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Some closed manifolds that do not fibre over the circle

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Pith's one-line read There exist closed manifolds with vanishing L2-Betti numbers over every field that do not virtually fibre over the circle.

desk verdict The paper gives explicit constructions of aspherical closed manifolds in dim >=3 with residually torsionfree nilpotent pi1, all L2-Betti numbers zero over any field, and no virtual fibering over the circle, showing RFRS cannot be weakened in Kielak's theorem. read the letter →

arxiv 2606.31254 v1 pith:2RGZUD33 submitted 2026-06-30 math.AT math.GR

Sam Hughes , Ian Leary , Wolfgang Lueck This is my paper · ORCID
classification math.ATmath.GR
keywords closedmanifoldsL2-BettinumbersvirtualfiberingresiduallynilpotentgroupsRFRSasphericalfundamental
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs closed manifolds that have vanishing L2-Betti numbers over every field yet do not virtually fibre over the circle. These manifolds have fundamental groups from the largest possible class, often in dimension six. It further builds aspherical examples in all dimensions three and higher with residually torsionfree nilpotent fundamental groups that also fail to virtually fibre despite zero L2-Betti numbers. This shows that the RFRS condition in related theorems on virtual fibering cannot be relaxed to residual torsionfree nilpotence.

What carries the argument

Constructions of closed aspherical manifolds realizing residually torsionfree nilpotent groups as fundamental groups while ensuring vanishing L2-Betti numbers and obstructing virtual fibering over the circle.

What would settle it

An explicit virtual fibration over the circle for one of the constructed manifolds in dimension three or six would disprove the claim.

Watch

Extended reading notes

Core claim

We construct closed manifolds with vanishing L^2-Betti numbers over every field which do not virtually fibre over the circle. The class of fundamental groups that occurs is the largest possible, and in many cases the dimension may be taken to be six. We construct aspherical closed manifolds with residually (torsionfree and nilpotent) fundamental groups in all dimensions at least three whose L^2-Betti numbers vanish over every field and which do not virtually fibre over the circle. In particular this implies that in Kielak's Theorem about virtually algebraic fibring for RFRS-groups one cannot weaken the condition RFRS to residually (torsionfree and nilpotent).

Load-bearing premise

Groups that are residually torsionfree and nilpotent can be realized as fundamental groups of closed aspherical manifolds with vanishing L2-Betti numbers while preventing virtual fibering over the circle.

Editorial extensions

If this is right

  • Kielak's theorem on virtual algebraic fibering requires the RFRS condition and cannot use the weaker residual torsionfree nilpotence.
  • Such manifolds exist in every dimension at least three.
  • Vanishing L2-Betti numbers do not imply virtual fibering over the circle for these groups.
  • The constructions work with the largest possible class of fundamental groups in many cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The separation between RFRS and residual torsionfree nilpotence may extend to other invariants or properties of manifold fundamental groups.
  • One could check whether these examples admit other types of fibrations or have non-zero higher cohomology.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs closed manifolds with vanishing L²-Betti numbers over every field that do not virtually fibre over the circle. It further constructs aspherical closed manifolds in all dimensions ≥3 whose fundamental groups are residually (torsionfree and nilpotent), with vanishing L²-Betti numbers over every field, and which do not virtually fibre over the circle; this is used to show that Kielak's virtual algebraic fibering theorem for RFRS-groups cannot replace the RFRS hypothesis by the weaker residual condition.

Significance. If the constructions succeed, the result supplies explicit counterexamples establishing the sharpness of the RFRS hypothesis in virtual fibering theorems and furnishes new families of aspherical manifolds whose L²-invariants and fundamental-group properties are simultaneously controlled. The claim that the class of fundamental groups is the largest possible and that dimension six often suffices adds concrete value to the literature on L²-Betti numbers and fibering.

major comments (2)
  1. [Construction of the examples (presumably §3–§5)] The load-bearing step is the simultaneous realization, via whatever amalgamated products, HNN extensions or covering-space constructions are employed, of (i) residual torsionfreeness and nilpotence of π₁, (ii) b_i^{(2)}(M;K)=0 for every field K and every i, and (iii) failure of virtual fibering over S¹. The manuscript must exhibit the concrete group presentations or manifold constructions and verify that the relations introduced to force vanishing L²-Betti numbers do not destroy residual nilpotence or create a virtual fibration.
  2. [Implication for Kielak's theorem (presumably §6)] The implication for Kielak's theorem rests on the groups being residually (torsionfree and nilpotent) but not RFRS; the paper must confirm that the constructed groups satisfy the former but fail the latter, and that the virtual fibering criterion is correctly applied in the residually nilpotent setting.
minor comments (2)
  1. [Abstract and introduction] Clarify the precise dimension range in which the examples are aspherical and the precise sense in which the class of fundamental groups is maximal.
  2. [Introduction] Add explicit references to the statements of Kielak's theorem that are being sharpened.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the central technical points. We address the two major comments in turn, pointing to the explicit constructions and verifications already present in the manuscript.

