A Steenrod square on Khovanov homology and a cup-i product

A STEENR OD SQUARE ON KHO V ANO V HOMOLOGY AND A CUP-I PR ODUCT AD VIKA RAJAP AKSE Abstract. Lipshitz-Sark ar deﬁned a stable homotopy t yp e reﬁning Khov anov homology , pro ducing cohomology op erations Sq i on the Khov anov homology K h ( L ) of a link L . Later, Mor´ an prop osed a sequence of cup-i pro ducts on the F 2 -co eﬃcien t co c hain complex of any augmen ted semi-simplicial ob ject in the Burnside category . Applied to the Kho v ano v func- tor, he obtained another sequence of op erations sq n on K h ( L ), where sq 0 , sq 1 agree with the usual Steenro d squares. W e prov e that Sq 2 , the ﬁrst Steenrod op eration that cannot be computed from merely homological data, agrees with Mor´ an’s sq 2 . Contents 1. In tro duction 1 1.1. Kho v anov homology and Khov anov stable homotopy 1 1.2. Higher Steenro d squares 2 1.3. Outline of the pro of 3 2. Bac kground and deﬁnitions 3 2.1. The cub e and the Burnside category 3 2.2. Augmen ted semi-simplicial sets 3 2.3. Mor´ an’s cup-i pro duct 4 2.4. Bac kground for the Sq 2 form ula 5 2.5. Fixing an ordering and a b oundary matc hing 6 3. Simplifying the expression ⟨ sq 2 ( α ) , z ⟩ 6 4. The diﬀerence of the tw o formulas is a cob oundary 13 App endix A. Basic co c hain iden tities 17 App endix B. Adv anced co c hain iden tities 20 References 26 1. Intr oduction 1.1. Kho v ano v homology and Kho v ano v stable homotop y. In [ Kho00 ], Kho v anov categoriﬁed the Jones polynomial. That is, for each oriented link diagram L , he assigned a bigraded ab elian group K h i,j ( L ) whose graded Euler characteristic is the (unnormalized) Jones p olynomial: χ ( K h i,j ( L )) = X i,j ( − 1) i q j rank K h i,j ( L ) = ( q + q − 1 ) V ( L ) . AR was supp orted b y NSF Grant DMS-2136090. 1 2 AD VIKA RAJAP AKSE F urthermore, K h i,j ( L ), which is called the Khovanov homolo gy of L , is an inv arian t of the underlying link. The discov ery of Kho v ano v homology has led to man y striking applications in low-dimensional top ology (see [ Ng05 , Pla06 , Ras10 , Pic20 ]). In [ LS14a ], Lipshitz-Sark ar gav e a space-level reﬁnement of Khov ano v homology , con- structing a space X ( L ) whose stable homotopy type is a link inv arian t, and whose reduced cohomology is K h i,j ( L ). This spatial lift has also lead to many topolological applications (see [ SZ24 , LS24 , BHS25 ]). F or example, this spatial lift allows us to deﬁne cohomology op erations on Kho v anov homology , coming from Steenro d squares Sq i . The ﬁrst couple are determined b y the identities Sq 0 = Id, and Sq 1 = β , the Bo ckstein. The next op eration, Sq 2 , is trickier to compute, since Sq 2 cannot simply b e determined by chain-complex-lev el-data. Nonetheless, Lipshitz-Sark ar [ LS14c ] ga ve a computable formula for Sq 2 , giving rise to new computable concordance inv arian ts [ LS14b ]. 