Coverings and Lifting Theory

Covering Theory: "Unfolding" a Space

A Covering as a "Multi-Sheeted" Mapping

Take the spiral ℝ and "wrap" it into the circle S¹: p(t) = e^{2πit}. Each point z ∈ S¹ corresponds to infinitely many pre-images in ℝ (sheets): p⁻¹(z) = {n + arg(z)/2π : n ∈ ℤ}. Locally, ℝ looks "like" S¹, but globally ℝ "covers" S¹ with infinitely many sheets.

This is a covering—a "multi-sheeted" locally homeomorphic mapping.

Definition of a Covering

A continuous mapping p: X̃ → X is a covering if for every point x ∈ X there exists an open neighborhood U ∋ x (elementary neighborhood), such that p⁻¹(U) = ⊔_α Ũ_α (disjoint union), and each Ũ_α is homeomorphic to U via p.

X̃ is the covering space, X is the base space. The fiber p⁻¹(x) is a discrete subset of X̃. The number of sheets |p⁻¹(x)| = deg(p) is the degree of the covering.

Examples

ℝ → S¹: p(t) = e^{2πit}. Infinite-sheeted covering. Fiber: p⁻¹(1) = ℤ.

S¹ → S¹: pₙ(z) = zⁿ. n-sheeted covering. Fiber: p⁻¹(1) = {e^{2πik/n} : k = 0,...,n−1}.

S² → ℝP²: p(x) = [x] (class of antipodal points). Double-sheeted covering. Fiber: {x, −x}.

SU(2) → SO(3): Double-sheeted. Each element of SO(3) corresponds to two quaternions q and −q.

Path Lifting Theorem

If p: X̃ → X is a covering, γ: [0,1] → X is a path, x̃₀ ∈ p⁻¹(γ(0)) is the initial point in X̃, then there exists a unique path γ̃: [0,1] → X̃ (lift of γ), such that γ̃(0) = x̃₀ and p ∘ γ̃ = γ.

Lifting a Homotopy: A homotopy of paths in X lifts to a homotopy in X̃. This means: π₁(X) acts on the fiber p⁻¹(x₀) (monodromy).

Fundamental Theorem of Coverings

For locally simply connected X: coverings of X (up to isomorphism over X) are in one-to-one correspondence with subgroups H ≤ π₁(X, x₀) (up to conjugation).

• H = π₁(X) → trivial covering (X → X).
• H = {e} → universal covering X̃ (simply connected).
• H is a normal subgroup → normal covering, Deck(X̃/X) ≅ π₁(X)/H.

Numerical Examples:

For S¹ (π₁ = ℤ): Subgroups of ℤ: nℤ for n = 0, 1, 2, .... n = 1: trivial covering (S¹ → S¹, id). n = 2: double-sheeted covering S¹ → S¹, z ↦ z². n = 0: universal covering ℝ → S¹. H = ℤ corresponds to π₁(S¹) = ℤ → trivial; H = {0} → universal X̃ = ℝ.

Computing π₁(S¹): Lift of the loop γₙ(t) = e^{2πint}: this is the path t ↦ nt in ℝ from 0 to n. For n ≠ 0: γₙ is not closed in ℝ → the loop cannot be contracted to ∅. Different n yield different classes in π₁(S¹). In total: π₁(S¹) ≅ ℤ.

Monodromy and Applications

Monodromy: The action of π₁(X, x₀) on the fiber p⁻¹(x₀): loop γ ↦ permutation of the fiber (σ(γ): x̃₀ ↦ γ̃(1), where γ̃ is the lift of γ). Homomorphism: π₁(X) → Sym(p⁻¹(x₀)).

Analytic continuation: A multi-valued function f (√z, ln z) defines the covering of a Riemann surface. Monodromy γ ↦ continuation matrix—"the action" of the loop on the space of solutions.

Crystallography: The 17 types of "wallpaper" on the plane correspond to 17 subgroups of the group of isometries of ℝ² (discrete). Each wallpaper type = quotient space ℝ² / Λ (Λ is a discrete group of translations and symmetries).

Quantum mechanics (Berry phase): During adiabatic motion along a loop γ in parameter space, the quantum state vector acquires a geometric phase e^{iγ_B}. This is the monodromy of the U(1) bundle over the parameter space. It has been measured experimentally and is used in quantum computers.

Deck group: Deck(X̃/X) is the group of automorphisms of the covering, that is, diffeomorphisms of X̃ commuting with p. For the universal covering: Deck(X̃/X) ≅ π₁(X, x₀). Example: Deck(ℝ/S¹) = ℤ—integer shifts t ↦ t + n. Normal coverings are those for which the Deck group acts transitively on the fibers; these correspond exactly to normal subgroups of π₁.

Covering Theory in Network Topology and Monodromy

Covering theory is applied in several important areas of mathematics and its applications. In Galois theory, coverings of algebraic curves correspond to field extensions: the fundamental group of the curve acts as the Galois group of the function field extension, linking algebraic geometry and group theory. This dictionary, developed by Grothendieck, underlies modern algebraic geometry and is used in number theory (Langlands program). In monodromy theory, analytic continuation of a multi-valued function (square root, logarithm) along a loop corresponds to an element of the fundamental group: the result of the continuation is determined by the homotopy class of the path, not its specific form. This explains the structure of branches: ln z on ℝ²{0} has countably many branches, corresponding to nontrivial coverings ℝ → S¹. In network traversal algorithms, the spanning tree of a graph and the first homology group H₁(G) describe independent cycles in the topological model of electric circuits (Kirchhoff's first law theorem). In condensed matter physics, anyons—particles with "fractional" statistics—are described by two-valued representations of the fundamental group of the configuration space, which is the foundation for topological quantum computing (qubits on anyons).

Problem: (a) For the covering pₙ: S¹ → S¹, z ↦ zⁿ: describe the fiber, the fundamental group of X̃ = S¹. To which subgroup of ℤ does this covering correspond? (b) Construct the lift of the path γ(t) = e^{2πit} (a full circle in S¹) in the covering ℝ, starting from x̃₀ = 0. Where does the endpoint land? (c) Double-sheeted covering S² → ℝP²: why does S² have no nontrivial coverings (S² is simply connected)?