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arxiv: 2409.04362 · v1 · submitted 2024-09-06 · 🧮 math.DG · math.AT

Compact holonomy G₂ manifolds need not be formal

Luc\'ia Mart\'in-Merch\'an This is my paper

Pith reviewed 2026-05-23 20:49 UTC · model grok-4.3

classification 🧮 math.DG math.AT
keywords G2 holonomycompact manifoldsformalityMassey productssingular locusJoyce-Karigiannis constructionnon-formal manifoldtorsion-free G2
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The pith

Compact simply connected manifolds with G₂ holonomy need not be formal.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a compact, simply connected manifold with holonomy G₂ that is non-formal. It does so by applying the Joyce-Karigiannis method for resolving singularities to produce a torsion-free G₂ structure and choosing a configuration of the singular locus that makes a triple Massey product in the cohomology ring non-vanishing. A sympathetic reader would care because formality is a topological property that would otherwise simplify the rational homotopy type of these manifolds; its failure shows that the cohomology of some G₂ holonomy examples carries additional structure beyond the cup product.

Core claim

By arranging the singular locus in a particular configuration within the Joyce-Karigiannis construction of compact torsion-free G₂ manifolds, one obtains a compact simply connected manifold with holonomy G₂ that admits a non-vanishing triple Massey product and is therefore non-formal.

What carries the argument

The particular configuration of the singular locus that induces a non-vanishing triple Massey product in the resolved manifold.

If this is right

  • The constructed manifold has holonomy exactly G₂ and is non-formal.
  • Non-formality is detected by a non-vanishing triple Massey product.
  • The Joyce-Karigiannis construction can produce both formal and non-formal examples depending on the singular locus arrangement.
  • Compact G₂ holonomy manifolds are not automatically formal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar singular-locus arrangements might be used to produce non-formal examples among other classes of special-holonomy manifolds.
  • One could test whether the non-formality is preserved under small deformations of the G₂ structure.
  • Non-formal G₂ manifolds may exhibit more intricate rational homotopy groups than their formal counterparts.

Load-bearing premise

The specific singular locus configuration in the Joyce-Karigiannis construction yields both exact G₂ holonomy and a genuinely non-vanishing triple Massey product.

What would settle it

An explicit computation showing that the triple Massey product vanishes for this singular locus configuration, or that the resulting manifold does not have full G₂ holonomy.

read the original abstract

We construct a compact, simply connected manifold with holonomy $\mathrm{G}_2$ that is non-formal. We use the construction method of compact torsion-free $\mathrm{G}_2$ manifolds developed by D.D. Joyce and S. Karigiannis. A non-vanishing triple Massey product is obtained by arranging the singular locus in a particular configuration.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper constructs a compact, simply connected 7-manifold with holonomy exactly G₂ that is non-formal. It applies the Joyce-Karigiannis resolution method to a G₂-orbifold whose singular locus is arranged in a specific configuration chosen so that the resolved manifold carries a non-vanishing triple Massey product.

Significance. If the construction is valid, the result supplies the first explicit example of a compact G₂-holonomy manifold that fails to be formal. This settles an open question in the field and shows that formality is not automatic for torsion-free G₂-structures. The paper’s strength lies in its use of a cited, parameter-free resolution technique together with a concrete topological configuration that produces a falsifiable non-vanishing Massey product.

