aa r X i v : . [ m a t h . A T ] A ug SZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE
MATTHIAS FRANZ
Abstract.
We prove that Szczarba’s twisting cochain is comultiplicative. Inparticular, the induced map from the cobar construction Ω C ( X ) of the chainson a 1-reduced simplicial set X to C (Ω X ), the chains on the Kan loop groupof X , is a quasi-isomorphism of dg bialgebras. We also show that Szczarba’stwisted shuffle map is a dgc map connecting a twisted Cartesian product withthe associated twisted tensor product. We apply our results to finite coveringspaces and to the Serre spectral sequence. Introduction
Let X be a simplicial set and G a simplicial group. Given a twisting function(1.1) τ : X > → G, Szczarba [21] has constructed an explicit twisting cochain(1.2) t : C ( X ) → C ( G ) . In [9, Thm. 6.2] we showed that it agrees with the twisting cochain obtained byShih [20, §II.1] using homological perturbation theory if one uses a slightly modifiedversion of the Eilenberg–Mac Lane homotopy.In this note we consider the associated map of differential graded algebras (dgas)(1.3) Ω C ( X ) → C ( G )where Ω C ( X ) is the reduced cobar construction of the differential graded coalgebra(dgc) C ( X ).The diagonal of C ( G ) is compatible with the multiplication, meaning that C ( G )is actually a dg bialgebra. By work of Baues [1] and Gerstenhaber–Voronov [10,Cor. 6], the same holds true for Ω C ( X ). Here the diagonal can be expressed via thehomotopy Gerstenhaber structure of C ( X ), that is, in terms of certain cooperations(1.4) E k : C ( X ) → C ( X ) ⊗ k ⊗ C ( X )with k ≥
0, see Section 3.The question arises as to whether the dga map (1.3) is comultiplicative, meaningcompatible with the diagonals. Hess–Parent–Scott–Tonks [13, Thm. 4.4] showedthat for 1-reduced X this is true up to homotopy in a strong sense, and theyobserved that it holds on the nose up to degree 3. In the case where X is a sim-plicial suspension the comultiplicativity was established by Hess–Parent–Scott [12,Thm. 4.11]. Our main result says that it is true in general. Theorem 1.1.
Let X be any simplicial set, and let G and τ be as above. The dgamap Ω C ( X ) → C ( G ) induced by Szczarba’s twisting cochain t is comultiplicative. Mathematics Subject Classification.
Primary 55U10; secondary 55R20, 55T10.The author was supported by an NSERC Discovery Grant.
This applies in particular to the canonical twisting cochain τ X : X > → Ω X ofa 1-reduced simplicial set where Ω X denotes the Kan loop group of X . This givesthe following. Corollary 1.2.
For -reduced X , the map Ω C ( X ) → C (Ω X ) induced by Szczarba’stwisting cochain t is a quasi-isomorphism of dg bialgebras. Using Hess–Tonks’ extended cobar construction, we generalize this to reducedsimplicial sets in Proposition 7.1.Given a left G -space F , one can consider the twisted tensor product(1.5) C ( X ) ⊗ t C ( F ) . Dualizing a construction due to Kadeishvili–Saneblidze [16], we turn the chaincomplex (1.5) into a dgc. Szczarba also defined a twisted shuffle map(1.6) ψ : C ( X ) ⊗ t C ( F ) → C ( X × τ F )and proved that it is a quasi-isomorphism of complexes [21, Thm. 2.4]. In [9,Prop. 7.1] we showed that ψ is in fact a morphism of left C ( X )-comodules, andalso of right C ( G )-modules in the case F = G . We strengthen the first aspect asfollows. Theorem 1.3.
Szczarba’s twisted shuffle map ψ is a morphism of dgcs. Content and structure of this paper are as follows: We review background ma-terial in Section 2 and homotopy Gerstenhaber coalgebras in Section 3. Afterestablishing a purely combinatorial result in Section 4 and discussing the Szczarbamaps in Section 5 we prove Theorem 1.1 in Section 6. The generalization of Corol-lary 1.2 mentioned above appears in Section 7. In Section 8 we explain how ho-motopy Gerstenhaber coalgebra structures give rise to dgc structures on twistedtensor products, and in Section 9 we prove Theorem 1.3. In Section 10 we compareSzczarba’s twisted tensor product with a similar one due to Shih [20]. We applyour results to finite covering spaces in Section 11 and to the (co)multiplicativestructure of the (co)homological Serre spectral sequence in Section 12. In the firstappendix we relate our diagonal on Ω C ( X ) to the one defined by Baues [1] for 1-reduced X . In the second we fill a gap in the literature by showing that Szczarba’stwisting cochain (1.2) and his twisted shuffle map (1.6) are actually well-defined onnormalized chain complexes. 2. Preliminaries
Generalities.
We write [ n ] = { , . . . , n } and n = { , . . . , n } for n ≥
0. Wework over a commutative ring k with unit; all tensor products and chain complexesare over k . Unless specified otherwise, all chain complexes are homological. Thedegree of an element c of a graded module C is denoted by | c | . We write 1 C for theidentity map of C and(2.1) T B,C : B ⊗ C → C ⊗ B, b ⊗ c ( − | b || c | c ⊗ b for the transposition of factors in a tensor product of graded modules. The sus-pension and desuspension operators are denoted by s and s − , respectively. Wesystematically use the Koszul sign rule, compare [8, Secs. 2.2 & 2.3].For clarity, we sometimes write 1 A for the unit of a dga A and 1 C for the unit ofa coaugmented dgc C . is a chain complex A that is both a dga and a dgc in such ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 3 a way that each pair of structure maps are morphisms with respect to the otherstructure.We write C ( X ) for the normalized chains on a simplicial set X . We also write˜ ∂ for the last face map, that is, ˜ ∂x = ∂ n x for x ∈ X n with n ≥ The cobar construction.
Let C be a dgc with coaugmentation ι : k ֒ → C ,so that C = k ⊕ ¯ C where ¯ C = ker ε . The (reduced) cobar construction of C is(2.2) Ω C = M k ≥ Ω k C where Ω k C = ( s − ¯ C ) ⊗ k , compare [15, Sec. II.3] or [1, §0]. We write elements of Ω C in the form(2.3) h c | . . . | c k i = s − c ⊗ · · · ⊗ s − c k with c , . . . , c k ∈ ¯ C . The cobar construction is an augmented dga with concate-nation as product and unit 1 = hi ∈ Ω C = k . The differential and augmentationare determined by(2.4) d h c i = −h dc i + ( s − ⊗ s − ) ¯∆ c and ε ( h c i ) = 0for h c i ∈ Ω C , where(2.5) ¯∆ c = ∆ c − c ⊗ − ⊗ c ∈ ¯ C ⊗ ¯ C is the reduced diagonal.2.3. Twisting cochains.
Let C be a coaugmented dgc and A an augmented dga.Recall that the complex Hom( C, A ) is an augmented dga via d ( f ) = d A f − ( − | f | f d C , Hom(
C,A ) = ι A ε C (2.6) f ∪ g = µ A ( f ⊗ g ) ∆ C , ε ( f ) = ε A f ι C (1)(2.7)for f , g ∈ Hom(
C, A ). Here ι A : k → A is the unit map, ι C is the coaugmentationof C , and ε C and ε A are the augmentations of C and A , respectively.A twisting cochain is a map t ∈ Hom(
C, A ) of degree − t ι C = 0 , ε A t = 0 , d ( t ) = t ∪ t. It canonically induces the morphism of dgas(2.9) Ω C → A, h c | . . . | c k i 7→ t ( c ) · · · t ( c k ) . For example, the canonical twisting cochain(2.10) t C : C → Ω C, c
7→ h c i ∈ Ω C corresponds to the identity map on Ω C .2.4. The shuffle map.
