Calculating fundamental groups is a key skill in algebraic topology. It involves analyzing loops in spaces and understanding how they relate to each other. This topic builds on earlier concepts of homotopy and introduces tools for computing these groups.

The Seifert-van Kampen theorem is a powerful method for calculating fundamental groups of complex spaces. By breaking spaces into simpler parts, we can piece together their fundamental groups. This approach connects to broader themes of decomposition and reconstruction in topology.

Calculating Fundamental Groups

Fundamental Group Definition and Properties

• The fundamental group of a space X, denoted π₁(X), describes the different ways of looping a path in X, starting and ending at the same point, up to homotopy equivalence
• The fundamental group is a homotopy invariant
  • If two spaces are homotopy equivalent, their fundamental groups are isomorphic
• The fundamental group of a convex set in Euclidean space is trivial, consisting of only the identity element

Fundamental Groups of Common Spaces

• The fundamental group of a circle, π₁(S¹), is isomorphic to the integers under addition, denoted (ℤ, +)
• The fundamental group of a figure eight (wedge sum of two circles) is the free group on two generators, denoted F₂
• The fundamental group of a torus, π₁(T²), is isomorphic to ℤ × ℤ, the direct product of two copies of the integers
• Higher-dimensional spheres, Sⁿ for n ≥ 2, have trivial fundamental groups
  • Example: The fundamental group of a 2-sphere (surface of a ball) is trivial

Seifert-van Kampen Theorem for Fundamental Groups

Theorem Statement and Conditions

• The Seifert-van Kampen theorem computes the fundamental group of a space that can be decomposed into simpler, overlapping subspaces
• The theorem applies when:
  • X is the union of two open, path-connected subspaces U and V
  • U ∩ V is also path-connected
• Under these conditions, the fundamental group of X is isomorphic to the free product of the fundamental groups of U and V, amalgamated along the fundamental group of their intersection

Free Product with Amalgamation

• The free product with amalgamation, denoted π₁(U) *_{π₁(U ∩ V)} π₁(V), is a quotient group of the free product π₁(U) * π₁(V)
  • It is quotiented by the normal subgroup generated by elements of the form i₁(g)i₂(g)⁻¹, where:
    • g ∈ π₁(U ∩ V)
    • i₁ and i₂ are induced homomorphisms from the inclusions of U ∩ V into U and V, respectively
• The Seifert-van Kampen theorem can be applied iteratively to decompose a space into multiple overlapping subspaces
  • This allows for the calculation of the fundamental group of more complex spaces

Fundamental Groups of Product Spaces

Fundamental Group of a Product

• The fundamental group of a product space X × Y is isomorphic to the direct product of the fundamental groups of X and Y
  • π₁(X × Y) ≅ π₁(X) × π₁(Y)
• This result follows from the fact that loops in a product space can be projected onto loops in each factor space
  • These projections uniquely determine the original loop up to homotopy

Application to n-Torus

• The fundamental group of an n-torus, the product of n circles, is isomorphic to the direct product of n copies of ℤ
  • π₁(T^n) ≅ ℤ^n
  • Example: The fundamental group of a 3-torus (T³) is isomorphic to ℤ × ℤ × ℤ

Fundamental Groups vs Covering Spaces

Covering Spaces and Covering Maps

• A covering space of a topological space X is a space C together with a continuous surjective map p: C → X, called the covering map
  • Each point x ∈ X has an open neighborhood U for which p⁻¹(U) is a disjoint union of open sets in C
  • Each open set in p⁻¹(U) is mapped homeomorphically onto U by p
• The fundamental group of X acts on the fiber of a covering space p: C → X over a basepoint x₀ ∈ X
  • The fiber is the set p⁻¹(x₀)
  • The action is by lifting loops in X based at x₀ to paths in C

Relationship between Fundamental Groups and Covering Spaces

• The action of π₁(X, x₀) on the fiber p⁻¹(x₀) is transitive if and only if the covering space is connected
• For a connected covering space p: C → X, the subgroup p_π₁(C, c₀) of π₁(X, x₀), where c₀ ∈ p⁻¹(x₀), is isomorphic to the stabilizer of c₀ under the action of π₁(X, x₀) on the fiber
• There is a one-to-one correspondence between:
  • Connected covering spaces of X (up to isomorphism)
  • Conjugacy classes of subgroups of π₁(X)
• The universal covering space of X is the unique simply connected covering space of X (up to isomorphism)
  • Its group of deck transformations is isomorphic to the fundamental group of X