A Steenrod square on Khovanov homology and a cup-i product
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915869708255232 |
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| author | Rajapakse, Advika |
| author_facet | Rajapakse, Advika |
| contents | Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations $\text{Sq}^i$ on the Khovanov homology $Kh(L)$ of a link $L$. Later, Morán proposed a sequence of cup-i products on the $\mathbb{F}_2$-coefficient cochain complex of any augmented semi-simplicial object in the Burnside category. Applied to the Khovanov functor, he obtained another sequence of operations $\mathfrak{sq}^n$ on $Kh(L)$, where $\mathfrak{sq}^0$, $\mathfrak{sq}^1$ agree with the usual Steenrod squares. We prove that Lipshitz-Sarkar's $\text{Sq}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Morán's $\mathfrak{sq}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Steenrod square on Khovanov homology and a cup-i product Rajapakse, Advika Geometric Topology Algebraic Topology 57K18 (Primary) 55P42, 55S10 (Secondary) Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations $\text{Sq}^i$ on the Khovanov homology $Kh(L)$ of a link $L$. Later, Morán proposed a sequence of cup-i products on the $\mathbb{F}_2$-coefficient cochain complex of any augmented semi-simplicial object in the Burnside category. Applied to the Khovanov functor, he obtained another sequence of operations $\mathfrak{sq}^n$ on $Kh(L)$, where $\mathfrak{sq}^0$, $\mathfrak{sq}^1$ agree with the usual Steenrod squares. We prove that Lipshitz-Sarkar's $\text{Sq}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Morán's $\mathfrak{sq}^2$. |
| title | A Steenrod square on Khovanov homology and a cup-i product |
| topic | Geometric Topology Algebraic Topology 57K18 (Primary) 55P42, 55S10 (Secondary) |
| url | https://arxiv.org/abs/2603.16221 |