We recall the definition of the shuffle map for an arbitrarynumber of factors. Given k ≥ q , . . . , q k with sum q , a( q , . . . , q k )-shuffle is a partition α = ( α , . . . , α k ) of the set [ q − − ( α ) = ( − ( α ,...,α k ) for its signature and Shuff( q , . . . , q k ) for the set of allsuch shuffles. Observe that for k = 1 there is only one ( q )-shuffle.For simplicial sets X , . . . , X k the shuffle map is given by ∇ X ,...,X k : C q ( X ) ⊗ · · · ⊗ C q k ( X k ) → C q ( X × · · · × X k ) , (2.11) x ⊗ · · · ⊗ x k X α ( − ( α ) ( s ¯ α x , . . . , s ¯ α k x k ) MATTHIAS FRANZ where the sum is over all α ∈ Shuff( q , . . . , q k ), and ¯ α s = [ q − r α s for 1 ≤ s ≤ k .Using the shuffle map, one turns the chain complex of a simplicial group G intoa dga. For m ≥
0, the m -fold iterated multiplication is given by(2.12) C ( G ) ⊗ m ∇ G,...,G −−−−−→ C ( G × · · · × G ) µ [ m ] ∗ −−−→ C ( G )where µ [ m ] : G × · · · × G → G is the m -fold product map. This gives the identitymap of C ( G ) for m = 1 and the unit map k ֒ → C ( G ) for m = 0.2.5. Twisted Cartesian products.
Twisted Cartesian products are simplicialversions of fibre bundles, compare [17, Sec. 18] or [21, Sec. 1]. More precisely, let X and F be simplicial sets, and assume that the simplicial group G acts on F fromthe left. The twisted Cartesian product X × τ F differs from the usual Cartesianproduct X × F only by the zeroeth face map, which is(2.13) ∂ ( x, y ) = (cid:0) ∂ x, τ ( x ) ∂ y (cid:1) . The twisting function(2.14) τ : X > → G is of degree − x ∈ X of dimension n > ∂ τ ( x ) = τ ( ∂ x ) − τ ( ∂ x ) , (2.15) ∂ k τ ( x ) = τ ( ∂ k +1 x ) for 0 < k < n ,(2.16) s k τ ( x ) = τ ( s k +1 x ) for 0 ≤ k < n ,(2.17)and for any x ∈ X of dimension n ≥ τ ( s x ) = 1 ∈ G n , (2.18)see [21, eq. (1.1)], [17, Def. 18.3] or [14, Sec. 1.3].2.6. Interval cut operations.
Let k , l ≥
0, and let u : k + l → k be a surjectionsuch that u ( i ) = u ( i + 1) for all 0 ≤ i < k + l . Berger–Fresse [2, Sec. 2] haveassociated to u an interval cut operation (2.19) AW u : C ( X ) → C ( X ) ⊗ k , natural in the simplicial set X . On an n -simplex x ∈ X , it is given by(2.20) AW u x = X p ( − pos( p )+perm( p ) x p ⊗ · · · ⊗ x p k . Here the sum runs over all decompositions p = (0 = p , p , . . . , p k + l = n ) of [ n ] into k + l intervals. If we think of these intervals as being labelled via u , then(2.21) x p s = x ( p i − , . . . , p i , p i − , . . . , p i , . . . , p i m − , . . . , p i m )where i , . . . , i m enumerate the intervals with label s . We refer to [2, §2.2.4]for the definitions of the position sign exponent pos( p ) and the permutation signexponent perm( p ).Whenever we talk about the length of an interval [ p i − , . . . , p i ] in this paper, wealways mean its naive length p i − p i − , not the possibly different length definedin [2, §2.2.3] to compute the position and permutation sign exponents. ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 5 Homotopy Gerstenhaber coalgebras
Homotopy Gerstenhaber coalgebras (hgcs) are defined such that their duals arehomotopy Gerstenhaber algebras (hgas), see Remark 3.2 below and also [16, p. 223].More precisely, an hgc is a coaugmented dgc C together with a family of coopera-tions(3.1) E k : C → C ⊗ k ⊗ C for k ≥ E = 1 C , (3.2) im E k ⊂ ( ¯ C ) ⊗ k ⊗ ¯ C for k > , (3.3) E k ( c ) = 0 for | c | < k .(3.4)Recall that ¯ C = ker ε is the augmentation ideal; we also write ¯ c = c − ι ε ( c ) for thecomponent of c in ¯ C . There are further conditions on the maps E k . Defining(3.5) E k : C → ( s − ¯ C ) ⊗ k ⊗ s − ¯ C = Ω k C ⊗ Ω C ⊂ Ω C ⊗ Ω C for k ≥ s ⊗ ( k +1) E k ( c ) = E k (¯ c ) , the assignment E : C → Ω C ⊗ Ω C, (3.7) c
7→ h ¯ c i ⊗ ∞ X k =0 E k (¯ c ) = h ¯ c i ⊗ ⊗ h ¯ c i + ∞ X k =1 E k ( c )is well-defined by (3.4). We require E to be a twisting cochain and the associateddga map(3.8) ∆ : Ω C → Ω C ⊗ Ω C, (cid:10) x (cid:12)(cid:12) · · · (cid:12)(cid:12) x k (cid:11) E ( x ) · · · E ( x k )to be coassociative, so that Ω C becomes a dg bialgebra.It will be convenient to rephrase these conditions in terms of the function E : C → Ω C ⊗ C, (3.9) c (1 ⊗ p C ) E ( c ) + 1 ⊗ ι ε ( c ) = 1 ⊗ c + (1 ⊗ p C ) ∞ X k =1 E k ( c )where(3.10) p C : Ω C −→ Ω C = s − ¯ C s −→ ¯ C ֒ → C is the composition of the canonical projection, the suspension map and the canonicalinclusion. Lemma 3.1.
Let E and E be as in (3.7) and (3.9) . (i) That E is a twisting cochain is equivalent to the two identities d ( E ) = ( µ Ω C ⊗ C )( t C ⊗ E ) ∆ C − ( µ Ω C ⊗ C ) (1 Ω C ⊗ T C, Ω C ) ( E ⊗ t C ) ∆ C , (1 Ω C ⊗ ∆ C ) E = ( µ Ω C ⊗ C ⊗ C ) (1 Ω C ⊗ T C, Ω C ⊗ C ) ( E ⊗ E ) ∆ C . (ii) Assume that E is a twisting cochain. The coassociativity of the diagonal (3.8) then is equivalent to the formula (∆ Ω C ⊗ C ) E = (1 Ω C ⊗ E ) E . MATTHIAS FRANZ
Proof.
For the first part, we note that both sides of the twisting cochain condition d ( E ) = E ∪ E only have components in Ω C ⊗ Ω l C with l ≤
2. We projectonto these components separately. The projections for l = 0 are always equal. Adirect calculation shows that the projections for l = 1 and l = 2 correspond to thetwo identities for E given above. It is helpful to distinguish the two cases c = 1and c ∈ ¯ C , and in the second one to split up the diagonal as ∆ c = c ⊗ ⊗ c + ¯∆ c where ¯∆ is the reduced diagonal (2.5). For the first identity one also uses d ( p C ) = 0.The second claim follows similarly by projecting the coassociativity condition(∆ ⊗
1) ∆ = (1 ⊗ ∆) ∆ to Ω C ⊗ Ω C ⊗ Ω C . (cid:3) Remark 3.2.
