Deck Transformations

Algebraic Topology · Axiom Academy

Automorphisms of covering spaces and their connection to the fundamental group 1. Definition of Deck Transformation A deck transformation (or covering transformation ) is a homeomorphism φ: E → E of the total space that "respects" the covering structure. This means φ permutes the sheets of the covering: if two points project to the same point in B, then their images under φ also project to the same point in B. The set of all deck transformations forms a group under composition, called the deck transformation group or automorphism group of the covering , denoted Aut(E/B) or Deck(E/B). Identity: id E is a deck transformation (trivially p ∘ id = p) Composition: If φ, ψ are deck transformations, so is φ ∘ ψ Inverses: If φ is a deck transformation, so is φ⁻¹ For the covering p: ℝ → S¹ with p(t) = e 2πit , the deck transformations are: So Aut(ℝ/S¹) ≅ ℤ, the group of integers under addition. The deck group acts on each fiber p⁻¹(b) by: φ · e = φ(e) for φ ∈ Aut(E/B) and e ∈ p⁻¹(b). This action has special properties: Well-defined: If p(e) = b, then p(φ(e)) = p(e) = b, so φ maps fibers to themselves Free: If φ(e) = e for some e ∈ E, then φ = id E (no non-trivial fixed points) Properly discontinuous: For each e ∈ E, there's a neighborhood U such that φ(U) ∩ U = ∅ for all φ ≠ id For a normal covering (one where the action is transitive on fibers), the deck group acts transitively: for any e₁, e₂ ∈ p⁻¹(b), there exists φ ∈ Aut(E/B) with φ(e₁) = e₂.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.