1.2. Higher Steenro d squares. In [ LLS20 ], La wson-Lipshitz-Sark ar give several refor- m ulations of the Kho v ano v sp ectrum X ( L ), in terms of a (strictly unitary , lax) 2-functor F K h ( L ) : 2 n → B from the cub e category to the Burnside category . In [ LLS17 ], they ask if, just lik e for Sq 1 and Sq 2 , there is a w a y to compute higher Steenro d squares using purely the data of F K h ( L ). In a p oten tial answ er to this question, Mor´ an [ CM20 ] deﬁned a sequence of cohomology op erations sq n : K h i,j ( L ; F 2 ) → K h i + n,j ( L ; F 2 ) , n ≥ 0 . These op erations are link in v ariants that indeed only dep end on F . T o deﬁne sq n , Mor´ an asso ciates to a 2-functor F : 2 N → B , an augmente d semi-simplicial obje ct X • = Λ( F ) in the Burnside category , which has a co chain complex C ∗ ( X • ) satisfying Σ C ∗ ( X • , F 2 ) ∼ = C ∗ (T ot F ; F 2 ) . The op erations sq n are deﬁned on the homology group H ∗ ( X • ; F 2 ), which, in the case F = F K h ( L ; F 2 ), is just (a grading shift of ) K h ( L ). So far, it was kno wn that sq 0 , sq 1 agree with Id and β , the identit y and Bockstein op erations resp ectively . The follo wing theorem and corollary conﬁrm that sq 2 do es indeed arise as the Steenro d square Sq 2 on the sp ectrum X ( L ). Therefore, we add evidence to the conjecture that sq n indeed do concide with Sq n for all n . Theorem 1.1. L et X • = Λ( F ) . With the c anonic al identiﬁc ation Σ C ∗ ( X • ; F 2 ) ∼ = C ∗ (T ot F ; F 2 ) , the op er ations sq 2 : H ∗ ( X • ; F 2 ) → H ∗ +2 ( X • ; F 2 ) , Sq 2 : H ∗ (T ot F ; F 2 ) → H ∗ +2 (T ot F ; F 2 ) agr e e. Corollary 1.2. Given an oriente d link diagr am L , and let X • = Λ( F K h ( L )) denote the augmente d semi-simplicial obje ct in the Burnside c ate gory, deﬁne d by [ CM20 ] . Identify Σ − n − +1 C ∗ ( X • ; F 2 ) ∼ = e C ∗ ( X ( L ); F 2 ) ∼ = Σ − n − C ∗ (T ot F ; F 2 ) ∼ = K c ( L ; F 2 ) . The op er ation sq 2 : K h i,j ( L ; F 2 ) → K h i,j ( L ; F 2 ) agr e es with the se c ond Ste enr o d squar e on the Khovanov stable homotopy typ e X ( L ) . A STEENROD SQUARE ON KHOV ANO V HOMOLOGY AND A CUP-I PRODUCT 3 Mor´ an deﬁnes the maps sq i using a combinatorially deﬁned family of op erations ⌣ i : C p ( X • ; F 2 ) ⊗ C q ( X • ; F 2 ) → C p + q − i ( X • ; F 2 ) , i ∈ Z , analogous to the cup-i products deﬁned in [ Ste47 , MM23 ]. Con tin uing the analogy , he deﬁnes sq i b y sq i ([ α ]) = [ α ⌣ n − i α ] for α any n -co cycle. With this construction in mind, we view sq 2 , for the rest of this pap er, as an op eration from co cycles to co cycles, stating, b y abuse of notation, sq 2 ( α ) = α ⌣ n − i α . On the other hand, Lipshitz-Sark ar’s [ LS14a ] deriv ation of sq 2 ( α ) comes from a more gometric strategy , using Brown represen tabilit y to compute sq 2 ( α ) as a pullbac k of a co cycle in the Eilenberg-Maclane space K ( Z / 2 , n + 2). 1.3. Outline of the pro of. Section 2 will cov er necessary bac kground and deﬁnitions of b oth form ulas sq 2 , sq 2 (whic h b oth dep end on a few choices). In Section 3 we will simplify Moran’s sq 2 form ula as m uch as we can. Section 4 will then simplify the diﬀerence of the t w o formulas sq 2 ( α ) − sq 2 ( α ), ev en tually proving it is a cob oundary . W e store many of our preliminary lemmas in App endix A and B , in order to highlight the main argumen t in Sections 3 and 4 . 