Stable systolic inequalities via mod n covering
Pith reviewed 2026-05-20 23:57 UTC · model grok-4.3
The pith
Mod n coverings improve the stable two-systolic bound to 2 for S²×S² and all oriented four-manifolds with b₂=2.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By prescribing a mod n cohomology class that guarantees a nonzero cup product and then constructing a short integral lift, one obtains a covering whose degree controls the stable systole. In rank two the covering constant is computed explicitly, producing the inequality stsys₂ ≤ 2 for every oriented four-manifold with b₂=2. When scalar curvature is at least 4, the same method combined with a sharp cowaist inequality for line bundles gives stsys₂ ≤ 8π on S²×S².
What carries the argument
The mod n covering, obtained by prescribing a cohomology class mod n that forces the desired cup product to be nonzero and then lifting it to a short integral class.
If this is right
- The stable two-systolic inequality holds with constant 2 for S²×S² and every oriented four-manifold with b₂=2.
- Every metric on S²×S² with scalar curvature at least 4 satisfies stsys₂ ≤ 8π.
- Odd-dimensional complex projective spaces obey the sharp stable two-systolic inequality.
- The product of m copies of S² with scalar curvature at least 2m satisfies an O(m log m) stable two-systolic bound.
Where Pith is reading between the lines
- The mod n lifting technique may extend to manifolds whose second cohomology has rank greater than two once covering constants can be estimated.
- Combining the method with other curvature-free systolic inequalities could produce volume-normalized bounds in additional topologies.
- The approach suggests a route to stable systolic bounds in dimensions higher than four by iterating the mod n construction.
Load-bearing premise
A short integral lift of the prescribed mod n cohomology class exists and the covering constant can be computed explicitly when the second Betti number is two.
What would settle it
An explicit Riemannian metric on S²×S² whose stable two-systole exceeds 2 would contradict the claimed bound.
read the original abstract
We introduce a mod $n$ covering based approach to stable systolic inequalities. The idea is to prescribe a cohomology class mod $n$ which forces the desired cup product or index to be nonzero, and then find a short integral lift of that class. The method is especially effective in rank two as we can compute the covering constant. As a curvature free application, we improve the stable two systolic bound for $S^2\times S^2$ to $2$. The same bound holds for every oriented four manifold with $b_2=2$. Under a positive scalar curvature lower bound, the mod $n$ covering method combined with a sharp cowaist inequality for line bundles gives stable two systolic bounds. This gives the sharp stable two systolic inequality for odd complex projective spaces and an $O(m\log m)$ bound for $(S^2)^m$ when scalar curvature is at least $2m$. For $S^2\times S^2$ one gets that every metric with scalar curvature at least $4$ has stable two systole at most $8\pi$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces a mod n covering approach to stable systolic inequalities. A mod n cohomology class is prescribed to force a nonzero cup product (or index), followed by an explicit short integral lift whose covering constant is computed in rank two. This yields a curvature-free improvement of the stable two-systole bound to 2 for S²×S² and all oriented 4-manifolds with b₂=2. Under a scalar-curvature lower bound the method is combined with a cowaist inequality for line bundles, producing the sharp stable two-systolic inequality for odd complex projective spaces, an O(m log m) bound for (S²)^m when scalar curvature is at least 2m, and the concrete bound stsys₂ ≤ 8π for metrics on S²×S² with scalar curvature at least 4.
Significance. If the lift construction and covering-constant computation are verified, the paper supplies a flexible new technique that improves known stable systolic bounds in dimension four and under positive scalar curvature. The rank-two case is presented as particularly effective because the covering constant can be computed explicitly; this is a concrete strength that could extend to other low-rank situations.
major comments (2)
- [mod n covering construction for rank two / curvature-free application] The improvement of the stable two-systole bound from its previous value to exactly 2 for S²×S² (and all oriented 4-manifolds with b₂=2) rests on the claim that a short integral lift of the prescribed mod n class exists and that the covering constant in rank two yields the factor 2. The abstract asserts that the constant is computable, but the explicit norm bound on the lift and the verification that it remains short enough to improve rather than recover a prior inequality must be supplied in the main text (see the construction in the section deriving the curvature-free bound).
- [scalar curvature applications] In the scalar-curvature section, the combination of the mod n method with the sharp cowaist inequality for line bundles is used to obtain stsys₂ ≤ 8π when scalar curvature ≥ 4 on S²×S² and the sharp bound for odd CP^n. The precise constant arising from the covering and the way the cowaist inequality is applied to the lifted class should be written out explicitly so that the claimed sharpness can be checked.
minor comments (2)
- [abstract] The phrase 'four manifold' in the abstract should be hyphenated as 'four-manifold'.
