Stable systolic inequalities via mod n covering
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866909053778657280 |
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| author | Kumar, Aditya |
| author_facet | Kumar, Aditya |
| contents | 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 $8π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26891 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stable systolic inequalities via mod n covering Kumar, Aditya Differential Geometry Metric Geometry 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 $8π$. |
| title | Stable systolic inequalities via mod n covering |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2604.26891 |