2. Ba ckgr ound and definitions 2.1. The cub e and the Burnside category . In this section, w e will giv e the necessary setup for the deﬁnitions of Steenro d squares in b oth Mor´ an’s work [ CM20 ], and the w ork of [ LS14c , Ra j25 ]. The original construction of Khov ano v homology [ Kho00 ] (see also [ BN02 ]), starts with an N -dimensional cub e, where each v ertex corresp onds to a resolution of an N -crossing link diagram L . Over each v ertex lie generators of the Khov ano v c hain complex K c ( L ), corresp onding to lab elings of circles, and a diﬀerential component on K c ( L ) is deter- mined b y an arro w trav ersing down an edge b et ween t wo generators. Lawson-Lipsih tz-Sark ar [ LLS17 , LLS20 ] pack age this data into a (strictly unitary , lax) 2-functor F K h ( L ) : 2 N → B from the cube category to the Burnside category . Applying the totalization functor and shifting the gradings do wn b y N − , they again obtain the Khov anov chain complex K c ( L ). The main adv antage of F K h ( L ) is that it deﬁnes a symmetric sp ectrum | F K h ( L ) | for whic h the stable homotopy t yp e of Σ − n − | F K h ( L ) | is a link inv ariant. W e deﬁne 2 N to hav e ob jects the subsets A of { 1 , . . . , N } , and morphisms the inclusions of subsets. F or a deﬁnition of the Burnside category B and a 2-functor F , w e refer the reader to [ LLS20 , CM20 ]. 2.2. Augmen ted semi-simplicial sets. In [ CM20 ], Mor´ an pack ages the data of a 2-functor F : 2 N → B into an augmente d semi-simplicial obje ct in the Burnside category B . Deﬁnition 2.1. Let ∆ inj ∗ denote the category of p ossibly empt y ﬁnite ordinals [ n ], with only injectiv e order-preserving maps as morphisms. An augmente d semi-simplicial obje ct X in a (weak) 2-category C is a (strictly unitary , lax) 2-functor from ∆ inj ∗ to C . As is customary , we write X = X • , X n = X ([ n ]), and ∂ n U = X ( δ n U ) for U ⊆ [ n ]. In practice, C will b e the Burnside category B , so face maps δ n i : [ n − 1] → [ n ] corresp ond with spans ∂ n i : X n → X n − 1 . W e state some notation that will help us deﬁne functors in B ∆ inj ∗ . Deﬁnition 2.2. Let A = ( a 0 , . . . , a n ) b e a subset of { 1 , . . . , N } , and let U ⊆ [ n ] b e ordered as ( u 1 , . . . , u k ). W e deﬁne A ( U ) = ( a u 1 , . . . , a u k ). 4 AD VIKA RAJAP AKSE W e can view a 2-functor F : (2 N ) op → B as an augmen ted semi-simplicial ob ject. Indeed, consider the map Λ : B (2 N ) op → B ∆ inj ∗ (whic h is, in fact, functorial), that takes F to the left Kan extension of F along the functor (2 N ) op → ∆ inj ∗ , A 7→ [ | A | − 1]. In other w ords, Λ tak es F to an augmen ted semi-simplicial ob ject X • = Λ( F ) deﬁned by X n = a | A | = n +1 F ( A ) , n ≥ − 1 ∂ n U = a | A | = n +1 F ( A \ A ( U ) ⊆ A ) , U ⊆ [ n ] X ( ∂ n −| V | W , ∂ n V ) = a | A | = n +1 F ( C ⊂ B , B ⊂ A ) V ⊂ [ n ] , W ⊂ [ n − | V | ] where B = A \ A ( V ), C = B \ B ( V ). Notation 2.3. Let ( p, q ) ∈ ∂ n −| V | W × X n −| V | ∂ n V . If X ( ∂ n −| V | W , ∂ n V ) takes ( p, q ) to s , then we say , b y abuse of notation, s = p ◦ q . 