Higher Steenrod squares for Khovanov homology

Abstract

We describe stable cup-i products on the cochain complex with ${\mathbb{F}}_2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

Introduction

In 2014 [7], Lipshitz and Sarkar defined a new invariant of knots and links valued in spectra that refined Khovanov homology. By using framed flow categories, they associated to each link a cellular spectrum $\mathcal{X}$ whose cellular cochain complex was the Khovanov complex, and therefore its cohomology was the Khovanov homology of the link. A consequence of this fact is that, when taking cohomology with coefficients in the field with two elements ${\mathbb{F}}_2$, Khovanov homology becomes endowed with Steenrod squares.

Shortly after, Lipshitz and Sarkar [8] were able to give a combinatorial formula for the second Steenrod square on the Khovanov homology of any link L

$${\mathrm{Sq}}^2:K{h}^{⁎}(L;{\mathbb{F}}_2)⟶K{h}^{⁎+2}(L;{\mathbb{F}}_2),$$

in terms of the Khovanov complex and an extra datum called ladybug matching. They also showed (see also [14]) that ${\mathrm{Sq}}^2$ distinguishes some pairs of knots that are not distinguished by Khovanov homology.

Three years later, together with Lawson [5], they gave two new constructions of Khovanov spectra that simplified the original construction of Lipshitz and Sarkar. In their second construction, they associated to each link diagram D a strictly unital lax 2-functor ${\mathcal{F}}_D$ from a cube poset to the Burnside 2-category, and associated a realisation spectrum $|{\mathcal{F}}_D|$ to each such 2-functor. The spectrum $|{\mathcal{F}}_D|$ is homotopy equivalent to the spectrum $\mathcal{X}$ first constructed by Lipshitz and Sarkar. This construction was revisited in [9] and in [6], where they asked the following question:

Are there nice formulations of the action of the Steenrod algebra on $K{h}^{⁎}(L;{\mathbb{F}}_2)$, purely in terms of the Khovanov functor to the Burnside category?

Symmetric multiplications. In order to make the question concrete, we introduce the following nice formulation of Steenrod squares on a cochain complex: A symmetric multiplication on a cochain complex $(C^{⁎},d)$ of ${\mathbb{F}}_2$-modules is a family of operations

$${⌣}_i:C^p\otimes C^q⟶C^{p+q-i},i\in \mathbb{Z},$$

satisfying that

$$\alpha {⌣}_i\beta =0\text{ for }i<0\text{,}$$

$$d(\alpha {⌣}_i\beta )=d\alpha {⌣}_i\beta +\alpha {⌣}_{i}d\beta +\alpha {⌣}_{i-1}\beta +\beta {⌣}_{i-1}\alpha \text{ for all }i\text{.}$$

Such structure endows the cohomology groups of the cochain complex with Steenrod squares, which are operations

$${\mathfrak{sq}}^i:H^n(C^{⁎})⟶H^{n+i}(C^{⁎}),{\mathfrak{sq}}^i([\alpha ])=[\alpha {⌣}_{n-i}\alpha ],$$

defined for $i⩾0$. As a consequence of (1.1), ${\mathfrak{sq}}^i([\alpha ])=0$ if $i>n$ and the 0th operation ${⌣}_0$ gives a well-defined graded multiplication on the cohomology of $(C^{⁎},d)$.

The prominent example of these structures appears in the normalised cochain complex $N^{⁎}(Y_{•};{\mathbb{F}}_2)$ of a simplicial set $Y_{•}$, which becomes endowed with a symmetric multiplication using the cup-i product formulas of Steenrod [15].

The normalisation process in the construction of $N^{⁎}(X_{•};{\mathbb{F}}_2)$ is done in two steps: first, kill the image of the degeneracies of the simplicial set $Y_{•}$ thus obtaining a semi-simplicial set $X_{•}$ (a simplicial set without degeneracies), and then take the dual of the chain complex $C_{⁎}(X_{•};{\mathbb{F}}_2)$ of alternating sums of face maps on the semi-simplicial set $X_{•}$. The cup-i products are defined out of the semi-simplicial structure and involve only face maps, so the cochain complex of any semi-simplicial set is also enhanced with cup-i products.

