What Is Fundamental Group Of A Curve?

The fundamental group of a curve is a profound invariant in algebraic topology, capturing the essential "holey" nature of the space.

The answer depends crucially on the nature of the curve and the base field over which it is defined.

In algebraic geometry, a "curve" is a one-dimensional variety. When the field is the complex numbers (k=Ck equals the complex numbers

𝑘=ℂ

), a smooth projective curve is a compact Riemann surface. The fundamental group is a topological invariant based on continuous loops, but in positive characteristic, the theory is more algebraic, involving finite étale covers.

1. The topological fundamental group of a complex curve

For a smooth complex curve, the fundamental group is a group constructed from the loops within the space.

The compact case: The fundamental group of a genus-g surface

A smooth, complex, projective curve of genus gg

𝑔

is topologically a compact, orientable surface with gg

𝑔

"handles". The fundamental group of this surface, denoted π1(Sg)pi sub 1 open paren cap S sub g close paren

𝜋1(𝑆𝑔)

, has a well-known presentation based on 2g2 g

2𝑔

generators and one relation.

• Generators: There are 2g2 g

  2𝑔

  generators: a1,b1,a2,b2,…,ag,bga sub 1 comma b sub 1 comma a sub 2 comma b sub 2 comma … comma a sub g comma b sub g

  𝑎1,𝑏1,𝑎2,𝑏2,…,𝑎𝑔,𝑏𝑔

  . These correspond to loops that go around each of the gg

  𝑔

  handles.

• Relation: The single relation is given by the commutator product of these generators:[a1,b1][a2,b2]⋯[ag,bg]=1open bracket a sub 1 comma b sub 1 close bracket open bracket a sub 2 comma b sub 2 close bracket ⋯ open bracket a sub g comma b sub g close bracket equals 1

  [𝑎1,𝑏1][𝑎2,𝑏2]⋯[𝑎𝑔,𝑏𝑔]=1

  where [a,b]=aba-1b-1open bracket a comma b close bracket equals a b a to the negative 1 power b to the negative 1 power

  [𝑎,𝑏]=𝑎𝑏𝑎−1𝑏−1

  . This relation arises from considering a single large loop that traverses the boundary of the surface's fundamental polygon, which is topologically trivial.

• **A concrete example: The torus (g=1g equals 1

  𝑔=1

  )**A torus is a surface of genus 1. Its fundamental group is given by the presentation ⟨a1,b1∣[a1,b1]=1⟩open angle bracket a sub 1 comma b sub 1 divides open bracket a sub 1 comma b sub 1 close bracket equals 1 close angle bracket

  ⟨𝑎1,𝑏1∣[𝑎1,𝑏1]=1⟩

  . This is the free abelian group on two generators, which is isomorphic to Z⊕Zthe integers circled plus the integers

  ℤ⊕ℤ

  .

The non-compact case: The fundamental group of an affine curve

An affine complex curve can be thought of as a compact curve with a finite number of points removed. For a smooth affine curve Ucap U

𝑈

obtained by removing rr

𝑟

points from a compact Riemann surface Xcap X

𝑋

of genus gg

𝑔

, the fundamental group has the following presentation:

• Generators: It has 2g+r2 g plus r

  2𝑔+𝑟

  generators: a1,b1,…,ag,bga sub 1 comma b sub 1 comma … comma a sub g comma b sub g

  𝑎1,𝑏1,…,𝑎𝑔,𝑏𝑔

  (from the compact part) and c1,…,crc sub 1 comma … comma c sub r

  𝑐1,…,𝑐𝑟

  (from the punctures).

• Relation: The single relation is:[a1,b1]⋯[ag,bg]c1⋯cr=1open bracket a sub 1 comma b sub 1 close bracket ⋯ open bracket a sub g comma b sub g close bracket c sub 1 ⋯ c sub r equals 1

  [𝑎1,𝑏1]⋯[𝑎𝑔,𝑏𝑔]𝑐1⋯𝑐𝑟=1

• Group structure: As a result, the fundamental group of such an affine curve is isomorphic to the free group on 2g+r−12 g plus r minus 1

  2𝑔+𝑟−1

  generators. This is a non-abelian group for most cases.

2. The algebraic (étale) fundamental group

In algebraic geometry, especially over fields other than Cthe complex numbers

ℂ

, the topological definition of the fundamental group based on loops is not possible. Alexander Grothendieck introduced the algebraic or étale fundamental group, which is a pro-finite group.

Definition via covering spaces

The étale fundamental group, denoted π1et(X)pi sub 1 raised to the e t power open paren cap X close paren

𝜋𝑒𝑡1(𝑋)

, is defined using finite étale covers of the curve Xcap X

𝑋

.

• Finite étale cover: A finite étale cover is a type of map between schemes that is a local isomorphism in the étale topology. For a smooth curve, this corresponds to a finite unramified covering space in the classical sense.
• Galois correspondence: The fundamental group is the group of deck transformations of the universal cover. In the algebraic setting, it is the group that classifies finite étale covers of the curve, analogous to the Galois group classifying finite field extensions.
• Pro-finite group: The étale fundamental group is constructed as the inverse limit of the Galois groups of all finite étale covers. This means it is a pro-finite group, which is a specific kind of topological group.

Case: characteristic zero

For a curve Ucap U

𝑈

over an algebraically closed field of characteristic 0 (like Cthe complex numbers

ℂ

), the étale fundamental group is the pro-finite completion of the topological fundamental group. This means it is an inverse limit of all finite quotients of the topological fundamental group.

Case: positive characteristic

In positive characteristic pp

𝑝

, the situation is more complex due to the existence of wildly ramified covers. The structure of the étale fundamental group is richer and carries more information about the curve.

• Abhyankar's Conjecture: For a smooth affine curve Ucap U

  𝑈

  of genus gg

  𝑔

  with n+1n plus 1

  𝑛+1

  punctures, a finite group Gcap G

  𝐺

  is a quotient of the fundamental group if and only if the group Gcap G

  𝐺

  modulo its quasi-pp

  𝑝

  subgroup is generated by 2g+n2 g plus n

  2𝑔+𝑛

  elements.

• Commutator subgroup: For an uncountable, algebraically closed field of characteristic pp

  𝑝

  , the commutator subgroup of the étale fundamental group of a smooth affine curve is a free pro-finite group of rank equal to the cardinality of the field.

3. Comparison of different approaches

Feature	Topological Fundamental Group (Complex Curves)	Étale Fundamental Group (General)
Foundation	Continuous loops.	Finite étale covers.
Group Type	Discrete, finitely generated.	Pro-finite, topological group.
Information Captured	Homotopy classes of loops, holes.	Finite cover information, including arithmetic data.
Relationship (Char 0)	Étale group is the pro-finite completion of the topological group.	The two theories align closely.
Relationship (Char p)	The topological group is not applicable.	The étale group has a richer and more complex structure.
Computability	Computable using tools like the Seifert-van Kampen theorem for certain decompositions.	Requires more abstract methods of Galois theory for algebraic schemes.

Summary

The fundamental group of a curve is a powerful invariant that takes different forms depending on the context. For complex curves, the topological approach defines it based on loops and captures the geometry of the surface. For more general algebraic curves, the étale fundamental group uses finite covering spaces, resulting in a pro-finite group that is particularly rich in characteristic pp

𝑝

. The fundamental group, in all its guises, provides a deep connection between the geometric or algebraic properties of a curve and the group-theoretic properties of its associated invariant.