monodromy in nLab

Context

Homotopy theory

• Idea
• In point-set topology
• Definition
• Properties
• In cohesive ∞\infty-Toposes
• References

Definition

(monodromy of a covering space)

Let XX be a topological space and E→pXE \overset{p}{\to} X a covering space. Write Π 1(X)\Pi_1(X) for the fundamental groupoid of XX.

Define a functor

Fib E:Π 1(X)⟶Set Fib_E \;\colon\; \Pi_1(X) \longrightarrow Set

to the category Set of sets as follows:

1. to a point x∈Xx \in X assign the fiber p −1({x})∈Setp^{-1}(\{x\}) \in Set;

2. to the homotopy class of a path γ\gamma connecting x≔γ(0)x \coloneqq \gamma(0) with y≔γ(1)y \coloneqq \gamma(1) in XX assign the function p −1({x})⟶p −1({y})p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\}) which takes x^∈p −1({x})\hat x \in p^{-1}(\{x\}) to the endpoint of a path γ^\hat \gamma in EE which lifts γ\gamma through pp with starting point γ^(0)=x^\hat \gamma(0) = \hat x

   p −1(x) ⟶ p −1(y) (x^=γ^(0)) ↦ γ^(1). \array{ p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y) \\ (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1) } \,.

This construction is well defined for a given representative γ\gamma due to the unique path-lifting property of covering spaces (this lemma) and it is independent of the choice of γ\gamma in the given homotopy class of paths due to the homotopy-lifting property (this lemma). Similarly, these two lifting properties give that this construction respects composition in Π 1(X)\Pi_1(X) and hence is indeed a functor.

Proposition

Given a homomorphism between two covering spaces E i→p iXE_i \overset{p_i}{\to} X, hence a continuous function f:E 1→E 2f \colon E_1 \to E_2 which respects fibers in that the diagram

E 1 ⟶f E 2 p 1↘ ↙ p 2 X \array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }

commutes, then the component functions

f| {x}:p 1 −1({x})⟶p 2 −1({x}) f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})

are compatible with the monodromy Fib EFib_{E} (def. ) along any path γ\gamma between points xx and yy from def. in that the following diagrams of sets commute

p 1 −1(x) ⟶f| {x} p 2 −1(x) Fib E 1([γ])↓ ↓ Fib E 2([γ]) p 1 −1(y) ⟶f| {y} p 2 −1({y}). \array{ p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x) \\ {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow && \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }} \\ p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\}) } \,.

This means that ff induces a natural transformation between the monodromy functors of E 1E_1 and E 2E_2, respectively, and hence that constructing monodromy is itself a functor from the category of covering spaces of XX to that of permutation representations of the fundamental groupoid of XX:

Fib:Cov(X)⟶Set Π 1(X). Fib \;\colon\; Cov(X) \longrightarrow Set^{\Pi_1(X)} \,.

Proof

For any x^∈p 1 −1(x)\hat x \in p_1^{-1}(x) let γ^\hat \gamma be the unique path in EE with γ^(0)=x^\hat \gamma(0) = \hat x and p∘γ^=γp \circ \hat \gamma = \gamma. By definition we have

Fib E 1([γ(f)])(x^)=γ^(1) Fib_{E_1}([\gamma(f)])(\hat x) = \hat \gamma(1)

and hence

f(Fib E 1([γ(f)])(x^))=f(γ^(1)) f(Fib_{E_1}([\gamma(f)])(\hat x)) = f(\hat \gamma(1))

Now f∘γ^f \circ \hat \gamma satisfies f∘γ^(0)=f(x^)f \circ \hat \gamma(0) = f(\hat x) and p∘f∘γ^=γp \circ f \circ \hat \gamma = \gamma by the fact that ff preserves fibers. Hence by uniqueness of path lifting (this lemma), f∘γ^f \circ \hat \gamma is the unique lift of γ\gamma with starting point f(x^)f(\hat x). By def. this means that

Fib E 2([γ])(f(x^))=f(γ^(1)). Fib_{E_2}([\gamma])(f(\hat x)) = f (\hat \gamma(1)) \,.

This is the equality to be shown.

Example

(fundamental groupoid of covering space)

Let E⟶pXE \overset{p}{\longrightarrow} X be a covering space.

Then the fundamental groupoid Π 1(E)\Pi_1(E) of the total space EE is equivalently the Grothendieck construction of the monodromy functor Fib E:Π 1(X)→SetFib_E \;\colon\; \Pi_1(X) \to Set

Π 1(E)≃∫ Π 1(X)Fib E \Pi_1(E) \;\simeq\; \int_{\Pi_1(X)} Fib_E

whose

• objects are pairs (x,x^)(x,\hat x) consisting of a point x∈Xx \in X and en element x^∈Fib E(x)\hat x \in Fib_E(x);

• morphisms[γ^]:(x,x^)→(x′,x^′)[\hat \gamma] \colon (x,\hat x) \to (x', \hat x') are morphisms [γ]:x→x′[\gamma] \colon x \to x' in Π 1(X)\Pi_1(X) such that Fib E([γ])(x^)=x^′Fib_E([\gamma])(\hat x) = \hat x'.

Proof

By the uniqueness of the path-lifting (this lemma) and the very definition of the monodromy functor.