As an example, here are formulas for ${⌣}_0$ and ${\mathfrak{sq}}^1$. If $\alpha \in C^p(X_{•};{\mathbb{F}}_2)$ and $\beta \in C^q(X_{•};{\mathbb{F}}_2)$, then $\alpha {⌣}_0\beta \in C^{p+q}(X_{•};{\mathbb{F}}_2)$ is the Alexander–Whitney product of α and β, whose value on a chain $\sigma \in C_{p+q}(X_{•};{\mathbb{F}}_2)$ is

$$(\alpha {⌣}_0\beta )(\sigma )=\alpha ({\partial }_{p+1}\cdots {\partial }_{p+1}\sigma )\cdot \beta ({\partial }_0\cdots {\partial }_0\sigma ).$$

If $\alpha \in C^n(X_{•};{\mathbb{F}}_2)$ is a cocycle, then the first Steenrod square ${\mathfrak{sq}}^1([\alpha ])$ of $[\alpha ]$ can be computed as $[\alpha {⌣}_{n-1}\alpha ]$, which is defined as

$$(\alpha {⌣}_{n-1}\alpha )(\sigma )=\sum _{\begin{array}{c}\hfill j<k\hfill \\ \hfill j,k\text{ even}\hfill \end{array}}\alpha ({\partial }_j\sigma )\cdot \alpha ({\partial }_k\sigma )+\sum _{\begin{array}{c}\hfill j>k\hfill \\ \hfill j,k\text{ odd}\hfill \end{array}}\alpha ({\partial }_j\sigma )\cdot \alpha ({\partial }_k\sigma ).$$

On the other hand, Steenrod squares on a topological space X can be defined as natural transformations ${\mathrm{Sq}}^i:H^{⁎}(X;{\mathbb{F}}_2)\to H^{⁎+i}(X;{\mathbb{F}}_2)$ that satisfy certain axioms. The fact that the Steenrod squares ${\mathfrak{sq}}^i$ for a simplicial set $X_{•}$ that arise from the cup-i products coincide with the axiomatic Steenrod squares ${\mathrm{Sq}}^i$ for the topological space $|X_{•}|$ is not immediate, and uses the singular chain functor of Eilenberg [3] to compare the cohomology operations in both settings.

Observe also that the simplicial structure is crucial to define the symmetric multiplication. In contrast, the cellular cochain complex of a CW-complex does not have in general a symmetric multiplication.

Stable symmetric multiplications. Condition (1.1) above implies that ${⌣}_0$ is a well-defined product on cohomology, but is not necessary for the definition of the Steenrod squares. A τ-shifted stable symmetric multiplication on a cochain complex $C^{⁎}$ of ${\mathbb{F}}_2$-modules is a family of operations

$${⌣}_i:C^p\otimes C^q⟶C^{p+q-i-\tau },i\in \mathbb{Z},$$

satisfying (1.2). Such a structure gives again operations

$${\mathfrak{sq}}^i:H^n(C^{⁎})⟶H^{n+i}(C^{⁎}),{\mathrm{Sq}}^i([\alpha ])=[\alpha {⌣}_{n-i-\tau }\alpha ]$$

defined for $i⩾0$. When $\tau =0$ we refer to it simply as a stable symmetric multiplication.

The Khovanov homology $K{h}_{⁎}(L;R)$ of an oriented link L with coefficients in a commutative ring with unit R is obtained as follows: From a diagram D of the link with c ordered crossings and $n_{-}$ negative crossings, Khovanov builds a contravariant functor $F_D:{\mathbf{2}}^{\mathbf{c}}\to \mathit{\text{R}}\text{-Mod}$ from the cube poset ${\mathbf{2}}^{\mathbf{c}}$ of dimension c to the category of R-modules (a cube of R-modules). Any such functor can be transformed functorially into a chain complex via the “totalisation functor” ${\mathrm{Tot}}_R$ and from the chain complex we can extract its cohomology:

The Khovanov cochain complex $C^{⁎}(D;R)$ of the link diagram D is the $n_{-}$-desuspension of dual cochain complex $C^{⁎}(F_D;R)$ and its cohomology does not depend on the diagram D, and thus defines an invariant $K{h}_{⁎}(L;R)$ of L, the Khovanov homology of L. If the link is connected, its Khovanov homology neither depends on the orientation of the link.

Lipshitz, Lawson and Sarkar went further by building a 2-functor ${\mathcal{F}}_D:{\left({\mathbf{2}}^{\mathbf{c}}\right)}^{\mathrm{op}}\to \mathcal{B}$ to the Burnside 2-category. The Burnside 2-category maps functorially to the category of R-modules, and $F_D$ can be recovered by postcomposing ${\mathcal{F}}_D$ with this functor:

Now, to each cube $\mathcal{F}$ in the Burnside 2-category they associated a spectrum $|\mathcal{F}|$ whose cohomology with coefficients in R is isomorphic to the graded R-module obtained from (1.3). Since the cohomology with coefficients in ${\mathbb{F}}_2$ of a spectrum is endowed with Steenrod operations ${\mathrm{Sq}}^i$, so does the cohomology with coefficients in ${\mathbb{F}}_2$ of any cube in the Burnside 2-category. In addition, the $n_{-}$-desuspension of the spectrum obtained from ${\mathcal{F}}_D$ does not depend on the link diagram D of L up to weak equivalence of spectra, and thus the Steenrod operations in Khovanov homology are link invariants as well.