Let A = Hom( C, k ) be the augmented dga dual to the coaugmenteddgc C . For k ≥ E k : A ⊗ k ⊗ A → A of the cooperation E k by(3.12) (cid:10) E k ( a , . . . , a k ; b ) , c (cid:11) = ( − k ( | a | + ··· + | a k | + | b | ) (cid:10) a ⊗ · · · ⊗ a k ⊗ b, E k ( c ) (cid:11) for c ∈ C , compare [8, eq. (2.4)]. The operations E k then form an hga structureon A that satisfies the analogues of the identities stated in [7, Sec. 6.1]. Note thatin [7] operations of the form E k ( a ; b , . . . , b k ) are used; see [7, Rem. 6.1] for theirrelation to the braces used by Gerstenhaber–Voronov [10]. The explicit signs giventhere remain unchanged, except for an additional overall minus sign in the formulafor d ( E k ).Let t : C → A be a twisting cochain, where C is an hgc and A a dg bialgebra. Wesay that t is comultiplicative if the induced dga map Ω C → A is a morphism of dgcsand therefore of dg bialgebras. This definition is dual to Kadeishvili–Saneblidze’snotion of a multiplicative twisting cochain [16, Def. 7.2]. For example, the canonicaltwisting cochain t C : C → Ω C is comultiplicative.The normalized chains C ( X ) on a simplicial set X form an hgc in a natural way.In terms of interval cut operations, the structure maps are given by(3.13) E k = ( − k AW e k , that is,(3.14) E k = ( − k ( k − / ( s − ) ⊗ ( k +1) AW e k where(3.15) e k = ( k + 1 , , k + 1 , , k + 1 , . . . , k + 1 , k, k + 1) . The sign difference in (3.14) compared to (3.13) stems from the fact that the (de)sus-pension operators have degree ±
1, so that(3.16) ( s − ) ⊗ ( k +1) s ⊗ ( k +1) = ( − k ( k +1) / because the sign changes each time an s − is moved past an s for a different tensorfactor. Note that AW e is the identity map as required by condition (3.2). A lookat the formula (2.21) moreover shows that the intervals labelled 1, . . . , k in thesurjection must have length at least 1 in order for the last factor x p k +1 of eachterm in the sum (2.20) for AW e k to be non-degenerate, which confirms (3.4) andalso (3.3). ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 7
Explicitly, the induced diagonal on Ω C ( X ) can be written as(3.17) ∆ h x i = E ( x ) = h x i ⊗ n X k =0 X p ( − ε ( p ) (cid:10) x p (cid:12)(cid:12) · · · (cid:12)(cid:12) x p k (cid:11) ⊗ (cid:10) x p k +1 (cid:11) for x ∈ X n , where p runs through the cuts of [ n ] prescribed by e k . The signexponent is given by(3.18) ε ( p ) = k ( k − p ) + pos( p ) + perm( p )where(3.19) des( p ) = k X s =1 ( k + 1 − s ) | x p s | = k X s =1 ( k + 1 − s )( p s − p s − )is the sign exponent incurred by the desuspension operators in (3.7).In Appendix A we show that for 1-reduced X the diagonal (3.17) on Ω C ( X )agrees with those defined by Baues [1] and Hess–Parent–Scott–Tonks [13]. Lemma 3.3.
Let k , n ≥ , and let p = ( p , . . . , p k +1 ) be an interval cut of [ n ] for the surjection e k such that all intervals with label k + 1 have length . Then ε ( p ) ≡ k X s =1 ( s − p s − p s − −
1) (mod 2) . Proof.
Modulo 2, we havepos( p ) = p + p + · · · + p k − , (3.20) perm( p ) = ( p − p ) + 2 · ( p − p ) + · · · + k · ( p k +1 − p k − )(3.21) ≡ p + p + · · · + p k − + n k, des( p ) = k X s =1 ( k + 1 − s )( p s − p s − )(3.22) ≡ n k + k X s =1 ( s − p s − p s − ) ,k ( k − k X s =1 ( s − , (3.23)which gives the desired result. (cid:3) Lemma 3.4.
Let ≤ m ≤ k and n ≥ , and let p : p k +1 p · · · m p m k +1 p m +1 m +1 · · · k p k k +1 p k +1 be an interval cut of [ n ] corresponding to the surjection e k . Assume that the intervalcorresponding to the ( m + 1) -st occurrence of k + 1 (highlighted above) has lengthat least . Let p ′ be the interval cut for e k +1 that is obtained from p by replacingthis interval by · · · m p m k +2 q m +1 q + 1 k +2 p m +1 m +2 · · · for some p m ≤ q < p m +1 . Then ε ( p ′ ) = ε ( p ) . MATTHIAS FRANZ
Proof.
One verifies directly that modulo 2 the exponent for the position sign changesby q , the one for the permutation sign by(3.24) p + · · · + p m + q + m + 1 , the one coming from desuspensions by(3.25) p + · · · + p m + k + m + 1and the one for the explicit sign by k . Hence there is no sign change in total. (cid:3) A bijection
For 0 ≤ l ≤ n we define S n,l = (cid:8) i = ( i , . . . , i l ) ∈ N l (cid:12)(cid:12) ≤ i s ≤ n − s for any 1 ≤ s ≤ l (cid:9) (4.1) = [ n − × [ n − × · · · × [ n − l ]as well as S n = S n,n . The degree of an element i ∈ S n,l is(4.2) | i | = i + · · · + i l . Note that S n, has the empty sequence ∅ as unique element.Let 1 ≤ k ≤ n and p = ( p , . . . , p k ) where 0 = p < p < · · · < p k = n . We set l = n − k and define(4.3) S n − ( p ) = (cid:8) i ∈ S n − ,l (cid:12)(cid:12) ∂ i l +1 · · · ∂ i +1 [ n ] = p (cid:9) . Here [ n ] denotes the standard n -simplex, to which the given face operators areapplied in the specified order. We also set q s = p s − p s − for 1 ≤ s ≤ k .We define a functionΨ p : S n − ( p ) → Shuff( q − , . . . , q k − × S q − × · · · × S q k − (4.4) i (cid:0) α = ( α , . . . , α k ) , j , . . . , j k (cid:1) as follows: Considering the condition (4.3), we think of an element i ∈ S n − ( p ) asdescribing a way of removing the l = n − k elements not appearing in the sequence p from the n -simplex [ n ]. For 1 ≤ s ≤ k the element j s ∈ S q s − similarly recordsthe order in which the elements between p s − and p s are removed by i , ignoring allother removed elements. The shuffle α keeps track of how the element removals ofthe intervals ( p s − , . . . , p s ) are interleaved. More precisely, we declare q − ∈ α s if and only if the face operator ∂ i q +1 in (4.3) removes an element between p s − and p s . Example 4.1.
Take n = 7, k = 3, p = (0 , , ,
7) and i = (5 , , , , , ,
7) are removed in the order 6, 1, 2, 5. Those missing in (0 , ,
7) in the order 6, 5. Wetherefore have j = (0 , j = ∅ and j = (1 ,
0) as well as α = { , } , α = ∅ and α = { , } .Note that for k = 1 the map Ψ p boils down to the identity map on S n − becauseShuff( n −
1) is a singleton. Moreover, for k = 2 we have S n − ,l = S n − ,n − ∼ = S n (since any i ∈ S n − ends in i n − = 0), and the maps Ψ p with 0 < p < n combineto the bijection(4.5) S n − ∼ = [ q + q = n Shuff( q − , q − × S q − × S q − described by Szczarba [21, Lemma 3.3]. ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 9
Proposition 4.2.
The map Ψ p is bijective, and in the notation of (4.4) we have | i | ≡ ( α ) + k X s =1 | j s | + k X s =1 ( s − q s −
1) (mod 2) . Remember from Section 2.4 that given a shuffle α = ( α , . . . , α k ) we write ( α )for the exponent of its signature. For k = 2 the above identity appears alreadyin [21, Lemma 3.3] and [14, Lemma 6]. Proof.
