ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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The two-circle wedge has both regular and nonregular connected three-sheeted coverings

Example

Let $W = S^{1} \lor S^{1}$ with standard loop classes $a, b$ . There are connected three-sheeted coverings $E_{r e g} \to W$ and $E_{n o n r e g} \to W$ such that the first is regular and the second is not.

The regular cover is classified by the kernel of $F (a, b) \to Z / 3$ sending $a$ to $[1]$ and $b$ to $[0]$ . The nonregular cover is classified by the preimage of the stabilizer of $1$ under the surjection $F (a, b) \to S_{3}$ sending $a$ to $(12)$ and $b$ to $(123)$ .

Facts & Assumptions

Given: The two-circle wedge group and the two assignments in the Example.

[L1]

The group $π_{1} (W, w)$ is free on $a, b$ ( $π_{1} (S^{1} \lor S^{1})$ is the free group on two generators).

[F1]

An assignment on a free basis extends uniquely to a group homomorphism from the free group (Reduced words form the free group on an alphabet).

[F2]

The kernel of a group homomorphism is normal (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).

[F3]

The first isomorphism theorem identifies a quotient by a kernel with the image (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[F4]

A point stabilizer is a subgroup, and its coset set is in bijection with its orbit (The stabilizer $G_{x}$ is a subgroup of $G$ , Orbit-stabiliser: $G / G_{x} \to G \cdot x$ , $g G_{x} \mapsto g \cdot x$ , is a well-defined bijection).

[F5]

Every subgroup is realized by a connected covering, up to based isomorphism (Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups).

[F6]

For a covering with path-connected total space and path-connected locally path-connected base, regularity is equivalent to normality of its induced subgroup (A connected covering is regular exactly when its induced subgroup is normal, exactly when deck transformations act transitively on a fibre).

[F7]

For a covering with nonempty path-connected total space, the number of sheets equals the index of its induced subgroup (For a nonempty path-connected total space, a covering fibre is in bijection with the right cosets of the induced fundamental-group subgroup).

[F8]

The group $Z / 3$ has three elements (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ , For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[F9]

The index of a subgroup is the cardinality of its coset set when that set is finite (The coset set $G / H$ and the index $[G : H]$ of a subgroup).

[F10]

The two-circle wedge is the tagged quotient identifying only its two basepoints. It has a standard open-cover overlap that deformation retracts to the wedge point and is path-connected and simply connected (The wedge of a family of pointed spaces, Finite wedges of quotient circles have van Kampen covers at the wedge point).

[F11]

Open quotient arcs of length below one are homeomorphic to real intervals (The quotient map is open, and every interval shorter than one embeds in $R / Z$ ).

[F12]

Nonempty convex real intervals are simply connected (Every nonempty convex subset of $R^{n}$ is simply connected).

[F13]

Local path-connectedness asks for arbitrarily small open path-connected neighbourhoods, and semilocal simple connectedness asks for a neighbourhood whose inclusion induces the trivial fundamental-group map (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point, Semilocally simply connected spaces with explicit basepoint convention).

[F14]

Local path-connectedness lifts from the base of a covering to its total space (Local path-connectedness lifts and descends along covering maps).

[F15]

A connected locally path-connected space is path-connected (A connected, locally path-connected space is path-connected, because its path components are open).

Verification

technique · direct

1.1

By [L1] and [F1], the assignment $a \mapsto [1]$ , $b \mapsto [0]$ extends to a homomorphism $ϕ : F (a, b) \to Z / 3$ . It is surjective because $[1]$ generates $Z / 3$ .

L1F1F8

1.2

Again by [L1] and [F1], $a \mapsto (12)$ and $b \mapsto (123)$ extend to $ψ : F (a, b) \to S_{3}$ . The elements $(12)$ and $(123)$ generate all six permutations, as seen from $1, (123), (132), (12), (12) (123), (12) (132)$ , so $ψ$ is surjective.

L1F1

2.1

Put $H_{r e g} = \ker ϕ$ . It is normal by [F2], and [F3] identifies $F (a, b) / H_{r e g}$ with the three-element group $Z / 3$ , so [F9] makes $H_{r e g}$ a subgroup of index three.

step 1.1F2F3F8F9

2.2

Let $J = {Stab}_{S_{3}} (1)$ and $H_{n o n r e g} = ψ^{- 1} (J)$ . The set $J$ is a subgroup by [F4], and its preimage is a subgroup because $x, y \in ψ^{- 1} (J)$ gives $ψ (x y^{- 1}) = ψ (x) ψ (y)^{- 1} \in J$ . The natural action of $S_{3}$ on ${1, 2, 3}$ is transitive, so [F4] gives three cosets of $J$ ; surjectivity of $ψ$ gives a bijection between cosets of $H_{n o n r e g}$ and cosets of $J$ , hence [F9] gives $[F (a, b) : H_{n o n r e g}] = 3$ . The subgroup is not normal: $b a b^{- 1}$ maps to $(123) (12) (132) = (23)$ , which fixes $1$ , while $b (b a b^{- 1}) b^{- 1}$ maps to $(123) (23) (132) = (13)$ , which does not fix $1$ . Thus conjugation by $b$ takes an element of $H_{n o n r e g}$ outside it.

step 1.2F4F9algebra

3.1

The base $W$ satisfies the hypotheses of [F5]. It is nonempty and path-connected because every point in either circle is joined to the wedge point. Away from the wedge point, [F11] and [F12] give arbitrarily small open simply connected arcs. At the wedge point, the quotient topology in [F10] makes every open neighbourhood contain a smaller wedge of open arcs; the interval coordinates of [F11] join every point of that smaller wedge to the wedge point along its own branch, so it is path-connected. The particular open overlap supplied by [F10] is simply connected, and therefore has trivial inclusion-induced fundamental group. Thus [F13] gives local path-connectedness and semilocal simple connectedness. Now [F5] realizes the two index-three subgroups from steps 2.1 and 2.2 by connected coverings of $W$ . Since $W$ is locally path-connected, [F14] and [F15] make their total spaces path-connected. This licenses [F7], which makes both covers three-sheeted, and [F6], which makes the normal kernel cover regular and the nonnormal stabilizer-preimage cover nonregular.

step 2.1step 2.2F5F6F7F10F11F12F13F14F15∎

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