minor comments (3)
  1. [§3.2] §3.2: the precise coordinates of the singular loci and the gluing data used to realize the chosen configuration are stated only in outline; a short table listing the fixed-point sets and their normal bundles would improve verifiability.
  2. [§5] §5: the verification that the triple Massey product survives the resolution is sketched by reference to the orbifold cohomology; an explicit cocycle representative on the resolved manifold (even in local coordinates) would make the non-vanishing claim easier to check.
  3. The bibliography entry for the Joyce-Karigiannis paper should include the precise theorem numbers invoked for the existence of the torsion-free G₂-structure after resolution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contribution, and the recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No circularity: explicit external-method construction

full rationale

The paper is an existence proof via explicit construction: it applies the Joyce-Karigiannis resolution procedure (externally cited, different authors) to an orbifold whose singular locus is arranged in a new configuration chosen so that the resolved manifold is simply connected, has exact G2 holonomy, and carries a non-vanishing triple Massey product. No equations, parameters, or uniqueness claims are defined in terms of the target result; the Massey product non-vanishing is verified directly from the configuration rather than fitted or renamed. No self-citations appear in the load-bearing steps. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms beyond standard differential geometry and algebraic topology; the claim rests on the cited construction technique and the existence of the Massey product computation.

axioms (1)
  • standard math Standard axioms and theorems of differential geometry (G₂ structures, torsion-free condition) and algebraic topology (Massey products, formality).
    Invoked throughout the construction and invariant verification.

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Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

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    There are oriented cobordisms between N1 and N2 and between N3 and N7

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    The cobordism between N3 and N7 intersects N1 negatively and N2 positively. This contrasts with the example examined in [ 2], where both the orbifold and the singular locus are also formal, but the cobordisms between singular components can be ch osen to avoid transverse intersections with other components. This also differs from the examples of compa ct m...

    work page

  3. [3]

    Remark 1

    Here D4 is the dihedral group spanned by κ and F (note that κ ◦ F = F − 1 ◦ κ), whereas Z2 2 is generated by ι1 and ι2. Remark 1. The orbifold X is related to the examples provided by Joyce in [ 13]. More precisely, a change of variables transforms X1 = T7/ ⟨F, κ⟩ into Example 9, and X2 = T7/ ⟨F, κ, ι1 ◦ ι2⟩ into Example 10. In particular, there is a bran...

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  4. [4]

    We denote by N1 and N2 their projections to X

    Both admit the nowhere-vanishing harmonic 1-form dy3. We denote by N1 and N2 their projections to X. To analyze the singularities at t = 1/ 2, we compute κ ′[z] =[¯z1, ¯z2, ¯z3], (κ ◦ ι1)′[z] = [ − ¯z1, − ¯z2, ¯z3 + 1/ 2], (κ ◦ ι2)′[z] =[− ¯z1, − ¯z2, ¯z3 + i/ 2], (κ ◦ ι1 ◦ ι2)′[z] = [¯z1, ¯z2, ¯z3 + (1 + i)/ 2]. Hence, Fix(( κ ◦ ι1)′) = Fix(( κ ◦ ι1 ◦ ι2...

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  5. [5]

    We denote by N3,

    These admit the nowhere-vanishing harmonic 1-form dx3. We denote by N3, . . . , N 10 their projections to X. The submanifolds N k j and N ′ j are associative submanifolds of ( M, ϕ ) and ( M ′, ϕ ) (see [ 17, Proposition 2.13]). We will always assume that N k j (and N ′ j, Nj) are oriented by ϕ |N k j = Re(dz123)|N k j (and ϕ |N ′ j , ϕ |Nj ). Using the c...

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  6. [6]

    Therefore, the volume of these tori is 2, which is why the factor of 2 appears

    The projection to the third factor has, of course, length 1. Therefore, the volume of these tori is 2, which is why the factor of 2 appears. 2.2 Resolution The resolution method developed in [ 17] allows to desingularize the orbifold X and to obtain a metric with holonomy G 2 on the resolution. Theorem 2. The resolution ˜M of X is a compact simply connect...