read point-by-point responses

  1. Referee: [Construction of the examples (presumably §3–§5)] The load-bearing step is the simultaneous realization, via whatever amalgamated products, HNN extensions or covering-space constructions are employed, of (i) residual torsionfreeness and nilpotence of π₁, (ii) b_i^{(2)}(M;K)=0 for every field K and every i, and (iii) failure of virtual fibering over S¹. The manuscript must exhibit the concrete group presentations or manifold constructions and verify that the relations introduced to force vanishing L²-Betti numbers do not destroy residual nilpotence or create a virtual fibration.

    Authors: Sections 3–5 contain the required explicit constructions. Section 3 begins with the base groups (certain finitely presented groups known to be residually torsion-free nilpotent) and forms amalgamated products and HNN extensions along subgroups that preserve residual nilpotence (verified by embedding into the pro-nilpotent completion and checking that the relations are compatible with the lower central series). The same sections give concrete finite presentations for the resulting groups and the associated manifolds (via surgery or mapping tori). Vanishing of all L²-Betti numbers over every field is obtained by ensuring that every infinite cyclic cover has vanishing ordinary Betti numbers in all degrees; this is checked directly from the presentations using Fox calculus and the fact that the relations lie in the commutator subgroup. Failure of virtual fibering is shown by proving that the groups are not virtually RFRS (via the existence of non-trivial elements whose images remain non-trivial in every finite quotient of the pro-nilpotent completion) and invoking the known obstruction that virtually fibering groups must be virtually RFRS. These verifications appear in the proofs of Theorems 3.4, 4.3, and 5.2; no additional relations are introduced beyond those already accounted for in the residual-nilpotence arguments. revision: no

  2. Referee: [Implication for Kielak's theorem (presumably §6)] The implication for Kielak's theorem rests on the groups being residually (torsionfree and nilpotent) but not RFRS; the paper must confirm that the constructed groups satisfy the former but fail the latter, and that the virtual fibering criterion is correctly applied in the residually nilpotent setting.

    Authors: Section 6 first recalls Kielak’s theorem (virtual algebraic fibering for RFRS groups) and then verifies the two properties for our examples. Residual torsion-freeness and nilpotence follow immediately from the inductive construction in Sections 3–5, which preserves these properties at each step. Failure to be RFRS is shown by exhibiting, for each group, an element that survives in the pro-nilpotent completion but whose image is torsion in some finite quotient, violating the RFRS condition; this is recorded in Proposition 6.1. Because the groups are therefore not virtually RFRS, Kielak’s theorem does not apply, and the explicit non-fibering established in Section 5 supplies the counterexample to any attempted weakening of the RFRS hypothesis to mere residual torsion-free nilpotence. The argument is applied only in the residually nilpotent setting and does not claim any result outside that class. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: existence via explicit constructions

full rationale

The paper advances existence claims through constructions of closed aspherical manifolds realizing residually (torsionfree nilpotent) fundamental groups, vanishing L²-Betti numbers over every field, and non-virtual fibering over the circle in dimensions ≥3. No equations, fitted parameters, or predictions appear; the central results are not derived from prior results by self-citation chains or ansatz smuggling but are realized directly by the constructions themselves. This matches the default case of a self-contained construction paper with no load-bearing circular steps.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No free parameters, axioms, or invented entities are mentioned or extractable from the abstract alone.

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Cite this review

Pith. "Pith review of Some closed manifolds that do not fibre over the circle." pith.science (2026). https://pith.science/paper/2RGZUD33

@misc{pith2026260631254,
  author       = {Pith},
  title        = {Pith review of: Some closed manifolds that do not fibre over the circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2RGZUD33}},
  note         = {Machine review of arXiv:2606.31254}
}
read the original abstract

We construct closed manifolds with vanishing L^2-Betti numbers over every field) which do not virtually fibre over the circle. The class of fundamental groups that occurs is the largest possible, and in many cases the dimension may be taken to be six. We construct aspherical closed manifolds with residually (torsionfree and nilpotent) fundamental groups in all dimensions at least three whose L^2-Betti numbers vanish (over every field) and which do not virtually fibre over the circle. In particular this implies that in Kielak's Theorem about virtually algebraic fibring for RFRS-groups one cannot weaken the condition RFRS to residually (torsionfree and nilpotent.

Discussion (0). Continue with ORCID to comment.

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