- [introduction] A brief comparison table or sentence recalling the best previously known stable two-systolic constants for S²×S² and for manifolds with b₂=2 would help readers assess the size of the improvement.
Simulated Author's Rebuttal
We thank the referee for the careful and constructive report, which identifies clear opportunities to strengthen the explicitness of our arguments. We address each major comment below and will revise the manuscript to incorporate the requested details while preserving the core contributions of the mod n covering method.
read point-by-point responses
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Referee: The improvement of the stable two-systole bound from its previous value to exactly 2 for S²×S² (and all oriented 4-manifolds with b₂=2) rests on the claim that a short integral lift of the prescribed mod n class exists and that the covering constant in rank two yields the factor 2. The abstract asserts that the constant is computable, but the explicit norm bound on the lift and the verification that it remains short enough to improve rather than recover a prior inequality must be supplied in the main text (see the construction in the section deriving the curvature-free bound).
Authors: We agree that the explicit norm bound and verification step benefit from greater prominence in the main text. The construction of the short integral lift and the computation of the covering constant for the rank-two case are already contained in the section on the curvature-free bound, but we will expand this section in the revision to include the full norm estimate, the explicit value of the covering constant, and a direct comparison demonstrating that the resulting bound improves upon prior inequalities rather than recovering them. This will make the factor-of-2 improvement fully verifiable from the main text. revision: yes
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Referee: In the scalar-curvature section, the combination of the mod n method with the sharp cowaist inequality for line bundles is used to obtain stsys₂ ≤ 8π when scalar curvature ≥ 4 on S²×S² and the sharp bound for odd CP^n. The precise constant arising from the covering and the way the cowaist inequality is applied to the lifted class should be written out explicitly so that the claimed sharpness can be checked.
Authors: We accept that the scalar-curvature applications require more explicit bookkeeping of constants. In the revised manuscript we will expand the relevant section to display the precise covering constant obtained from the mod n lift, the manner in which the cowaist inequality is applied to the lifted class, and the arithmetic that produces the bound stsys₂ ≤ 8π for metrics on S²×S² with scalar curvature at least 4. The same explicit treatment will be given for the sharp inequality on odd complex projective spaces, allowing direct verification of sharpness. revision: yes
Circularity Check
No significant circularity; derivation relies on independent cohomology and covering constructions
full rationale
The paper introduces a mod n covering method that prescribes a cohomology class forcing nonzero cup product, then seeks a short integral lift with explicitly computed covering constant in rank two. This is applied to improve the stable two-systole bound for S²×S² and b₂=2 manifolds to 2 (curvature-free) and to obtain scalar-curvature bounds such as 8π for S²×S². No quoted step reduces a claimed prediction or bound to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation chain or ansatz smuggled from prior author work. The central claims rest on standard algebraic topology and covering-space arguments that remain externally verifiable and do not presuppose the target systolic inequalities.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Cup product in cohomology is nonzero when forced by a suitable mod n class
- domain assumption Existence of short integral lifts of mod n classes
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We introduce a mod n covering based approach to stable systolic inequalities... improve the stable two systolic bound for S²×S² to 2
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
Cited by 1 Pith paper
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Sharp systolic inequalities for K\"ahler manifolds
Sharp systolic inequalities for Kähler manifolds with positive scalar curvature attain equality on CP^n with Fubini-Study metric and imply Gromov's rational-essentialness conjecture.
Reference graph
Works this paper leans on
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[1]
Stable $2$-systoles, scalar curvature and spin$^c$ comass bounds
arXiv:2604.25900. [GHK23] T. Goodwillie, J. Hebda, and M. Katz. Extending Gromov’s optimal systolic inequality.J. Geom., 114:Paper No. 23,
work page internal anchor Pith review Pith/arXiv arXiv
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[2]
[Gro23] M. Gromov. Four lectures on scalar curvature. InPerspectives in scalar curvature. Vol. 1, pages 1–514. World Sci. Publ., Hackensack, NJ, [2023]©2023. [Str26] D. Stryker. Stable 2-systole bounds in positive scalar curvature,
work page 2023
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[3]
Stable 2-systole bounds in positive scalar curvature
arXiv:2604.22106. Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA
work page internal anchor Pith review Pith/arXiv arXiv
discussion (0)
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