2.3. Mor´ an’s cup-i pro duct. Let C ∗ denote the c hain complex C ∗ ( X • ; F 2 ), deﬁned in [ CM20 ]. Mor´ an deﬁnes sq i ([ α ]) = [ α ⌣ n − i α ], where ⌣ i : C ∗ ⊗ C ∗ → C ∗ shifts grading down b y i and is deﬁned as the dual to a homomorphism ∇ i : C ∗ → C ∗ ⊗ C ∗ . Mor´ an’s deﬁnition of ∇ i is analogous to deﬁnitions in [ Ste47 , MM23 ], where they deﬁne a similar operation on the level of simplicial sets. Let P q ( n ) denote the set of q -tuples of in tegers 0 ≤ a 1 ≤ a 2 ≤ . . . ≤ a q ≤ n where every num ber app ears at most twice (think of a staircase with unevenly spaced steps), and let U + , U − ⊆ U b e the subsets of U of p ositive index elements (see [ CM20 ] for deﬁnitions). Mor´ an deﬁnes a copro duct of spans (1) ∇ ( n q ) = a U ∈P q ( n ) ∂ U − ∧ ∂ U + , where ∂ U − ∧ ∂ U + are spans of the form X n × X n ← − ∂ U − ∧ ∂ U + − → X n −| U − | × X n −| U + | . Finally , he deﬁnes ∇ i := X n A F 2  ∇ ( n n − i ) ◦ ∆ n  , where ∆ n : X n → X n × X n is the diagonal span and A F 2 denotes the F 2 -linearization functor (see [ CM20 ]). Th us, for α an n -co cycle and z ∈ X n +2 view ed as a co chain, we un wind deﬁnitions to compute ⟨ sq 2 ( α ) , z ⟩ = ⟨ α ⌣ n − 2 , z ⟩ = D α ⊗ α, A F 2  ∇ ( n +2 4 ) ◦ ∆ n +2  ( z ) E = D α ⊗ α, A F 2  ∇ ( n +2 4 )  ( z ⊗ z ) E , A STEENROD SQUARE ON KHOV ANO V HOMOLOGY AND A CUP-I PRODUCT 5 Finally , w e observ e that α ⊗ α ∈ C n ( X • ; F 2 ) ⊗ C n ( X • ; F 2 ), whic h implies that we can replace ∇ ( n +2 4 ) with the smaller span ∇ ( n +2 4 ) [2 , 2] = a U ∈P 4 ( n +2) | U − | = | U + | =2 ∂ U − ∧ ∂ U + , and write ⟨ sq 2 ( α ) , z ⟩ =  α ⊗ α, X U ∈P 4 ( n +2) | U − | = | U + | =2 A F 2 ( ∂ U − ∧ ∂ U + )( z ⊗ z )  . 2.4. Bac kground for the Sq 2 form ula. W e use a con v enien t ob ject to deﬁne sq 2 ( α ) called a cubic al sp e cial gr aph structur e Γ( z , α ), used in [ Ra j25 ]. These ob jects are based of the construction of a sp e cial gr aph structur es , ﬁrst deﬁned in [ LOS20 ] to give a general Steenro d square formula for framed ﬂow categories. W e recall a com binatorial formula of the second Steenro d square deﬁned in [ Ra j25 ], which is equiv alent to the original combinatorial form ula [ LS14c ]. But ﬁrst, w e introduce some notation Deﬁnition 2.4. Given a span X m ← − ∂ m U − → X l , ( l = m − | U | ) and subsets S ⊂ X m , T ⊂ X l , we deﬁne S ← − ∂ U ( S ,T ) − → T to b e the span consisting of elements s ∈ ∂ n U with source in S , and target in T . In the case where S is a singleton set { z } , w e write ∂ U ( { z } , T ) as ∂ U ( z , T ). In practice, w e often view a co cycle α ∈ C l ( X • ; F 2 ) as a subset α ⊂ X l , and we use the term ∂ U ( z , α ). Finally , we deﬁne m U ( S, T ) = # ∂ U ( S, T ). W riting { a, b } = ab , w e ha v e m ab ( z ,α ) = #( ∂ ab ( z , α )). Theorem 2.5 ([ LS14c ],[ Ra j25 ]) . L et α ∈ C ∗ (T ot F ; F 2 ) b e a c o cycle. A ny fac ewise b oundary matching m for α determines a unique c o cycle sq 2 ( α ) = sq 2 m ( α ) , deﬁne d by ⟨ sq 2 ( α ) , z ⟩ := X a b } + # { a → ← − b → c | a < b }  . Given the c anonic al identiﬁc ation of e H ∗ ( | F | ; F 2 ) with H ∗ (T ot F ; F 2 ) , we have Sq 2 ([ α ]) = [sq 2 ( α )] . A facewise b oundary matc hing is the following data: F or y ∈ X n +1 , w e group ∂ ( y , α ), in to ordered pairs of the form ( s, t ) such that if ( s, t ) ∈ ∂ a × ∂ b is an ordered pair, then a ≤ b . W e ﬁx a facewise b oundary matc hing m for α using the following conv en tion: Con v en tion 2.6. F or eac h y , w e view ∂ ( y , α ) as a subset of ∂ n +1 to obtain an induced or- dering s 1 < s 2 < . . . s r . W e deﬁne m to ha ve the ordered pairs ( s 1 , s 2 ) , ( s 2 , s 3 ) , . . . , ( s r − 1 , s r ). 6 AD VIKA RAJAP AKSE 2.5. Fixing an ordering and a b oundary matc hing. The diﬃculty in proving the iden- tit y sq 2 = Sq 2 is that b oth formulas for sq 2 and Sq 2 are deﬁned on co cycles α by computing a representativ e ( α ⌣ n − 2 α and sq 2 ( α ) resp ectively), whic h creates some am biguity . On one hand, α ⌣ n − 2 α dep ends on an ordering of all spans ∂ n U , and on the other hand, sq 2 ( α ) dep ends on a facewise b oundary matc hing for α . W e ﬁx once and for all an ordering of spans ∂ n U and a facewise b oundary matc hing for α . Deﬁnition 2.7. F or the rest of this pap er, w e ﬁx orderings on our spans ∂ n U . W e start b y ordering spans of the form ∂ n a arbitrarily . No w we order spans of the form ∂ n ab . F or a span ∂ n ab , we ﬁx an ordering once and for all called the left-br e ak ordering. The ordering is deﬁned as follows: (1) Given a < b and t w o span elements s, s ′ ∈ ∂ n +2 ab , w e write s = p ◦ q , where q ∈ ∂ n +2 a , p ∈ ∂ n +1 b − 1 , and we write s ′ = p ′ ◦ q ′ , where q ′ ∈ ∂ n +2 a , p ′ ∈ ∂ n +1 b − 1 . (2) W e use the lexicographic order on ∂ n +2 a × ∂ n +1 b − 1 , and deﬁne s < s ′ if and only if ( q , p ) < ( q ′ , p ′ ) with resp ect to this lexicographic ordering. Generalizing the ordering on eac h ∂ n a , we adopt an ordering on ∂ n := F a ∂ n a to b e the unique ordering that extends the ordering on each ∂ n a suc h that ∂ n 0 < ∂ n 1 < . . . . In other words, we declare s < t if s ∈ ∂ n a , t ∈ ∂ n b , a < b . W e deﬁne the right-br e ak ordering on ∂ n +2 ab similarly , but w e write s = p ′ ◦ q ′ for q ′ ∈ ∂ n +2 b , p ′ ∈ ∂ n +1 a and ordering lexicographically . R emark 2.7.1 . The term left-br e ak ordering comes from viewing a span element s ∈ ∂ n ab as an in terv al I in the mo duli space f − 1 M ( w , u ) (see [ LS14a , LLS20 ] for equiv alen t deﬁnitions). W riting s = p ◦ q corresp onds to ﬁnding the “leftmost” end of I . 3. Simplifying the expression ⟨ sq 2 ( α ) , z ⟩ Let α b e an n -co cycle. In this section, we try to simplify ⟨ sq 2 ( α ) , z ⟩ as m uc h as p ossible b efore comparing it with ⟨ sq 2 ( α ) , z ⟩ . Recall from Section 2.3 our formula ⟨ sq 2 ( α ) , z ⟩ =  α ⊗ α, X U ∈P 4 ( n +2) | U − | = | U + | =2 A F 2 ( ∂ U − ∧ ∂ U + )( z ⊗ z )  . Mor´ an’s construction of ∂ U − ∧ ∂ U + is slightly diﬀerent based on whether n is even or odd. F or ease of reading, w e follow the computation in the case that n is ev en. The computation for n o dd follows similarly and results in an almost iden tical formula. W e no w expand the in terior term (2) X U ∈P 4 ( n +2) | U − | = | U + | =2 A F 2 ( ∂ U − ∧ ∂ U + ) , and in the following equalities, w e drop the “ A F 2 ” term to lighten the notation, lea ving it implicit. The term ( 2 ) can b e written as X U ∈P 0 4 ( n +2) | U + | = | U − | =2 ∂ U − ∧ ∂ U + + X U ∈P 1 4 ( n +2) | U + | = | U − | =2 ∂ U − ∧ ∂ U + + X U ∈P 2 4 ( n +2) | U + | = | U − | =2 ∂ U − ∧ ∂ U + A STEENROD SQUARE ON KHOV ANO V HOMOLOGY AND A CUP-I PRODUCT 7 = X a b 8 AD VIKA RAJAP AKSE W e only use the letters a, b, c, d in this notation throughout the pap er. The conv enience of this deﬁnition is seen in by the isomorphism of spans ∂ n − 1 a b ◦ ∂ n b ∼ = ∂ n ab ∼ = ∂ n − 1 b a ◦ ∂ n a . No w w e can rewrite I I as I I = X c  X a,b  = c ab a, b o dd m a c ( s out ) m b c ( t out ) ! = X s,t ∈ ∂ c sb a, b o dd m a ( s out ) m b ( t out ) ! . Referring directly to the deﬁnition of ∂ ab ∧ ∂ ab in [ CM20 ], we ﬁnd (3) I I I := X a t (see [ CM20 ]). Therefore, it suﬃces to prov e that for any s, t suc h that s < t , the num ber of p ositiv e maximal chains of ( s, t )-go o d pairs is 1 (mo d 2) if and only if the left-break order of { s, t } disagrees with the righ t-break order. By our ordering con v ention, the latter statemen t is equiv alent to the the right-break order b eing ( t, s ). Our lemma is thus a consequence of the following statement: # { p ositiv e maximal chains of ( s, t )-go o d pairs } ≡  1 if right-break order is ( t, s ) 0 if right-break order is ( s, t ) (mo d 2) . A STEENROD SQUARE ON KHOV ANO V HOMOLOGY AND A CUP-I PRODUCT 9 ∂ n +2 a ∂ n +2 b ∂ n +1 a ∂ n +1 b − 1 s t X n +2 X n +1 X n +1 X n q s q t q ′ t q ′ s p ′ t p ′ s p s p t X n +2 X n +1 X n +1 X n Figure 1. Left: the elements s, t in the span ∂ ab = ∂ ab ( z ) The thick arro ws denote the spans ∂ n +2 a , ∂ n +2 b , ∂ n +1 b − 1 , ∂ n +1 a , . Right: W e write b oth s and t as comp ositions in ∂ b − 1 ◦ ∂ a (comp osing along the left side of the diagram) or in ∂ a ◦ ∂ b (comp osing along the righ t side of the diagram). W rite s = p s ◦ q s ∈ ∂ b − 1 ◦ ∂ a and s = p ′ s ◦ q ′ s ∈ ∂ a ◦ ∂ b . Similarly , we write t = p t ◦ q t ∈ ∂ b − 1 ◦ ∂ a and t = p ′ t ◦ q ′ t ∈ ∂ a ◦ ∂ b (see Figure 1 for a diagram). Now w e measure the num b er of p ositiv e maximal c hains of ( s, t )-go od pairs. This measurement dep ends on whether q s = q t , and q ′ s = q ′ t . There are 4 total cases, but w e will only consider tw o, as the rest are a similar v eriﬁcation. W e ﬁrst introduce notation that will help our measurement. First consider the case where none of the equalities hold. Then the only p ossible p ositiv e maximal chains are ( ∅ , { a } ) ≺ ( ∅ , { a, b } ), and ( ∅ , { b } ) ≺ ( ∅ , { a, b } ). The ﬁrst chain is p ositiv e if and only if (w e are using the fact that n is ev en here), q s < q t . The second chain is positive if and only if q ′ s < q ′ t . But note s < t implies q s < q t b y Deﬁnition 2.7 . Therefore, the num b er of p ositive maximal chains of ( s, t )-go o d pairs is 1 + 1 | q ′ s b } + # { a → ← − b → c | a < b } ) + X s,t ∈ ∂ c s b } + # { a → ← − b → c | a < b } ) mo d cob oundary . No w recall