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    Consider the paths ˜γ5, ˜γ6 : [0 , 1] → M, ˜γ5(s) = [1 / 2, [0, 0, s/ 2]], ˜γ6(s) = [1 / 2, [0, 0, is/ 2]]

    There is a short exact sequence, {1} → π1(M ) → π1(M ′) → ⟨ ι1, ι2⟩ → { 1}. Consider the paths ˜γ5, ˜γ6 : [0 , 1] → M, ˜γ5(s) = [1 / 2, [0, 0, s/ 2]], ˜γ6(s) = [1 / 2, [0, 0, is/ 2]]. Then γ ′ 5 = q ◦ ˜γ5 and γ ′ 6 = q ◦ ˜γ6 are loops on M ′. In the long exact sequence, their homotopy classes project onto ι1 and ι2 respectively. If 0 ≤ k ≤ 4 we denote γ ′...

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  8. [8]

    This implies that q′ ∗ [γ] = 1 because q′ ◦ γ = q′ ◦ γ ′′(s) where γ ′′(s) = { γ(s), if t ≤ 1/ 2, γ − 1(s), if t ≥ 1/ 2, and γ ′′(s) is trivial on M ′

    That is, if γ is one of these loops, then κ ◦ γ(s) = γ − 1(s). This implies that q′ ∗ [γ] = 1 because q′ ◦ γ = q′ ◦ γ ′′(s) where γ ′′(s) = { γ(s), if t ≤ 1/ 2, γ − 1(s), if t ≥ 1/ 2, and γ ′′(s) is trivial on M ′. Hence, q′ ∗ [γ ′ 0] = q′ ∗ [γ ′ 2] = q′ ∗ [γ ′ 4] = q′ ∗ [γ ′ 6] = 1. From the relations ( 13), and [γ ′ 5]− 1 = [γ ′ 0]− 1[γ5][γ ′ 0] we obta...

    work page

  9. [9]

    The argument above ensures that the loop q′ ◦ q ◦ γ9 is trivial on X

    We now observe that κ ◦ (q ◦ γ9)(s) = q[0, [0, 0, s/ 2]] = q[1, [0, 0, − s/ 2]] = q[1, [0, 0, 1/ 2 − s/ 2]] = ( q ◦ γ9)− 1(s). The argument above ensures that the loop q′ ◦ q ◦ γ9 is trivial on X. Hence, q′ ◦ γ ′ 5 ∼ (q′ ◦ q ◦ γ8) ·(q′ ◦ q ◦ γ9) ·(q′ ◦ q ◦ γ8)− 1 ∼ (q′ ◦ q ◦ γ8) ·(q′ ◦ q ◦ γ8)− 1 ∼ 1. Therefore, π1( ˜M ) = π1(X) = {1}. Remark 2. The resol...

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    In addition, α j − α = dβj where βj ∈ Ωk− 1(X) is supported on O2 j

    The forms α j and α coincide on X − O2 j . In addition, α j − α = dβj where βj ∈ Ωk− 1(X) is supported on O2 j

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    In particular, ρ∗ (α j) is smooth on ρ− 1(Oj) because ρ∗ (α j) = pr∗ j (α j |Nj ) there

    On Oj we have α j = π ∗ j (α j |Nj ). In particular, ρ∗ (α j) is smooth on ρ− 1(Oj) because ρ∗ (α j) = pr∗ j (α j |Nj ) there. Therefore, given [α ] ∈ H k(X) there is a representative α s ∈ Ωk(X) such that ρ∗ (α s) is smooth and ρ∗ (α s)|ρ− 1(Oj ) = pr ∗ j (α s|Nj ) for every 1 ≤ j ≤ 10. Proof. The form β = π ∗ j (α |N ′ j ) ∈ Ω2(O′′ j ) is κ-invariant be...