The purpose of this paper is to endow the Khovanov cochain complex $C^{⁎}(D;{\mathbb{F}}_2)$ of a link diagram with a shifted stable symmetric multiplication, and, as a consequence, obtain Steenrod operations ${\mathfrak{sq}}^i$ on Khovanov homology. With this in mind, we will work in the more general setting of augmented semi-simplicial objects in the Burnside category, i.e., contravariant 2-functors from ${\mathrm{\Delta }}_{\mathrm{inj}⁎}$ (the category of finite ordinals and order-preserving injective maps) to the Burnside 2-category. In this case, we have a diagram similar to (1.3)

where M denotes the Moore complex construction. From a finite dimensional augmented semi-simplicial object in the Burnside category $X_{•}$ one can obtain its realisation spectrum ${\Vert X_{•}\Vert }_{\mathbb{S}}$ mimicking the realisation of a cube in the Burnside category given in [5]. Again, the cohomology of this spectrum is isomorphic to the graded abelian group obtained from (1.4), and therefore the cohomology of any augmented semi-simplicial object in the Burnside category becomes endowed with Steenrod squares.

In Definition 2.10, Definition 6.3 we introduce a category $o{\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$ of ordered augmented semi-simplicial objects in the Burnside category and free order-preserving maps between them that maps surjectively on objects to the category ${\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$. Additionally, the category of simplicial sets includes into $o{\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$:

Finally, we will relate cubes and semi-simplicial objects via a certain functor Λ obtaining the following commutative diagram

where the composition of the bottom row sends a simplicial set to its Moore complex.

Our first result constructs a stable symmetric multiplication on the cohomology with ${\mathbb{F}}_2$-coefficients of any ordered augmented semi-simplicial object in the Burnside category that is functorial on $o{\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$:

Theorem A

The cochain complex of an ordered augmented semi-simplicial object in the Burnside category $X_{•}$ with coefficients in ${\mathbb{F}}_2$ has a natural stable symmetric multiplication, i.e., there are explicit operations

$${⌣}_i:C^p(X_{•};{\mathbb{F}}_2)\otimes C^q(X_{•};{\mathbb{F}}_2)⟶C^{p+q-i}(X_{•};{\mathbb{F}}_2),i\in \mathbb{Z}$$

satisfying (1.2), and, for every free order-preserving map $f:X_{•}\to Y_{•}$,

$$f^{⁎}(\alpha {⌣}_i\beta )=f^{⁎}(\alpha ){⌣}_i{f}^{⁎}(\beta ).$$

If $X_{•}$ is a semi-simplicial set, then these operations are the Steenrod cup-i products.

The explicit formulas for the ${⌣}_i$ products are given in Section 3 and the naturality is proven in Proposition 6.4. In Section 5 we prove that the Steenrod squares ${\mathfrak{sq}}^i$ induced by the symmetric multiplication are invariant under suspension, satisfy a Cartan formula and the first square ${\mathfrak{sq}}^1$ is the Bockstein homomorphism.

In Theorem 6.6 we show that these Steenrod squares are canonically defined for augmented semi-simplicial objects in the Burnside category and are functorial on the category ${\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$:

Theorem B

Let $Y_{•}$ be an ordered augmented semi-simplicial object in the Burnside category mapping to $X_{•}\in {\mathcal{B}}^{{\mathit{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$. The Steenrod operations ${\mathfrak{sq}}^i:H^{⁎}(Y_{•};{\mathbb{F}}_2)\to H^{⁎+i}(Y_{•};{\mathbb{F}}_2)$ associated to the stable symmetric multiplication depend only on $X_{•}$, thus they give canonical operations

$${\mathfrak{sq}}^i:H^{⁎}(X_{•};{\mathbb{F}}_2)\to H^{⁎+i}(X_{•};{\mathbb{F}}_2).$$

Additionally, these operations are natural with respect to maps of augmented semi-simplicial objects in the Burnside category.

If D is a oriented link diagram with $n_{-}$ negative crossings, choosing a lift of $\mathrm{\Lambda }({\mathcal{F}}_D)$ to $o{\mathcal{B}}^{{\mathrm{\Delta }}_{\mathrm{inj}⁎}^{\mathrm{op}}}$ allows to define, via Theorem A, a $(1-n_{-})$-shifted stable symmetric multiplication on the Khovanov cochain complex $C^{⁎}({\mathcal{F}}_D;{\mathbb{F}}_2)$. The following is obtained in Corollary 7.10:

Corollary C

There is an explicit $(1-n_{-})$-shifted stable symmetric multiplication on the Khovanov complex of D with coefficients in ${\mathbb{F}}_2$

$${⌣}_i:C^p(D;{\mathbb{F}}_2)\otimes C^q(D;{\mathbb{F}}_2)⟶C^{p+q-i-1+n_{-}}(D;{\mathbb{F}}_2)i\in \mathbb{Z}.$$

Therefore Khovanov homology becomes endowed with the Steenrod squares

$${\mathfrak{sq}}^i:K{h}^n(D;{\mathbb{F}}_2)⟶K{h}^{n+i}(D;{\mathbb{F}}_2),{\mathfrak{sq}}^i([\alpha ])=[\alpha {⌣}_{n+n_{-}-1-i}\alpha ]$$

associated to this stable symmetric multiplication, which are invariant under Reidemeister moves and reordering of the crossings.