from Lemma 3.5 that I I I = X ab m a ( s out ) m b ( t out ) + X ab ( ab ) m a ( s out ) m b ( t out ) = X a,b  = c ab ( a c b c ) m a c ( s out ) m b c ( t out ) 14 AD VIKA RAJAP AKSE = X a,b  = c ab ( a + 1 | cb ( ab + b | c b ( ab + b | c b ( b | c b ( b | c b ( b | c b ( a | c a + b | c>b + 1 | c a + b | c>b ) m a c ( s out ) m b c ( s out ) + X c, s,t ∈ ∂ c ( z ) sa + b | c>b b y adding the (identically zero) term in Equation ( 11 ). F rom Lemma A.3 , the top t w o sums add to 0 mo d cob oundary . So w e are left with the b ottom tw o sums, which equal X c, s,t ∈ ∂ c ( z ) sc s ∈ ∂ a c ( t out ,α ) ← − − − order( s ) , simply from the fact that  k 2  = 0 + 1 + 2 + . . . + ( k − 1). W e are left with ⟨ sq 2 ( α ) − sq 2 ( α ) , z ⟩ = X c, t ∈ ∂ c X a  = c, s ∈ ∂ a c ( t out ,α ) ← − − − order( s ) · a + X c, t ∈ ∂ c X c 0 } . W e call a semicircle in H centered on the real axis a chor d . In other w ords, c hords are precisely the lines that do not reach inﬁnit y when viewing H as P oincar ´ e’s half-plane mo del. Deﬁnition B.1. Given an augmented semi-simplicial ob ject X • , an elemen t z ∈ X n +2 , and a set of elemen ts S ⊂ X n , we deﬁne the de gener ate chor d pr esentation C ( z , S ) as the following set of chords C em b edded in H : W e view each s ∈ ∂ ab ( z , S ) as a copy of the chord C s ha ving ends on a and b . W e no w deﬁne the (nonde gener ate) chor d pr esentation C ( z , S ) as a similar presen tation of c hords, but w e p erturb the endp oin ts so no tw o endp oints meet at the same p oint (although their index stays the same. (See Figure 3 ). W e c ho ose a sp eciﬁc p erturbation, that relies A STEENROD SQUARE ON KHOV ANO V HOMOLOGY AND A CUP-I PRODUCT 21 on a lexicographic order on a pro duct of spans. F or eac h index a , w e p erturb the c hords meeting a by the following pro cedure: (P-1) W e write the chords that intersect a as the union (12) G s ∈ ∂ a ( z ) ∂ ( s out , α ) × { s } ⊂ ∂ n +1 × X n +1 ∂ n +2 . (P-2) Order the ab ov e union according to the lexicographic order of ∂ n +2 × ∂ n +1 , where the ordering of ∂ n +2 , ∂ n +1 is b y Deﬁnition 2.7 (notice the switch of factors here). In particular, we ha ve (13) ∂ ( s out , α ) × { s } < ∂ ( t out , α ) × { t } if s < t. (P-3) Perturb the chords meeting a according to this ordering. That is, higher ordered c hords should b e to the right of low er ordered c hords. The disjoin t union in ( 12 ) can b e view ed as a partition on the set of c hord ends on a . And b y ( 13 ), eac h partition o ccupies its own (small) arc in S 1 . W e sa y that c hords in ∂ ( s out , α ) ◦ { s } the me et s . R emark B.1.1 . By picking a biholomorphism H → D , w e can view degenerate and nonde- generate chord presen tations as lying inside D . W e now pro v e some (progressively more c hallenging) lemmas, whic h will help us gain familiarit y with chord presen tations. Lemma B.2. L et α b e a c o cycle in C n ( X • ; F 2 ) , and let z ∈ X n +2 . Both C ( z , α ) and C ( z , α ) have an even numb er of chor ds c oming out of e ach index a . F urthermor e, b oth chor d pr esen- tations have an even numb er of