It is clear how to reverse the construction to obtain the inverse of Ψ p .Regarding the claimed formula, we assume first that i is of the form(4.6) i = (cid:0) , . . . , | {z } q − , , . . . , | {z } q − , . . . , k − , . . . , k − | {z } q k − (cid:1) . Then the shuffle α = ( α , . . . , α k ) is the identity map on [ l −
1] and j s = (0 , . . . , s , from which we conclude that the formula holds.Consider two elements from [ l −
1] that are removed one right after the other.Changing the order of the removals changes changes the degree of i by ±
1. If thetwo removed values belong to the same, say the s -th, interval of p , then the degreeof j s also changes by ±
1, and α remains fixed. If the values belong to differentintervals, then all j s remain the same, but the sign of the shuffle changes. Hencein any case the claimed identity is preserved.Starting from (4.6), we can reach any i ∈ S n − ( p ) by repeating this swappingprocedure. This completes the proof. (cid:3) The Szczarba operators
The twisting cochain.
We review the definition of Szczarba’s twisting cochain[21, pp. 200–201] in the formulation given by Hess–Tonks [14, Sec. 1.4]. Let X bea simplicial set and G a simplicial group, and let(5.1) τ : X > → G be a twisting function. It will be convenient in what follows to write σ ( x ) = τ ( x ) − for x ∈ X > .Szczarba [21, Thms. 2.1] has introduced the operatorsSz i : X n → G n − (5.2) x D i , σ ( x ) D i , σ ( ∂ x ) · · · D i ,n − σ (( ∂ ) n − x )for n ≥ i ∈ S n − . In particular, one has Sz ∅ x = σ ( x ). We follows Hess–Tonks [14, Def. 5] in using the symbol Sz i and the name Szczarba operator . Interms of these operators, Szczarba’s twisting cochain t : C ( X ) → C ( G ) is givenfor x ∈ X n by(5.3) t ( x ) = n = 0 , Sz ∅ x − σ ( x ) − n = 1 , P i ∈ S n − ( − | i | Sz i x if n ≥ . Recall that Szczarba writes the signature of the shuffle ( ν, µ ) as sgn( µ, ν ) and also from [14,p. 1866] that his sign exponent ε ( i, n + 1) equals n + | i | . Also note that the subscripts of thedegeneracy operators s µ and s ν in [14] should be swapped. In Appendix B we recall the definition of the simplicial operators D i ,k , and we showthat t is well-defined on normalized chains. Example 5.1.
In low degrees, Szczarba’s twisting cochain looks as follows. Sim-plices are indicated by vertex numbers. For example, a 2-simplex x ∈ X is writtenas 012 and s ∂ x as 122. Note that the products are taken in the simplicialgroup G , not in the dga C ( G ). t (01) = + σ (01) − , i = ∅ (5.4) t (012) = + σ (012) σ (122) , i = (0) (5.5) t (0123) = + σ (0123) σ (1223) σ (2333) i = (0 , (5.6) − σ (0113) σ (1233) σ (2333) , i = (1 , t (01234) = + σ (01234) σ (12234) σ (23334) σ (34444) i = (0 , , (5.7) − σ (01224) σ (12224) σ (23344) σ (34444) i = (0 , , − σ (01134) σ (12334) σ (23334) σ (34444) i = (1 , , + σ (01114) σ (12344) σ (23344) σ (34444) i = (1 , , + σ (01124) σ (12224) σ (23444) σ (34444) i = (2 , , − σ (01114) σ (12244) σ (23444) σ (34444) i = (2 , , We need to understand how the Szczarba operators relate to the bijection Ψ p introduced in Section 4. Let n = k + l with 1 ≤ k ≤ n . We can write any i = ( i , . . . , i n − ) ∈ S n − in the form(5.8) i = ( i , i ) = ( i , , . . . , i ,l , i , , . . . , i ,k − )with i ∈ S n − ( p ) and i ∈ S k − , where(5.9) p = ( p , . . . , p k ) = ∂ i l +1 · · · ∂ i +1 [ n ] . Lemma 5.2.
Using this notation, we have ( ∂ ) l Sz i x = Sz i x ( p , p , . . . , p k ) , ˜ ∂ k − Sz i x = s ¯ α Sz j x ( p , . . . , p ) · · · s ¯ α k Sz j k x ( p k − , . . . , p k ) where Ψ p ( i ) = ( α , j , . . . , j k ) and ¯ α s = [ l − r α s for ≤ s ≤ k .Proof. The case l = 0 of the first identity is void. Given the definition (4.3), itreduces for l = 1 to the formula(5.10) ∂ Sz i x = Sz ( i ,...,i n − ) ∂ i +1 x, which is stated in [14, Lemma 6]. The case l ≥ k = 1, compare the discussion of Ψ p followingExample 4.1. For k = 2 it is again given in [14, Lemma 6]. For larger k it followsby induction:Assume the identity proven for k and l and consider k ′ = k + 1 and l ′ = l − p ′ , i ′ = ( i ′ , i ′ ) and Ψ p ′ ( i ′ ) = ( α ′ , j ′ , . . . , j ′ ). ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 11
Let p = ∂ i ′ , +1 p ′ , and let ˆ p be the removed value. We split i ′ as i ′ = ( i , i )with i = ( i ′ , , . . . , i ′ ,l − , i ′ , ) and i = ( i ′ , , . . . , i ′ ,k − ) and corresponding val-ues α and j , . . . , j k . Then˜ ∂ k Sz i ′ x = ˜ ∂ ˜ ∂ k − Sz i ′ x (5.11) = ˜ ∂ (cid:16) s ¯ α Sz j x ( p , . . . , p ) · · · s ¯ α k Sz j k x ( p k − , . . . , p k ) (cid:17) By the definition of the shuffle α , we have l − ∈ α r if ∂ i l +1 removes an elementbetween p r − and p r . Hence l − / ∈ α s for s = r and therefore= s ¯ α r { l − } Sz j x ( p , . . . , p ) · · · s ¯ α r ˜ ∂ Sz j r x ( p r − , . . . , p r ) · · · s ¯ α k r { l − } Sz j k x ( p k − , . . . , p k ) . Set ˆ q = ˆ p − p r − and ˆ q = p r − ˆ p . Again by the case l = 2 we have(5.12) ˜ ∂ Sz j r x ( p r − , . . . , p r ) = s ¯ β Sz k x ( p r − , . . . , ˆ p ) · s ¯ β Sz k x (ˆ p, . . . , p r )for a (ˆ q − , ˆ q − β , β ) and sequences k ∈ S ˆ q − , k ∈ S ˆ q − . We thusobtain the desired formula since j ′ = j , . . . , j ′ r = k , j ′ r +1 = k , . . . , j ′ k +1 = j k , (5.13) α ′ = ( α , . . . , α r − , γ , γ , α r +1 , . . . , α k )(5.14)where the subsets γ , γ ⊂ [ l −
2] are defined by (cid:3) (5.15) s ¯ γ = s ¯ α r s ¯ β , s ¯ γ = s ¯ α r s ¯ β . The twisted shuffle map.
