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Fundamental groupoids of free topological quotients

PDF p. 7: §2.4, first paragraph

Statement

Let $X$ be a nonempty, path-connected, locally path-connected and semilocally simply connected space. Thus any two points of $X$ can be joined by a path; each point has a basis of path-connected neighbourhoods; and each point has a neighbourhood whose inclusion sends every based loop in that neighbourhood to a null-homotopic loop in $X$ . Let a group $N$ act on $X$ by homeomorphisms, freely and properly discontinuously, in the following sense: for each $x \in X$ there is an open neighbourhood $U$ of $x$ such that $n U \cap U = \emptyset$ for every nonidentity $n \in N$ . Write $p : X \to X / N$ for the orbit map, with the quotient topology on $X / N$ .

Then $p$ is a normal covering: it is a covering map and its deck transformations, the homeomorphisms $h : X \to X$ satisfying $p \circ h = p$ , act transitively on each fibre. The induced functor $p_{*} : Π_{1} (X) \to Π_{1} (X / N)$ is a groupoid covering: for every $x \in X$ , it bijects the arrows starting at $x$ with those starting at $p (x)$ . Here $Π_{1}$ has points as objects and endpoint-fixed path-homotopy classes as arrows, composed by concatenation.

The action of $N$ on points and path classes induces an action by groupoid automorphisms on $Π_{1} (X)$ . Its quotient $Π_{1} (X) / N$ has object and arrow orbits, with composition defined by aligning representatives. Projection induces a canonical isomorphism ${\overline{p}}_{*} : Π_{1} (X) / N \to Π_{1} (X / N)$ , and $p_{*} = {\overline{p}}_{*} \circ Q$ , where $Q$ is the orbit functor. Thus the two groupoid coverings agree under this canonical identification.

Statement references

Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

Covering-space actions by disjoint translates of neighbourhoods

Fundamental groupoid of a space

Semilocally simply connected spaces with explicit basepoint convention

Free actions and quotient groupoids

Groupoid coverings, universal and normal covers

Covering-space actions and fundamental-groupoid quotients

Facts & Assumptions

[F1]

A group action by homeomorphisms with the stated disjoint-neighbourhood condition is a covering-space action. Covering-space actions by disjoint translates of neighbourhoods

[F2]

For such an action on a nonempty path-connected, locally path-connected space, the orbit map is a regular covering, its induced fundamental-groupoid functor is a groupoid covering, and projection induces an isomorphism from the orbit groupoid to the fundamental groupoid of the quotient. Covering-space actions and fundamental-groupoid quotients

[F3]

The fundamental groupoid consists of points and endpoint-fixed path-homotopy classes. For a free object action, its quotient has object and arrow orbits and composition by aligned representatives. A groupoid covering has unique arrow lifts from a prescribed source. Fundamental groupoid of a space Free actions and quotient groupoids Groupoid coverings, universal and normal covers

Proof

Given: $X$ , $N$ and $p$ satisfy the hypotheses in the Statement.

Proof technique: direct.

Goal: Prove the topological covering, the induced groupoid covering, and the stated identification of quotient groupoids and functors.

1.1F1F2given

Apply the covering-space result. The Given supplies homeomorphisms and the disjoint-neighbourhood condition, so [F1] applies. The nonempty, path-connected and locally path-connected hypotheses satisfy [F2]. Hence $p$ is a regular, equivalently normal, covering in the sense stated: deck transformations act transitively on each fibre. The semilocal simple connectedness hypothesis is retained from the article but is not needed for this application.

2.1F2F3step 1.1∎

Identify the groupoid maps. By [F2], translation induces the action on points and path classes, projection gives an isomorphism ${\overline{p}}_{*} : Π_{1} (X) / N \to Π_{1} (X / N)$ , and $p_{*}$ is a groupoid covering. The quotient and covering conventions are those of [F3]. By construction, $Q$ sends a point or path class to its orbit, and ${\overline{p}}_{*}$ sends that orbit to its projection; their composite therefore equals $p_{*}$ on both objects and arrows. This gives the asserted equality of the two coverings after the canonical identification.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 7, §2.4, first paragraph