chor ds me eting s ∈ ∂ a . W e remark that the pro of of this lemma is iden tical to the pro of of Lemma A.1 , but stated in the language of chord presen tations. Pr o of. Fix an s ∈ ∂ a . The n um b er of c hords meeting s is equal to # ∂ ( s out , α ), whic h is 0 mo d 2 by virtue of α b eing a co cycle. Adding up ov er all s ∈ ∂ a ( z ), w e obtain an even n um b er of chords coming out of a . □ Lemma B.3. We have (14) X a b } + # { a → ← − b → c | a < b }  mo d cob oundary . Pr o of. Adding X s ∈ ∂ c X c b } + # { a → ← − b → c | a < b } for any cycle K ⊂ Γ( z , α ) e 1 e ′ 1 e 2 e ′ 2 . . . e ′ 3 e k e ′ k e 1 , with facet cycle orien ted as a 1 → a 2 → . . . → a m → a 1 . No w ﬁx a cycle K and view the c hords of C ( K ) in the upp er half-plane (see Figure 6 ). By reparametrizing our cycle if necesssary , w e can assume that the left end of C ( e ′ 1 ) lies on a 1 and is the leftmost end of an y c hord in C ( K ). Since # { a → − → b → c | a > b } + # { a → ← − b → c | a < b } = # { a → − → b → c | c < b } + # { a → ← − b → c | c > b } (mo d 2) , it suﬃces to pro ve (17) #(crossings in C ( K )) = 1 + # { a → − → b → c | c < b } + # { a → ← − b → c | c > b } . 24 AD VIKA RAJAP AKSE e ′ 2 e ′ 1 e ′ 3 a 1 a 2 a 3 a 4 Figure 6. W e coun t the quan tit y X ( k ) in ( 18 ) for k = 4. If e ′ 4 w as the dashed c hord, then X (4) = 2 + 0 = 0 (mo d 2), since a k +1 > a k but a k − 1 → − → a k → a k +1 . If e ′ 4 w as the dotted chord, then X (4) = 1 + 1 = 0 (mo d 2). The dot highlights the a k endp oin t of e ′ k − 1 . T o prov e this iden tit y , consider the quan tit y (18) X ( k ) = # {C ( e ′ k ) ∩ C ( e ′ j ) : 1 ≤ j < k } +      1 if a k − 1 → − → a k → a k +1 and a k +1 < a k 1 if a k − 1 → ← − a k → a k +1 and a k +1 > a k 0 otherwise, where in the second term, we deﬁne a 0 := a m , a m +1 := a 1 . By summing these terms from 1 to m , we obtain all the terms in ( 17 ) except the term that is 1. F or all k where 1 < k < m , the quantit y X ( k ) = 0. Indeed, if either of the ﬁrst t w o conditions in Equation ( 18 ) hold, then the chord C ( e ′ k ) must ha v e endp oints to the right and left of the a k -endp oin t of C ( e ′ k − 1 ). Therefore, the in tersection num b er of C ( e ′ k ) with the chords C ( e ′ 1 ) ∪ . . . ∪ C ( e ′ k − 1 ) is 1. It remains to show that X (1) + X ( m ) = 1. T o this end, observe that X (1) = 0 + 1 = 1, b ecause a m → ← − a 1 → a 2 and a 2 > a 1 . No w counting the in tersection num ber of C ( e ′ m ) with C ( e ′ 1 ) ∪ . . . ∪ C ( e ′ m − 1 ), we determine # {C ( e ′ m ) ∩ C ( e ′ j ) : 1 ≤ j < m } =  1 if a m − 2 → − − → a m − 1 → a m 0 otherwise. This is equal to the second summand of ( 18 ), as a m +1 = a 1 < a m . Therefore, X ( m ) = 0. □ Before the next prop osition, we giv e a necessary deﬁnition. Deﬁnition B.5. Let y ∈ X n +1 . W e deﬁne a total ordering < on the set ∂ ( y , α ) to b e the order induced by our total ordering on ∂ n +1 . Then w e deﬁne order( s ) = # { t ∈ ∂ ( y , α ) : t < s } , ← − − − order( s ) = # { t ∈ ∂ ( y , α ) : t > s } . Prop osition B.6. We have X a

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