Let F be a left G -space. We recall the definitionof Szczarba’s twisted shuffle map [21, Thms. 2.3](5.16) ψ = ψ F : C ( X ) ⊗ t C ( F ) → C ( X × τ F )in a notation inspired by Hess–Tonks. For any n ≥ i ∈ S n we define theoperator b Sz i : X n → ( X × τ G ) n = X n × G n , (5.17) x (cid:0) D i , x, D i , σ ( x ) D i , σ ( ∂ x ) · · · D i ,n σ (( ∂ ) n − x ) (cid:1) , which is interpreted as b Sz ∅ x = ( x, ∈ X × G for n = 0 and i = ∅ . Based onthis we define the map(5.18) ψ ( x ⊗ y ) = X i ∈ S n ( − | i | (id X , µ F ) ∗ ∇ (cid:0) b Sz i x ⊗ y (cid:1) , where ∇ : C ( X × τ G ) ⊗ C ( F ) → C ( X × τ G × F ) is the shuffle map and µ F : G × F → F the group action. For a proof that ψ descends to normalized chains see againProposition B.2.Given a decomposition n = k + l with k , l ≥
0, we can write any i ∈ S n in theform i = ( i , i ) with i ∈ S n ( p ) and i ∈ S k , where(5.19) p = (0 = p , p , . . . , p k +1 = n + 1) = ∂ i l +1 · · · ∂ i +1 [ n + 1] . We also write q s = p s − p s − for 1 ≤ s ≤ k + 1. In the definition of ψ in [21, p. 201] the upper summation index should read “ p !”. Lemma 5.3.
In the notation above, we have ( ∂ ) l b Sz i x = b Sz i x ( p − , . . . , p k +1 − , ˜ ∂ k b Sz i x = s ¯ α b Sz j x (0 , . . . , p − · s ¯ α Sz j x ( p − , . . . , p − · · · s ¯ α k +1 Sz j k +1 x ( p k − , . . . , p k +1 − , where Ψ p ( i ) = ( α , j , . . . , j k +1 ) and ¯ α s = [ l − r α s for ≤ s ≤ k + 1 .Proof. Apart from the trivial case l = 0, the first formula follows by induction fromthe case l = 1, that is,(5.20) ∂ b Sz i x = b Sz ( i ,...,i n ) ∂ i x, which can be found in [21, pp. 205–206] as the discussion of the “first term of (4.1)”there.The second formula is also trivial for k = 0, and for k = 1 it is containedin [21, eq. (4.5)]. The extension to larger k follows again by induction, based onthe case k = 2 of the present claim as well as the case k = 2 of Lemma 5.2, usingthe same kind of reasoning as given there. (cid:3) Proof of Theorem 1.1
Let X be a simplicial set, G a simplicial group and τ : X > → G a twistingfunction. Explicitly, the Szczarba map (1.3) is given by(6.1) Sz : Ω C ( X ) → C ( G ) , (cid:10) x | · · · | x m (cid:11) t ( x ) · · · t ( x m )where t : C ( X ) → C ( G ) is Szczarba’s twisting cochain as defined in (5.3). Since weare looking at a multiplicative map between bialgebras, we only have to show(6.2) ∆ C ( G ) Sz h x i = (Sz ⊗ Sz) ∆ Ω C ( X ) h x i for any x ∈ X , say of degree n . If n = 0, then h x i is primitive and annihilatedby Sz, so that (6.2) holds. We therefore assume n ≥ h x i = ∆ t ( x ) = n X k =1 ˜ ∂ k − t ( x ) ⊗ ( ∂ ) l t ( x )(6.3) = n X k =1 X i ∈ S n − ( − | i | ˜ ∂ k − Sz i x ⊗ ( ∂ ) l Sz i x where we have again used the abbreviation l = n − k . Using the explicit for-mula (3.17) for the diagonal, we can write the right-hand side of (6.2) in the form(6.4) (Sz ⊗ Sz) ∆ h x i = t ( x ) ⊗ n X k =0 X p ( − ε ( p ) t ( x p ) · · · t ( x p k ) ⊗ t ( x p k +1 )where p = ( p , p , . . . , p k +1 ) ranges over the cuts of [ n ] into 2 k + 1 intervals cor-responding to the surjection e k . We are going to pair off the summands of theexpressions (6.3) and (6.4). We write q s = p s − p s − for 1 ≤ s ≤ k and ℓ ( p ) forthe sum of the lengths of the intervals in p corresponding to the final value k + 1. ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 13
Assume ℓ ( p ) = 0, so that the k intervals labelled 1, . . . , k cover the wholeinterval [ n ]. From the definition of t we get(6.5) t ( x p k +1 ) = X i ∈ S k − ( − | i | Sz i x p k +1 , and together with that of the shuffle map (2.11) also(6.6) ( − ε ( p ) t ( x p ) · · · t ( x p k ) = X ( − ε ( p )+( α )+ P s | j s | s ¯ α Sz j x p · · · s ¯ α k Sz j k x p k + additional terms with fewer than k factors.Here the sum is over all ( q − , . . . , q k − α = ( α , . . . , α k ) as well asover all j ∈ S q − , . . . , j k ∈ S q k − . The additional terms indicated above arisewhenever we have q s = 1 for some s because of the extra term − ∈ C ( G ) producedby t in the case of a 1-simplex.Consider the case k >
1. According to Lemma 5.2, the expressions ( ∂ ) l t ( x )in (6.3) that give terms of the form (6.5) are indexed by the i = ( i , i ) ∈ S n − with i ∈ S n − ( p ) and i ∈ S k − . By the same lemma, the terms ˜ ∂ k − t ( x )for all such i give exactly the terms in the sum formula of (6.6). Lemma 3.3and Proposition 4.2 show that also the signs work out correctly since | i | = | i | + | i | and(6.7) | i | = ( α ) + k X s =1 | j s | + k X s =1 ( s − q s −
1) = ε ( p ) + ( α ) + k X s =1 | j s | . If k = 1, then x p = x , x p = x (0 , n ) is of degree 1, and ε ( p ) = 0. In addition tothe terms discussed in the preceding paragraph, we get a − − t ( x ) ⊗ p with ℓ ( p ) > k = 0) lead to terms in the sum (6.4) that cancel outwith the additional terms in (6.6) for ℓ ( p ) = 0.Given two decompositions p and p ′ for the surjections e k and e k ′ , we write p ′ ≥ p if p ′ can be obtained from p by zero or more applications of the “refinementprocedure” described in Lemma 3.4. This gives a partial order on the set of all suchdecompositions.For any decomposition p there are exactly 2 ℓ ( p ) decompositions p ′ ≥ p . In themaximal such p ′ , all intervals with the final label k + 1 in p have been subdividedinto intervals of length 1 and relabelled with non-final values, separated by intervalsof length 0 labelled k + 1. In particular, ℓ ( p ′ ) = 0. Conversely, there are exactly2 ℓ ( p ) decompositions p ′ ≤ p , where ℓ ( p ) is the number of intervals of length 1having non-final labels. The minimal such p ′ has no intervals of this kind. Example 6.1.
