Fundamental Group

Fundamental Group: Algebra of Loops

Idea: Loops as Probes of "Holes"

Take a torus (the surface of a bagel). Wind a thread around a "hole" (along one of the two independent cycles). Can this thread be "shrunk" to a point while staying on the torus? No! But on a sphere, any loop can be contracted.

The fundamental group π₁(X, x₀) formalizes this observation: it counts the "number of ways to wind a loop" which cannot be reduced to each other by a continuous deformation.

This is a topological invariant: homeomorphic spaces have isomorphic fundamental groups. Different π₁ → not homeomorphic.

Loops and Homotopy

A loop at a point x₀ ∈ X: a continuous map γ: [0,1] → X with γ(0) = γ(1) = x₀.

Homotopy of loops: γ₀ and γ₁ are homotopic (γ₀ ≃ γ₁) if there exists a continuous "deformation" H: [0,1]² → X with H(t,0) = γ₀(t), H(t,1) = γ₁(t), H(0,s) = H(1,s) = x₀ for all s. Subtlety: the ends remain at x₀ throughout the deformation process.

Homotopy class [γ] is the set of all loops homotopic to γ.

The Fundamental Group π₁(X, x₀)

Elements: Homotopy classes of loops at x₀.

Operation: Concatenation: (γ₁ * γ₂)(t) = γ₁(2t) for t ≤ 1/2 and γ₂(2t−1) for t ≥ 1/2 (first traversing γ₁, then γ₂).

Identity element: The constant loop c(t) = x₀.

Inverse: γ⁻¹(t) = γ(1−t) (the loop traversed in the reverse direction).

The group π₁(X, x₀) is well-defined (homotopic loops yield homotopic concatenations). On changing the base point x₀ → x₁ (in connected X): π₁(X, x₀) ≅ π₁(X, x₁) (isomorphism depends on the choice of path).

Numerical Examples

π₁(ℝⁿ) = 0: ℝⁿ is contractible (any loop is contracted via H(t,s) = (1−s)γ(t) + s·x₀).

π₁(S¹) = ℤ: The loop eⁿ: t ↦ e^{2πint} performs n revolutions around S¹. Homotopy classes are indexed by integer n — the "winding number". This is the most important example!

π₁(Sⁿ) = 0 for n ≥ 2: Spheres of dimension ≥ 2 are simply connected.

π₁(T²) = ℤ × ℤ: The torus has two independent cycles (the meridian m and the parallel p). [m] and [p] commute: π₁ = ℤ².

π₁(ℝP²) = ℤ₂: The projective plane: there is one nontrivial loop (a double traversal returns to the original point).

π₁(S¹ ∨ S¹) = ℤ * ℤ: The "figure eight" — wedge of two circles: free group on two generators (non-commutative!).

van Kampen's Theorem

For X = A ∪ B (open, connected, A ∩ B connected):

π₁(X, x₀) = π₁(A, x₀) *_{π₁(A∩B, x₀)} π₁(B, x₀)

This is an amalgamated free product — "gluing" of groups along a common subgroup.

Application: π₁(T²) is computable: T² = (ℝ²{point})/ℤ² — a square with sides identified. Fundamental polygon: π₁(T²) = ⟨a,b | aba⁻¹b⁻¹⟩ (a and b commute) = ℤ².

π₁(wedge S¹ ∨ S¹): ⟨a, b⟩ — free group on a, b. Noncommutative: ab ≠ ba.

Topological Invariants and Applications

Poincaré Conjecture (proved by Perelman, 2003): Every closed simply connected 3-manifold is homeomorphic to S³. This is one of the "Millennium Problems". Perelman declined the $1 million prize.

Application to physics: The fundamental group of the order parameter space classifies linear defects (vortices, dislocations): π₁(M) = ℤ → stable vortices (superconductors, liquid crystals). π₁(M) = 0 → no stable linear defects.

Higher Homotopy Groups

The groups πₙ(X) generalize the fundamental group: πₙ(X, x₀) — classes of continuous maps Sⁿ → X with base point. For n ≥ 2 all πₙ are abelian. Hopf fibration (1931): η: S³ → S² — a surjective map in which each fiber is a circle S¹. This gives rise to π₃(S²) = ℤ — an unexpected result: the 2-sphere has a nontrivial 3rd homotopy group! In quantum chromodynamics, instantons are classified by π₃(SU(2)) = ℤ: the topological charge of an instanton is an integer, defining the vacuum structure of the quantum field and connected with the index of the Dirac operator via the Atiyah–Singer theorem.

Fundamental Group in Physics and Cryptography

The fundamental group captures topological obstructions to contracting loops and has direct applications. In solid state physics, quasiparticle "defects" — dislocations in crystals, vortices in superfluid, skyrmions in magnetics — are classified by the first homotopy group π₁ of the order parameter space. Vortices in liquid helium correspond to nontrivial elements of π₁(S¹) = ℤ: each vortex carries an integer topological charge, which cannot annihilate without merging with an antivortex. In quantum computation, topological qubits, realized on the basis of Majorana fermions, use π₁(BG) = G (the fundamental group of the classifying space) as decoherence-protected quantum memory. In cryptography, protocols based on the discrete logarithm problem in Lie groups and the braid group problem (braid group is a generalization of the fundamental group) are used in post-quantum cryptography. The Poincaré theorem on homogeneous spaces, proved by Perelman (2003), resolved a century-old problem by linking π₁ = 0 to homeomorphism to S³ — a key result on the relationship between algebra and geometry of three-dimensional manifolds.

The theory of braids (braid group Bₙ) — a generalization of the fundamental group of the configuration space of n distinguishable points in the plane — is used in cryptography (the conjugacy search problem in the braid group is potentially post-quantum secure) and in the description of anyons in the quantum Hall effect, where exchange statistics is given by irreducible representations of the braid group instead of the symmetric group.

Exercise: (a) Compute π₁(ℝ² {0, 1}) (the plane with two punctured points). Use van Kampen's theorem. (b) For the fundamental polygon of the Klein bottle (sides ab a b⁻¹): write π₁ as a presentation. Is this group abelian? (c) Show: π₁(S¹) = ℤ using the lifting of loops to the universal cover ℝ → S¹.