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A surface is a synonym for a 2-dimensional manifold. Complex 2-dimensional (real 4-dimensional) complex manifolds are also called surfaces. This article deals with real, compact, connected surfaces. Unless stated otherwise (Sections 2 and 3), surfaces without boundary are considered.
1 First construction: connected sum
All orientable surfaces are homeomorphic to the connected sum of tori (). The case refers to the 2-sphere . All non-orientable surfaces are homeomorphic to the connected sum of real projective spaces (). For example, the Klein bottle is homeomorphic to . The numbers and are called the genus, if the surface is orientable, and genus or cross-cap number, if the surface is non-orientable.
The connected sum of with is homeomorphic to . Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by and , with the sole relation .
2 Surfaces with boundary
The boundary of a surface is a disjoint union (possibly empty) of circles. Surfaces with boundary can be constructed by removing open discs from surfaces without boundary.
Compact, connected surfaces are classified by orientability (yes/no), the number of boundary components (a nonnegative integer) and the genus after filling the bounday circles by disks (an integer in the orientable case, in the non-orientable case). Instead of the genus, also e. g. the Euler characteristic can be used in the classification. The classifications up to homotopy equivalence, homeomorphism, PL-equivalence and diffeomorphism coincide.
4 More constructions
4.1 By polygons
Each orientable surface of genus can be constructed by identifying pairs of edges in a regular polygon with sides. Label the edges by the sequence . Also orient the edges such that those labeled without an overbar are oriented in one direction (e. g. clockwise) and those with an overbar are oppositely oriented. Now identify corresponding edges, respecting the orientation.
The 2-sphere can be obtained from a 2-gon with edges labeled .
Each non-orientable surface of genus can be obtained from a -gon with edges labeled .
Reference: [Massey1977, Section 1.5]
4.2 By gluing handles and crosscaps
An orientable surface of genus can be obtained by successively gluing 1-handles to the 2-sphere such that the embeddings of in are in each case either orientation-preserving or orientation-reversing on both components and .
A non-orientable surface of genus can be obtained by gluing crosscaps to . For this, embed in (or from the second crosscap on), remove the interior and glue in the Möbius strip, which also has boundary . The result of attaching a non-orientable handle to or any handle to a non-orientable surface is diffeomorphic to the surface with two additional crosscaps.
Reference: [Hirsch1994, Section 9.1]
4.3 By branched coverings
Every orientable surface of genus can be obtained as the branched double covering of the 2-sphere with branching points.
Proof. The branched double covering of the 2-sphere, with branch points, has Euler characteristic . Thus every orientable closed surface is such a branched covering.
4.4 As complex curves
A smooth, irreducible, plane, complex projective-algebraic curve of order (i. e. the zero set of a non-constant, homogeneous, irreducible polynomial of degree in whose gradient vector does not vanish in any point of the zero set) is a compact, connected, orientable, real surface of genus
Furthermore, surfaces of every degree can be obtained as normalizations of plane, complex projective-algebraic curves with only double points as singularities.
Reference: [Brieskorn&Knörrer1986, Lemma 9.2.1]
5.1 Orientable surfaces
Let denote an oriented surface of genus .
- By the polygon construction above, each orientable surface has a cell decomposition with one 0-cell, 1-cells and one 2-cell. All differentials in the chain complex are zero maps.
- Therefore, the homology groups with any coefficients are given by , , .
- The integral cohomology ring is completely determined by the intersection form on , which is necessarily isomorphic to the hyperbolic form. A basis for , for which the intersection form is the standard hyperbolic form, is given by the loops in the polygon construction above.
- Homotopy groups
- For the homotopy group of , see the article about spheres.
- All surfaces of higher genus are aspherical with fundamental group . In particular, the fundamental group of the 2-torus is isomorphic to the abelian group .
- All orientable surfaces can be embedded in .
- Every surface can be given a complex structure. Together with the complex structure, it is a Riemann surface
- Characteristic classes
- All Stiefel-Whitney classes vanish.
- All Pontrjagin classes vanish.
- The Euler characteristic is .
- Given a complex structure on the surface, the first Chern class is equal to the Euler class.
- All orientable surfaces admit metrics with constant curvature: the standard metric on the unit sphere in has constant positive curvature, the torus can be given a flat metric, and all surfaces of higher genus admit metrics of constant negative curvature.
- All surfaces are smoothly amphicheiral.
5.2 Non-orientable surfaces
Let denote an non-orientable surface of genus .
- Again, each non-orientable surface has a cell decomposition with one 0-cell, 1-cells and one 2-cell. The differential is the zero map, while the differential with respect to the basis given by the loops the loops in the polygon construction is the matrix .
- Therefore, the integral homology groups are isomorphic to , , .
- The mod-2 homology groups are , , since the chain complex above is acyclic mod 2. The intersection form on with respect to this cellular basis is given by the identity matrix.
- Homotopy groups
- The fundamental group of is .
- The orientation double covering of is . Therefore, all higher homotopy groups of equal those of , and the non-orientable surfaces of higher genus are aspherical.
- The orientation double covering also determines the curvature properties: can be given a metric with constant positive curvature, the Klein bottle a flat metric, and all nonorientable surfaces can be given metrics with constant negative curvature.
- Non-orientable surfaces cannot be embedded in . However, they do embed in .
- Characteristic classes
- The first Stiefel-Whitney class is the orientation character. It can be described by the homomorphism which maps each generator to the generator of . The second Stiefel-Whitney class is zero if is even and is the non-zero element of if is odd. In all cases .
- The Euler characteristic is .
There are purely point-set topology characterizations of surfaces; see [Wilder1949].
- [Ahlfors&Sario1960] L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton University Press, Princeton, N.J., 1960. MR0114911 (22 #5729) Zbl 0508.01017
- [Brieskorn&Knörrer1986] E. Brieskorn and H. Knörrer, Plane algebraic curves, Birkhäuser Verlag, Basel, 1986. MR886476 (88a:14001) Zbl 0588.14019
- [Hirsch1994] M. W. Hirsch, Differential topology, Springer-Verlag, New York, 1994. MR1336822 (96c:57001) Zbl 0804.57001
- [Massey1977] W. S. Massey, Algebraic topology: an introduction, Springer-Verlag, New York, 1977. MR0448331 (56 #6638) Zbl 0725.55001
- [Moise1977] E. E. Moise, Geometric topology in dimensions and , Springer-Verlag, New York, 1977. MR0488059 (58 #7631) Zbl 0349.57001
- [Wilder1949] R. L. Wilder, Topology of Manifolds, American Mathematical Society, 1949. MR0598636 (82a:57001) Zbl 0511.57001