Take k = 1 and the decomposition(6.8) p : 0 . The maximal p ′ ≥ p and the minimal p ′′ ≤ p are as follows. Subdivided orcombined intervals are indicated in boldface. p ′ : 0 k ′ = 3) , (6.9) p ′′ : 0 k ′′ = 0) . (6.10)Note that we have(6.11) x p k +1 = x p ′ k ′ +1 whenever p and p ′ are comparable. We therefore look at a minimal p in ourordering and the term x p k +1 it produces. As the added intervals of any p ′ ≥ p areall of length 1, the corresponding terms t ( x p ′ s ) in(6.12) ( − ε ( p ′ ) t ( x p ′ ) · · · t ( x p ′ k ′ )all contain − ∈ C ( G ). The summand(6.13) ( − ε ( p ′ )+( α ′ )+ P s ′ | j ′ s ′ | + ℓ ( p ′ ) Y ≤ s ′ ≤ k ′ q ′ s ′ =1 s ¯ α s ′ Sz j ′ s ′ x p ′ s ′ =: ( − ℓ ( p ′ ) a therefore appears in the product (6.12). We claim that the expression a only de-pends on p . More precisely, we have(6.14) a = ( − ε ( p )+( α )+ P s | j s | s ¯ α Sz j x p · · · s ¯ α k Sz j k x p k . This is because an interval of length q ′ s ′ = 1 leads to α ′ s ′ = ∅ and j ′ s ′ = ∅ , while theremaining α ′ s ′ and j ′ s ′ are not affected and appear as α s and j s for some index s ≤ s ′ .Moreover, we have ε ( p ′ ) = ε ( p ) by a repeated application of Lemma 3.4.If ℓ ( p ) >
0, then we get 2 ℓ ( p ) terms with alternating signs, so that(6.15) X p ′ ≥ p ( − ℓ ( p ′ ) a ⊗ t ( x p ′ k ′ +1 ) = X p ′ ≥ p ( − ℓ ( p ′ ) a ⊗ t ( x p k +1 ) = 0 . The only terms in (6.4) not appearing in such a sum are t ( x ) ⊗ p with ℓ ( p ) = 0, and we have seen already that they add up to (6.3).This completes the proof.7. The extended cobar construction and the loop group
Let X be a reduced simplicial set (that is, having a unique 0-simplex), and let GX be its Kan loop group, compare [17, Def. 26.3]. Let τ : X > → GX be thecanonical twisting function, and let t be Szczarba’s twisting cochain associated toit. Hess–Tonks have defined an extended cobar construction ˜Ω C ( X ) such that thecanonical dga map Ω C ( X ) → C ( GX ) extends to a dga map(7.1) φ : ˜Ω C ( X ) → C ( GX ) , see [14, Thm. 7]. They moreover showed that φ is a strong deformation retract ofchain complexes such that all maps involved are natural in X [14, Thm. 15].Let us recall the definition of ˜Ω C ( X ) in the form given by Rivera–Saneblidze [19,Sec. 4.2]. Write C = C ( X ), and let G be the free group on generators g x where x runs through the non-degenerate 1-simplices of X . We define a new dgc ˜ C by ˜ C n = C n for n = 1 and ˜ C = k [ G ], the group algebra of G . We set d g = 0, ε ( g ) = 0 and ∆ g = g ⊗ C + 1 C ⊗ g for any g ∈ G . We embed C into ˜ C bysending x as before to g x − G . The dga ˜Ω C ( X ) is the quotient of the usual cobarconstruction Ω ˜ C by the two-sided dg ideal generated by the cycles h a | b i − h ab i for a , b ∈ ˜ C . By abuse of notation, we write elements of ˜Ω C ( X ) like those of Ω ˜ C . ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 15
We extend Szczarba’s twisting cochain t to a linear map ˜ t : ˜ C → C ( GX ) bydefining ˜ t ( g x ) = σ ( x ) for any non-degenerate 1-simplex x ∈ X and taking itsmultiplicative extension to G ⊂ ˜ C . The result is again a twisting cochain. Theinduced dga morphism Ω ˜ C → C ( GX ) descends to ˜Ω C ( X ), where it defines themap φ from (7.1).We extend the augmentation and the diagonal from Ω C ( X ) to Ω ˜ C by setting(7.2) ε ( h g i ) = 1 and ∆ h g i = h g i ⊗ h g i for any g ∈ G . This induces well-defined maps on ˜Ω C ( X ). Proposition 7.1.
Let X be a reduced simplicial set. With the structure maps givenabove, ˜Ω C ( X ) becomes a dg bialgebra and φ a quasi-isomorphism of dg bialgebras.Proof. The maps (7.2) are compatible with φ because analogous formulas hold forthe 0-simplices φ ( h g i ) ∈ GX . Since φ is a deformation retract, it is an injectivequasi-isomorphism and its image a direct summand of C ( GX ). Because the latteris a dg bialgebra, so is ˜Ω C ( X ), and φ is a morphism of dg bialgebras. (cid:3) Twisted tensor products
Let C be an hgc and A a dg bialgebra, and let M be an A -dgc. By the latter wemean a dgc M that is also a left A -module such that the diagonal ∆ M : M → M ⊗ M and the augmentation ε M : M → k are A -equivariant. (Recall that A acts on M ⊗ M via its diagonal ∆ A : A → A ⊗ A and on k via its augmentation ε A : A → k .)Let t : C → A be a twisting cochain. The differential of the twisted tensorproduct C ⊗ t M is given by(8.1) d t = d C ⊗ ⊗ d M − δ t where(8.2) δ t = (1 ⊗ µ M ) (1 ⊗ t ⊗
1) (∆ C ⊗ µ M : A ⊗ M → M is the structure map of the A -module M . In the Sweedlernotation this is expressed as(8.3) d t ( c ⊗ m ) = d c ⊗ m + ( − | c | c ⊗ d m − X ( c ) ( − | c (1) | c (1) ⊗ t ( c (2) ) m for c ⊗ m ∈ C ⊗ t M .The purpose of this section is to observe that C ⊗ t M can again be turnedinto a dgc if t is comultiplicative. The dual situation of a multiplication on thetwisted tensor product of an hga and a dg bialgebra has already been consideredby Kadeishvili–Saneblidze [16, Thm. 7.1].Let f : Ω C → A be the map of dg bialgebras induced by the comultiplicativetwisting cochain t . Based on f and on the map E introduced in (3.9), we introducethe map of degree 0(8.4) F : C E −→ Ω C ⊗ C f ⊗ −−−→ A ⊗ C. The diagonal of C ⊗ t M then is defined as(8.5) ∆ = (1 C ⊗ µ M ⊗ C ⊗ M )(1 C ⊗ A ⊗ T C,M ⊗ M ) (cid:0) C ⊗ F ⊗ M ⊗ M (cid:1) (∆ C ⊗ ∆ M ) where µ M : A ⊗ M → M is the action. In terms of the Sweedler notation this means(8.6) ∆( c ⊗ m ) = X ( c ) , ( m ) X i ( − | c i || m (1) | (cid:0) c (1) ⊗ a i · m (1) (cid:1) ⊗ (cid:0) c i ⊗ m (2) (cid:1) for c ⊗ m ∈ C ⊗ t M and F ( c (2) ) = P i a i ⊗ c i ∈ A ⊗ C . Proposition 8.1.
Let t : C → A be a comultiplicative twisting cochain and M an A -dgc. Then the twisted tensor product C ⊗ t M is a dgc with the diagonal givenabove and the augmentation ε C ⊗ ε M .Proof. This is a lengthy computation based on the analogues d ( F ) = ( µ A ⊗ C )( t ⊗ F ) ∆ C − ( µ A ⊗ C ) (1 A ⊗ T C,A ) ( F ⊗ t ) ∆ C , (8.7) (1 A ⊗ ∆ C ) F = ( µ A ⊗ C ⊗ C ) (1 A ⊗ T C,A ⊗ C ) ( F ⊗ F ) ∆ C , (8.8) (∆ A ⊗ C ) F = (1 A ⊗ F ) F . (8.9)of the identities for E stated in Lemma 3.1. One additionally uses the formula(8.10) ∆ A t = (1 ⊗ t ) F + t ⊗ ι A , which can be seen as follows: Since f is a morphism of coalgebras, one has(8.11) ∆ A t = ∆ A f t C = ( f ⊗ f ) ∆ Ω C t C = ( f ⊗ f ) E . The image of E lies in Ω C ⊗ Ω l C with l ≤
1. Considering the terms for l = 0and l = 1 separately as in the proof of Lemma 3.1 gives (8.10).In order to prove that ∆ = ∆ C ⊗ M as given in (8.5) is a chain map, it is convenientto use the tensor product differential d ⊗ = d C ⊗ ⊗ d M on C ⊗ M and analogouslyon ( C ⊗ M ) ⊗ ( C ⊗ M ) and to show that(8.12) d ⊗ (∆ C ⊗ M ) − ( δ t ⊗ C ⊗ M ) ∆ C ⊗ M − (1 C ⊗ M ⊗ δ t ) ∆ C ⊗ M + ∆ C ⊗ M δ t = 0 . With respect to these differentials, F is the only map appearing in (8.5) that is nota chain map. The boundary d ⊗ (∆) therefore has two summands coming from theright-hand side of (8.7). The first of them cancels with ( δ t ⊗
1) ∆. Using (8.10),the term ∆ δ t splits up into two. Taking (8.8) into account, the first one cancelswith (1 ⊗ δ t ) ∆ and the second one with the second summand in d ⊗ (∆).The coassociativity of ∆ C ⊗ M is a consequence of (8.8) and (8.9). The propertiesinvolving the augmentation follow directly from the definitions. (cid:3) Corollary 8.2.