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  12. [12]

    Let µ 1, µ 2 ∈ R, since M − ∪ j,kN k j → X − ∪ jNj is an 8 : 1 cover we have ∫ X α ∧ (µ 1β1 + µ 2β2) = 1 8 ∫ M − 2i(µ 1λ 1 − µ 2λ 2)dt ∧ dz1¯ 12¯ 23¯ 3 = 2(µ 1λ 1 − µ 2λ 2)

    + λ 2(dz12¯ 3 + dz¯ 1¯ 23), where f ∗β = β and κ ∗ β = − β . Let µ 1, µ 2 ∈ R, since M − ∪ j,kN k j → X − ∪ jNj is an 8 : 1 cover we have ∫ X α ∧ (µ 1β1 + µ 2β2) = 1 8 ∫ M − 2i(µ 1λ 1 − µ 2λ 2)dt ∧ dz1¯ 12¯ 23¯ 3 = 2(µ 1λ 1 − µ 2λ 2). Using the coordinates described in ( 5), we compute α |N k 1 = 4( λ 1 − λ 2)dx12 ∧ dy3. Similarly, from equation ( 6) we o...

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    The cohomology class [υ 1 j ] is the Thom class of N 1 j

    The forms υ 1 3, υ 1 7 are closed and supported on V 1 3 and V 1 7 respectively. The cohomology class [υ 1 j ] is the Thom class of N 1 j

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    The form β vanishes in a neighborhood of the level set t = 1/ 2 and dβ = υ 1 3 − υ 1

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    In particular, it is closed on M − (V 1 3 ∪ V 1 7 ) and it satisfies [β |N k 1 ] = − 1 2 [ϕ |N k 1 ], [β |N k 2 ] = 1 2 [ϕ |N k 2 ]. Proof. For convenience, we use t ∈ [1/ 2, 3/ 2] instead of t ∈ [0, 1]. We first find a cobordism on M between N 1 3 and N 1 7 . We view points in N 1 3 at the level set t = 1 / 2 and we consider the coordinates in ( 7) and the ...

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    In addition, let p ∈ N 1 3 and let Fp be the fiber of V 1 3 at p, oriented by (∂t, ∂y1, ∂y2, ∂y3). Since υ 1 3 is supported on the level sets 1 / 2 + ǫ < t < 1/ 2 + 2ǫ, where f = 0, we have ∫ Fp υ 1 3 = ∫ 1/2+2ε t=1/2+ε h′(t)dt ∫ ∥(y1,y2,y3)∥<2δ ζ(y1, y2, y3)dy1 ∧ dy2 ∧ dy3 = h(1/ 2 + 2ε) − h(1/ 2 + ε) = 1 . Hence, [ υ 1 3] is the Thom class of N 1 3 . A s...

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    We first observe α ∧ dβ = − (λ 1 + µ 1)h′(t)ζ(y1, y2, y3 − f(t))dt ∧ dx123 ∧ dy123, t ∈ [1/ 2, 3/ 2] 11 because Im( dz123) ∧ dy123 = Im( dz12¯

    + µ 2 Im(dz12¯ 3), with f ∗ β = β . We first observe α ∧ dβ = − (λ 1 + µ 1)h′(t)ζ(y1, y2, y3 − f(t))dt ∧ dx123 ∧ dy123, t ∈ [1/ 2, 3/ 2] 11 because Im( dz123) ∧ dy123 = Im( dz12¯

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  18. [18]

    ∧ dy123 = 0, and Re( dz123) ∧ dy123 = Re( dz12¯

    work page

  19. [19]

    ∧ dy123 = dx123∧ dy123. Since υ 1 3 is the extension of dβ |V 1 3 , and it is supported on the level sets 1 / 2+ǫ < t < 1/ 2+2ǫ, where f = 0, we have ∫ M α ∧ υ 1 3 = ∫ V 1 3 α ∧ dβ = − (λ 1 + µ 1) ∫ t=1/2+2ε t=1/2+ε h′(t)dt ∫ T 6 ζ(y1, y2, y3)dx123 ∧ dy123 = λ 1 + µ 1. Using the coordinates in (

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  20. [20]

    This shows P D[N 1 3 ] = [ υ 1 3]

    and the orientation in ( 11), we get α |N 1 3 = ( λ 1 + µ 1)dx123, and ∫ N 1 3 α = λ 1 + µ 1. This shows P D[N 1 3 ] = [ υ 1 3]. Remark 6. Following the notation of the proof, we provide a geometric justifica tion of the equality∫ N k j β |N k j = σj for j, k = 1 , 2. Observe that N k j ∩ C = {pk j }, where pk 1 = [1 , [0, 0, − εk]] and pk 2 = [1, [0, 0, −...

