface.<ref name="fw"/>

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.

Surfaces with boundary

Compact surfaces, possibly with boundary, are simply closed surfaces with a finite number of holes (open discs that have been removed). Thus, a connected compact surface is classified by the number of boundary components and the genus of the corresponding closed surface – equivalently, by the number of boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface.

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 homeomorphism group acts k-transitively on any connected manifold of dimension at least 2.

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 <math>\Sigma_{g,k},</math> for example in the study of the mapping class group.

Riemann surfaces

A closely related example to the classification of compact 2-manifolds is the classification of compact Riemann surfaces, i.e., compact complex 1-manifolds. (Note that the 2-sphere and the torus are both complex manifolds, in fact algebraic varieties.) Since every complex manifold is orientable, the connected sums of projective planes are not complex manifolds. Thus, compact Riemann surfaces are characterized topologically simply by their genus. The genus counts the number of holes in the manifold: the sphere has genus 0, the one-holed torus genus 1, etc.

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 Cantor set in the sphere, otherwise known as the Cantor tree surface. However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the Jacob's ladder and the Loch Ness monster, which are non-compact surfaces with infinite genus.

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 th 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.

Surfaces that are not even second countable

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 Prüfer manifold, which can be described by simple equations that show it to be a real-analytic surface. The Prüfer manifold may be thought of as the upper half plane together with one additional "tongue" Tx hanging down from it directly below the point (x,0), for each real x.

In 1925, Tibor Radó proved the theorem that non-compact Riemann surfaces (i.e., one-dimensional complex manifolds) are necessarily second countable. By contrast, the existence of the Prüfer surface shows that there exist two-dimensional complex manifolds (which are necessarily 4-dimensional real manifolds) with no countable base. (This is because any n-real-dimensional real-analytic manifold Q can be extended to an n-complex-dimensional complex manifold W that contains Q as a real-analytic submanifold.)

Proof

The classification of closed surfaces has been known since the 1860s,<ref name="fw"></ref> 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 simplicial complex, which is of interest in its own right. The most common proof of the classification is ,<ref name="fw"/> which brings every triangulated surface to a standard form. A simplified proof, which avoids a standard form, was discovered by John H. Conway circa 1992, which he called the "Zero Irrelevancy Proof" or "ZIP proof" and is presented in .

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.


Surface sections
Intro  Definitions and first examples  Extrinsically defined surfaces and embeddings  Construction from polygons  Connected sums   Closed surfaces   Surfaces in geometry  See also  Notes  References  External links  

Closed surfaces
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