Theorems · Definition · algebraic topology

IsQuotientCoveringMap.fundamentalGroupToMulOpposite

{E : Type u_1} →
  {X : Type u_2} →
    [inst : TopologicalSpace E] →
      [inst_1 : TopologicalSpace X] →
        {p : E → X} →
          {G : Type u_4} →
            [inst_2 : Group G] →
              [inst_3 : MulAction G E] →
                IsQuotientCoveringMap p G → {x : X} → ↑(p ⁻¹' {x}) → FundamentalGroup X x →* Gᵐᵒᵖ

Choosing an arbitrary basepoint e ∈ f ⁻¹' {x} induces a bijection f ⁻¹' {x} ≃ G, and the G-action on f ⁻¹' {x} corresponds to left multiplication. The monodromy action commutes with the G-action, so each monodromy must corresponds must correspond to a right multiplication.

Defined in

Mathlib.Topology.Homotopy.Lifting

Cited by

7 results in Mathlib

Foundations

Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceGroupMulAction