nLab surface

Contents

Context

Manifolds and cobordisms

Idea
Properties

Classification
Homotopy
CoHomology

Examples
Related concepts
References

Idea

A surface is a space of dimension 2, usually understood to be connected.

In differential topology/differential geometry this means a 2-dimensional (differentiable/smooth) manifold.

In complex analytic geometry this usually means a complex manifold of complex dimension 2 (hence real dimension 4).

Similarly and more generally, in algebraic geometry an algebraic surface is a variety of algebraic dimension 2.

Properties

Classification

The oriented closed real surfaces are classified, up to isomorphism, by a natural number $g \in \mathbb{N}$ called the genus (intuitively its “number of holes”, in a sense):

genus	oriented surface
$0$	2-sphere $S^2$
$1$	2-torus $\mathbb{T}^2$
$2$	double torus $\Sigma^2_2$

(cf. Kinsey 1991 Thm. 4.14, Gallier & Xu 2013 Thm 6.3, Bray et al. Thm. 4.14)

Here the closed oriented surface $\Sigma^2_g$ of genus $g$ may be obtained from the regular $4g$ -gon by identifying (cf. Giblin 1977 Thm 2.8):

all the boundary vertices with a single point,

and, going clockwise for $k \in \{0, \cdots, g-1\}$ ,

the $4k+1$ st boundary edge with the reverse of the $4k +3$ rd,
the $4k+2$ nd edge with the reverse of the $4k+4$ th.

(This and the following graphics are taken from SS25.)

Similarly, the non-orientable closed real surfaces are classified by a positive natural number $h \in \mathbb{N}_{\geq 1}$ , also called the (non-orientable) genus or the number of crosscaps

# crosscaps	non-orientable surface
$1$	projective plane $\mathbb{R}P^2$
$2$	Klein bottle $\mathbb{R}P^2 # \mathbb{R}P^2$

Here the non-orientable surface $\Sigma^2_{\overline{g}}$ with $h$ crosscaps may be obtained from the regular $2h$ -gon by identifying

all the boundary vertices with a single point,

and, going clockwise for $k \in \{0, \cdots, h-1\}$ ,

the $2k+1$ st boundary edge with the $2k + 2$ nd.

(This is a conveniently concise but not the most intuitively visualized choice of fundamental polygons – other choices are possible and often discussed, cf. also MO:q/172784.)

Homotopy

The above classification may be restated in algebro-topological terms by saying that (the homotopy type of) the oriented closed genus $=g$ surface has a 2-dimensional CW-complex-structure exhibited by the following pointed cell attachment:

Here $\bigvee_{2g} S^1$ is the wedge sum of $2g$ circles, whose fundamental group is the free group on $2g$ generators $(a_i, b_i)_{i=1}^g$ , and the attaching map is a representative in this free group of the composition of group commutators

[a_i, b_i] \;\coloneqq\; a_i \cdot b_i \cdot a_i^{-1} \cdot b_i^{-1} \,,

as indicated.

Similarly for (the homotopy type of) the non-orientable surface $\Sigma^2_{\overline{h}}$ of crosscap number $h$ :

It follows that:

Proposition

The fundamental group of the oriented closed real surface $\Sigma^2_g$ of genus $g \in \mathbb{N}$ (see above) is the quotient group of the free group on $2g$ generators $(a_1, \cdots, a_g, \, b_1, \cdots, b_g)$ by the normal subgroup generated by the group product of the group commutators $[a_i, b_i]$ of the sequence of pairs of generators:

\pi_1\big( \Sigma^2_g \big) \;\simeq\; \big\langle a_1, \cdots, a_g ,\, b_1, \cdots, b_g \big\rangle \big/ \Big( \textstyle{\prod_i} [a_i, b_i] \Big) \,.

Similarly, the fundamental group of the non-orientable closed real surface $\Sigma^2_{\overline{h}}$ with $h \in \mathbb{N}_{\geq 1}$ crosscaps is the quotient group of the free group on $h$ generators $(a_1, \cdots, a_h)$ by the normal subgroup generated by the group product of the squares $a^2_ \,\coloneqq\, a_i \cdot a_i$ of the sequence of pairs of generators:

\pi_1\big( \Sigma^2_{\overline{h}} \big) \;\simeq\; \big\langle a_1, \cdots, a_h \big\rangle \big/ \Big( \textstyle{\prod_i} a_i^2 \Big) \,.

(cf. Gallier & Xu 2013 p 100, Actipes 2013 Thm. 6.3).

CoHomology

Proposition

The ordinary homology with integer coefficients of a closed oriented surface $\Sigma_g$ of genus $G$ is the free abelian group on $2g$ generators:

H_1\big( \Sigma_g ;\, \mathbb{Z} \big) \;\simeq\; \mathbb{Z}^{2g} \,.

(e.g. Giblin 1977 Thm 6.13)

By the universal coefficient theorem the analogous result holds for the integral cohomology

H^1\big( \Sigma_g ;\, \mathbb{Z} \big) \;\simeq\; \mathbb{Z}^{2g} \,.

It follows for instance that:

Proposition

On a closed oriented surface of genus $g \in \mathbb{N}$ there exist precisely $2^{2g}$ distinct spin structures.

Proof

Since spin structures are classified by $H_1(-;C_2)$ (see there) this follows by Prop. and the universal coefficient theorem.

Cf. Atiyah 1971, Lawson & Michelson 1989 Ex. 2.6.

Examples

Related concepts

Riemann surface
punctured surface
differential geometry of curves and surfaces
analog for dimension 1: curve, algebraic curve
genus of a surface
area
moduli space of curves
complex surface
surface knot
hypersurface
minimal surface

manifolds in low dimension:

References

Monographs:

L. Christine Kinsey: Topology of Surfaces, Spinger (1991) [doi:10.1007/978-1-4612-0899-0, pdf]
Richard E. Schwartz: Mostly Surfaces, American Mathematical Society, Student Mathematical Library 60 (2011) [draft: pdf, endmatter:pdf]
Jean Gallier, Dianna Xu: A Guide to the Classification Theorem for Compact Surfaces, Springer (2013) [doi:10.1007/978-3-642-34364-3, Wikipedia entry]

In the broader context of algebraic topology:

P. J. Giblin: Graphs, Surfaces and Homology – An Introduction to Algebraic Topology, Chapman and Hall (1977) [doi:10.1007/978-94-009-5953-8]
Allen Hatcher: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]
Clark Bray, Adrian Butcher, Simon Rubinstein-Salzedo, chapters 2 & 4 of: Algebraic Topology, Springer (2021) [doi:10.1007/978-3-030-70608-1, pdf]