Let t : C ( X ) → C ( G ) be Szczarba’s twisting cochain determined bya twisting function τ : X > → G , and let F be a left G -space. Then C ( X ) ⊗ t C ( F ) is a dgc. The diagonal is independent of the chosen coaugmentation of C ( X ) and looksexplicitly as follows: For x ∈ X n and y ∈ F m we have(8.13) ∆ ( x ⊗ y ) = n X i =0 m X j =0 n − i X k =0 X p ( − ε ( p )+ i +( m − j − | z p k +1 | · (cid:16) ˜ ∂ i x ⊗ t ( z p ) · · · t ( z p k ) ˜ ∂ j y (cid:17) ⊗ (cid:16) z p k +1 ⊗ ( ∂ ) m − j y (cid:17) where z = ( ∂ ) n − i x , and the last sum is over all interval cuts p of [ i ] correspondingto e k . (Recall that the unit 1 ∈ Ω C is annihilated by the map p C implicit in F anddefined in (3.10), hence so is the term h z i ⊗ h z i by 1 ⊗ p C .) ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 17 Proof of Theorem 1.3
This proof is similar to the one for Theorem 1.1 given in Section 6. We start byobserving that it is enough to consider the case F = G because we can write thetwisted shuffle map ψ F in the form(9.1) C ( X ) ⊗ t C ( F ) = C ( X ) ⊗ t C ( G ) ⊗ C ( G ) C ( F ) ψ G ⊗ −−−−→ C ( X × τ G ) ⊗ C ( G ) C ( F ) ∇ −−→ C (cid:0) X × τ G × G F (cid:1) = C ( X × τ F ) . Hence if ψ G is a dgc map, then so is ψ F . (Recall from [5, (17.6)] that the shufflemap ∇ is a morphism of dgcs. This also implies that the tensor product of a leftand a right A -dgc over a dg bialgebra A is again a dgc, compare [6, p. 848].)The diagonal on the right C ( G )-module C ( X × τ G ) is C ( G )-equivariant, andinspection of the formula (8.13) shows that so is the diagonal on C ( X ) ⊗ t C ( G ).Because ψ = ψ G is also C ( G )-equivariant, we may assume y = 1 ∈ C ( G ). In otherwords, it suffices to consider elements of the form x ⊗ ∈ C ( X ) ⊗ t C ( G ) whenchecking the claimed identity(9.2) ∆ ψ = ( ψ ⊗ ψ ) ∆ . We therefore need to look at ∆ ψ ( x ) = ( − | i | ∆ b Sz i x . Combining Lemma 5.2with Proposition 4.2, we have(9.3) ( ∂ ) l b Sz i x = b Sz i x ( p − , . . . , p k +1 − X i ∈ S n ( p ) ( − | i | ˜ ∂ k b Sz i x = X ( − ε b Sz j x (0 , . . . , p − · Sz j x ( p − , . . . , p − · · · Sz j k +1 x ( p k − , . . . , p k +1 − , where the sum on the right-hand side is over all j ∈ S q − , . . . , j k +1 ∈ S q k +1 − ,and(9.5) ε = | j | + · · · + | j k | + k X s =1 ( s − q s − . Also, formula (8.13) for the diagonal on C ( X ) ⊗ t C ( G ) boils for x ⊗ x ⊗
1) = n X i =0 n − i X k =0 X p ( − ε ( p )+ i −| z p k +1 | · (cid:16) ˜ ∂ i x ⊗ t ( z p ) · · · t ( z p k ) (cid:17) ⊗ (cid:16) z p k +1 ⊗ (cid:17) where x ∈ X n , z = ( ∂ ) n − i x ∈ X i , and the last sum is over all interval cuts p of [ i ]corresponding to e k . To this expression we have to apply the map ψ ⊗ ψ . Notethat the first tensor factor above is of the form(9.7) ˜ ∂ i x ⊗ t ( z p ) · · · t ( z p k ) = ˜ ∂ i x ⊗ Sz z p · · · Sz z p k + additional terms with fewer than k factors in the second component.As in Section 6, these additional terms arise whenever a z p s with 1 ≤ s ≤ k is ofdegree 1 because of the extra term − ∈ C ( G ) in the definition of t in this case. We first consider the cuts p in (9.6) with ℓ ( p ) = 0, that is, where the intervalswith labels 1 to k cover all of [ i ]. In this case we conclude the following from (9.3)and (9.4): If we apply ψ ⊗ ψ to the terms in (9.6) that correspond to the first lineof (9.7), then we exactly get the terms appearing in(9.8) X i ∈ S n ( p ) ( − | i | ˜ ∂ k b Sz i x ⊗ ( ∂ ) l b Sz i x if we set i = p − z = x ( n − i, . . . , n ). Moreover, the formula (9.5) tells usthat the sign above corresponds with the one in (9.6).We now proceed to showing that the decompositions p with ℓ ( p ) > p with ℓ ( p ) = 0. The variable i ∈ [ n ] in (9.6) is fixed during the following discussion.We look at a minimal decomposition p of [ i ] according to the partial orderingintroduced in Section 6 and at the 2 ℓ ( p ) decompositions p ′ ≥ p . They all lead tothe same z p ′ k ′ +1 = z p k +1 , hence to the same second tensor factor b Sz z p k +1 in(9.9) ( ψ ⊗ ψ ) ∆ ( x ⊗ . For each such p ′ , the first tensor factor in (9.6),(9.10) ( − ε ( p ′ )+ i + | z p ′ k ′ +1 | ˜ ∂ i x ⊗ t ( z p ′ ) · · · t ( z p ′ k ′ ) , contains the term(9.11) ( − + ℓ ( p ′ ) (cid:16) ( − ε ( p )+ i + | z p k +1 | ˜ ∂ i x ⊗ Sz z p · · · Sz z p k (cid:17) , because of the contributions − ∈ C ( G ) of each interval of length 1, and alsobecause we have ε ( p ′ ) = ε ( p ) by Lemma 3.4. As before, these terms add up to 0for ℓ ( p ) >
0, which completes the proof.10.