The techniques of this paper do not allow us to prove that the Steenrod squares of Corollary C coincide with the Steenrod squares

$${\mathrm{Sq}}^i:H^n(|{\mathcal{F}}_D|;{\mathbb{F}}_2)⟶H^{n+i}(|{\mathcal{F}}_D|;{\mathbb{F}}_2),i⩾0$$

of the realisation spectrum of ${\mathcal{F}}_D$. Such comparison will be developed in the companion paper [2], where we will construct a “singular chain functor” from the category of spectra to the category of augmented semi-simplicial objects in the Burnside category. The results in the present paper are purely combinatorial, and have to be compared with the constructions of Steenrod in [15] for simplicial complexes, whereas the results of the companion paper [2] are mainly homotopy-theoretic and have to be compared with the constructions of Eilenberg [3] for topological spaces. In particular, spectra will be essentially absent from this paper, and will only be barely mentioned in some examples in Section 8.

The formulas for the Steenrod squares in Khovanov homology of [8] were first used by Lipshitz and Sarkar themselves to find non-trivial second Steenrod squares in knots up to 11 crossings. They also found a pair of links with the same Khovanov homology but different ${\mathrm{Sq}}^2$. Shortly after, Seed extended these computations to knots up to 14 crossings [14] and found many pairs of knots with the same Khovanov homology but different ${\mathrm{Sq}}^2$. Later, Lobb, Orson and Schütz [11] gave a new algorithm to compute ${\mathrm{Sq}}^2$ using simplifications developed in [10] and [4]. They first associated to each framed flow category a framed 1-flow category, which is a simpler object that keeps enough information to compute ${\mathrm{Sq}}^2$. This framed 1-flow category is then further simplified using “flow category moves”. Because of the Adem relations, the first and second Steenrod squares determine the third. As a consequence, any of these algorithms is able to compute ${\mathrm{Sq}}^i$ for $i⩽3$, and extensive computations have been made for these squares. It is therefore natural to ask, for each $i⩾4$, the following:

Are there links whose Khovanov homology supports a non-trivial Steenrod square ${\mathrm{Sq}}^i$? Are there pairs of links with the same Khovanov homology but different ${\mathrm{Sq}}^i$?

The formulas for ${\mathrm{Sq}}^2$ in [8] do not agree with the formulas of Theorem A at the cochain level. How do the formulas of Lipshitz and Sarkar for ${\mathfrak{sq}}^2$ compare to the ones in this article? More concretely,

is there an explicit cocycle whose coboundary is the difference between both formulas?

The formulas of [8] for ${\mathrm{Sq}}^2$ were proven to be well-defined using topological methods, and a purely combinatorial proof was later given in [13].

Outline of the paper. In Section 2, we first explain how to translate the framework of [5], which is expressed in terms of cubes in the Burnside category, to our framework in terms of augmented semi-simplicial objects in the Burnside category. Then we introduce several definitions and constructions that will be used through the paper. In Section 3, we present formulas for cup-i products, and we prove in Section 4 that they endow the cohomology of any augmented semi-simplicial object in the Burnside category with a stable symmetric multiplication. In Section 5 we define Steenrod squares and we prove that they are stable under suspension, that they satisfy a Cartan formula and that the first square is the Bockstein homomorphism. The proofs of naturality are deferred to Section 6. In Section 7 we apply the previous results to the Khovanov functor of Lawson, Lipshitz and Sarkar and we prove Corollary C. The paper finishes with several examples in Section 8. The reader only interested on explicit formulas for operations on Khovanov homology will find them in Section 3 after having got used to the terminology introduced in Section 2, and may afterward safely skip Sections 4, 5 and 6 and proceed directly to Section 7.3 and the examples in Section 8.

Acknowledgments. The author is especially grateful to Aníbal Medina-Mardones for the inspiration received while reading his paper [12]. He is also grateful to Fernando Muro and David Chataur, and to Javier Gutiérrez, Carles Casacuberta, Joana Cirici and Marithania Silvero from the Topology group at Barcelona. He thanks Tyler Lawson, Clemens Berger, and Oscar Randal-Williams for their feedback during his stay at the Isaac Newton Institute for Mathematical Sciences.