[论文解读] A Steenrod square on Khovanov homology and a cup-i product

Advika Rajapakse|arXiv (Cornell University)|Mar 17, 2026

Homotopy and Cohomology in Algebraic Topology被引用 0

一句话总结

本论文证明Khovanov同调上的第二 Steenrod 开平方 Sq^2 与 Morán 在相关增广半简单对象上的 cup-i 构造一致，从而将两种在 Khovanov 理论中高阶 Steenrod 运算的方法联系起来。

ABSTRACT

Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations $ ext{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 $ ext{Sq}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Morán's $\mathfrak{sq}^2$.

研究动机与目标

• Motivate and review the Lipshitz-Sarkar stable homotopy refinement of Khovanov homology and its induced cohomology operations.
• Introduce Morán’s cup-i product construction and define the associated sq^n operations on Kh(L) with F2 coefficients.
• Prove that Sq^2 and Morán’s sq^2 coincide via a coboundary argument, strengthening the conjecture that sq^n agrees with Sq^n for all n.

提出的方法

• Describe the cube/Burnside category framework underlying F_Kh(L) and the Tot(F_Kh) construction.
• Define Morán’s cup-i product via the coproduct of spans and the diagonal map, and express sq^2 via alpha smash_{n-2} alpha.
• Present a cubical special graph structure Gamma(z, alpha) to formulate the Sq^2 computation.
• Fix an ordering of spans and a boundary matching to align Morán’s and Lipshitz-Sarkar’s constructions.
• Compute and simplify the expression for <sq^2(alpha), z> and compare it with <Sq^2(alpha), z> by identifying a coboundary.
• Utilize auxiliary lemmas and coboundary equivalences to show the difference is a coboundary.

实验结果

研究问题

• RQ1 Does Morán’s sq^2 coincide with Lipshitz-Sarkar’s Sq^2 on Kh(L) when coefficients are in F2?
• RQ2 Can the second Steenrod square be expressed purely from Morán’s cup-i framework and augmented semi-simplicial data?
• RQ3 Do the two constructions agree up to coboundary for all cocycles in the Tot(F) complex associated to Kh(L;F2)?

主要发现

• Sq^2 and Morán’s sq^2 agree on Kh(L; F2) after canonical identification with the Tot(F) complex.
• A coboundary is exhibited that accounts for the difference between the two constructions, establishing their equivalence.