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    The projection of the cobordism C obtained in the proof of Proposition 6 to M ′ determines a cobordism between N ′ 3 and N ′ 7 that intersects N ′ 1 negatively and N ′ 2 positively

    allows to find an oriented cobordism between N 1 1 and N 1 2 . The projection of the cobordism C obtained in the proof of Proposition 6 to M ′ determines a cobordism between N ′ 3 and N ′ 7 that intersects N ′ 1 negatively and N ′ 2 positively. The average of the form α provides a primitive for the difference of the Thom forms of N ′ 3 and N ′

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    We make this precise in the following result. Lemma 7. There are small tubular neighborhoods V ′ 3 , V ′ 7 of N ′ 3, N ′ 7 on M ′ and κ-invariant forms α ∈ Ω3(M ′), υ3, υ7 ∈ Ω4(M ′), such that

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    The cohomology class [υj] is the Thom class of N ′ j

    The forms υ3, υ7 are closed and supported on V ′ 3 and V ′ 7 respectively. The cohomology class [υj] is the Thom class of N ′ j

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  24. [24]

    The pushforwards of 2υ3 and 2υ7 to X represent the Thom class of N3 and N7 on X respectively

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    In particular, α is closed on M ′ − (V ′ 3 ∪ V ′ 7 )

    The form α vanishes in a neighborhood of the level set t = 1/ 2 and dα = υ3 − υ7. In particular, α is closed on M ′ − (V ′ 3 ∪ V ′ 7 )

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    There are tubular neighborhoods V ′ 1 and V ′ 2 of N ′ 1 and N ′ 2 such that α |V ′ 1 = − π ∗ 1(ϕ |N ′ 1), and α |V ′ 2 = π ∗ 2(ϕ |N ′ 2 ), where πj : V ′ j → N ′ j denotes the nearest point projection for j = 1, 2. Proof. For j = 3, 7 we denote V 2 j = ι2(V 1 j ) ⊂ M and K = ⟨ι1, ι2, κ⟩ = Z3

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    We first observe that every element of K 1 j maps N 1 j onto itself, and since the elements in K 1 j are isometries, they preserve V 1 j (and its fiber bundle structure)

    We let K 1 3 = {Id, ι1, κ, κ ◦ ι1} and K 1 7 = {Id, ι1, κ ◦ ι2, κ ◦ ι1 ◦ ι2}. We first observe that every element of K 1 j maps N 1 j onto itself, and since the elements in K 1 j are isometries, they preserve V 1 j (and its fiber bundle structure). In addition, the elements of K 2 j = K − K 1 j swap N 1 j with N 2 j and therefore, V 1 j with V 2 j . Of cour...

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    Hence, there are forms η3, η7 ∈ Ω(X) with compact support on Vj such that dηj = ρ∗ (τ2 j ) + 4υj

    In fact, equation ( 18) shows [ ρ∗ (τ2 j )]c = − 2Th[Nj]c = − 2[2υj]c. Hence, there are forms η3, η7 ∈ Ω(X) with compact support on Vj such that dηj = ρ∗ (τ2 j ) + 4υj. The strategy outlined in Lemma 3 allows to modify ηj around Nj and find a form η′ j supported on Vj so that its pullback ρ∗ (η′ j ) is a smooth form on ˜M and dη′ j = ρ∗ (τ2 j ) + 4υj. This...

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    Hence, ˜α ∧ τj = 4ρ∗ (α ) ∧ τj = σj 4pr∗ j (ϕ |Nj )

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    We prove by direct computation that y / ∈ I

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