Comparison with Shih’s twisted tensor product
We have mentioned in the introduction already that Szczarba’s twisting cochainagrees with the one constructed by Shih [20, §II.1] using homological perturbationtheory. In [9, Sec. 7] we pointed out that despite this agreement their approacheslead to different twisted tensor products and different twisted shuffle maps.Recall that given any cochain t : C → A , one can define the twisted tensorproducts(10.1) C ⊗ t M and M ⊗ t C for a left or, respectively, right A -module, see [15, Def. II.1.4] for instance. Thetwisted tensor products considered so far have been of the first kind.In Section 9 we have proven that Szczarba’s twisted shuffle map(10.2) ψ : C ( X ) ⊗ t C ( F ) → C ( X × τ F )is a morphism of dgcs, and and it is not difficult to see that for F = G it is also amorphism of right C ( G )-modules [9, Prop. 7.1].Shih on the other hand uses the twisted tensor product C ( F ) ⊗ t C ( X ) (wherethe fibre F is considered as a right G -space). His twisted shuffle map(10.3) ∇ τ : C ( F ) ⊗ t C ( X ) → C ( F × τ X )is part of a contraction that is a homotopy equivalence of right C ( X )-comodulesand, in the case F = G , of left C ( G )-modules, see [20, Props. II.4.2 & II.4.3] and [11, ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 19
Lemma 4.5 ∗ ]. In this sense his result is stronger because it is not known whetherSzczarba’s map ψ is part of such a homotopy equivalence. On the other hand, there does not seem to be a dgc structure on C ( F ) ⊗ t C ( X ).The “mirror image” of (8.5) gives a chain map(10.4) C ( F ) ⊗ t C ( X ) → (cid:16) C ( F ) ⊗ t C ( X ) (cid:17) ⊗ (cid:16) C ( F ) ⊗ t C ( X ) (cid:17) , but it is not coassociative in general because of the asymmetry inherent in thedefinition of the cooperations E k . We expect, however, that (10.4) extends to an A ∞ -coalgebra structure.There is a different definition of an hgc, based on cooperations(10.5) ˜ E k : C → C ⊗ C ⊗ k , which for simplicial sets is realized by the interval cut operations ˜ E k = AW ˜ e k based on the surjections ˜ e k = (1 , , , . . . , , k, cf. [8, Sec. 4]. In this setting C ( F ) ⊗ t C ( X ) would become a dgc with the diagonal (10.4) if Szczarba’s twistingcochain t were comultiplicative with respect to this new hgc structure. This is notthe case, however, as can be seen for h x i ∈ Ω C ( X ) with x ∈ X already.11. Finite covering spaces
In this section we derive a dga model for fibre bundles, meaning twisted Cartesianproducts. For the following purely algebraic reason we restrict to a very special classof bundles including principal bundles for finite groups.The dual C ∗ of a dgc C with coproduct ∆ is a dga with the transpose ∆ ∗ asmultiplication, or more precisely, with the composition(11.1) C ∗ ⊗ C ∗ → ( C ⊗ C ) ∗ ∆ ∗ −−→ C ∗ . However, the dual of a dga A is not a dgc in general, but it is so if C is finitelygenerated free k -module in each degree. The coproduct is the transpose µ ∗ of themultiplication or rather its composition with the isomorphism ( A ⊗ A ) ∗ ∼ = A ∗ ⊗ A ∗ .To apply this to fibre bundles, we assume that the structure group G has onlyfinitely many non-degenerate simplices in each degree. Then C ∗ ( G ) is a dgc, and C ∗ ( X ) is a dga for any X , as mentioned above. Because of the definition(11.2) d C ∗ = − d ∗ C of the differential on a dual complex as the negative of the transpose of the originalone (compare [8, eq. (2.12)]), the transpose(11.3) t ∗ : C ∗ ( G ) → C ∗ ( X )of Szczarba’s twisting cochain satisfies d ( t ∗ ) = d C ∗ ( X ) t ∗ + t ∗ d C ∗ ( G ) = − (cid:0) d ∗ C ( X ) t ∗ + t ∗ d ∗ C ( G ) (cid:1) (11.4) = − (cid:0) t d C ( X ) + d C ( G ) t (cid:1) ∗ = − d ( t ) ∗ = − ( t ∪ t ) ∗ = − t ∗ ∪ t ∗ . In other words, u = − t ∗ is again a twisting cochain in our sense.If the fibre F has also only finitely many non-degenerate simplices in each degree,then we have an isomorphism of complexes(11.5) (cid:0) C ( X ) ⊗ t C ( F ) (cid:1) = C ∗ ( X ) ⊗ u C ∗ ( F ) , Since the underlying complexes are free and defined for k = Z , the map ψ is at least ahomotopy equivalence of complexes, cf. [4, Prop. II.4.3]. which is now a twisted tensor product of the second form in (10.1). The minus signin u = − t ∗ arises again from (11.2) and also reflects the sign difference between thetwo kinds of twisted tensor products, see again [15, Def. II.1.4].The product on (11.5) is as described by Kadeishvili–Saneblidze [16, eq. (12)].With our sign convention and in Sweedler notation it is(11.6) ( a ⊗ b ) · ( a ′ ⊗ b ′ ) = X k ≥ X ( b ) ( − k a E k ( u ( b (1) ) , . . . , u ( b ( k ) ); a ′ ) ⊗ b ( k +1) b ′ for a , a ′ ∈ C ∗ ( X ) and b , b ′ ∈ C ∗ ( F ). The transposes(11.7) E k = (cid:0) E k (cid:1) ∗ : C ∗ ( X ) ⊗ k ⊗ C ∗ ( X ) → C ∗ ( X )are the structure maps of the hga C ∗ ( X ), see Remark 3.2. Note that the sumover k in (11.6) is in fact only over 0 ≤ k ≤ | b | + | a ′ | because of the vanishingcondition (3.4). 12. The Serre spectral sequence
Theorem 1.3 allows for a short proof of the product structure in the cohomolog-ical Serre spectral sequence. The same applies to the comultiplicative structure inthe homological setting considered by Chan [3, Thm. 1.2]. We assume throughoutthis section that k is a principal ideal domain.Recall that if the homology H ( C ) of a dgc C is free over k , then it is a gradedcoalgebra with diagonal(12.1) H ( C ) −→ H ( C ⊗ C ) ∼ = −→ H ( C ) ⊗ H ( C )where the last map is the inverse of the Künneth isomorphism. Proposition 12.1.
Let E = X × τ F be a twisted Cartesian product with thesimplicial group G as structure group. (i) Assume that H ( X ) and H ( F ) are free over k and that G acts triviallyon H ( F ) . The homological Serre spectral sequence is a spectral sequence ofcoalgebras with the componentwise coproduct on E pq = H p ( X ) ⊗ H q ( F ) , converging to H ( E ) as a coalgebra. (ii) Assume that F is of finite type, that H ∗ ( X ) or H ∗ ( F ) is flat over k andthat G acts trivially on H ∗ ( F ) . The cohomological Serre spectral sequence isa spectral sequence of algebras with the componentwise product on E pq = H p ( X ) ⊗ H q ( F ) , converging to H ∗ ( E ) as an algebra.Proof. By Theorem 1.3, the dgc C ( E ) is quasi-isomorphic to M = C ( X ) ⊗ t C ( F )with the coproduct (3.17). We filter M by increasing degree in C ( X ) and then M ⊗ M via the tensor product filtration. Let E r be the associated spectral sequenceconverging to H ( M ) and F r the one converging to H ( M ⊗ M ).Since G acts trivially on H ( F ), the definition (5.3) of Szczarba’s twistingcochain tells us that this module is annihilated by t ( x ) for any x ∈ X . There-fore, E pq = C p ( X ) ⊗ C q ( F ) , d = 1 ⊗ d, (12.2) ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 21 E pq = C p ( X ) ⊗ H q ( F ) , d = d ⊗ , (12.3) E pq = H p ( X ) ⊗ H q ( F )(12.4)and similarly F pq = M p + p = p M q + q = q C p ( X ) ⊗ H q ( F ) ⊗ C p ( X ) ⊗ H q ( F ) , (12.5) F pq = M p + p = p M q + q = q H p ( X ) ⊗ H q ( F ) ⊗ H p ( X ) ⊗ H q ( F ) . (12.6)Inspection of the formula (3.17) shows that the coproduct is filtration-preserv-ing and that the induced maps between the first and second pages of the spectralsequences are the componentwise diagonals: In the notation of Sections 6 and 9,summands corresponding to partitions p with ℓ ( p ) > t mentioned above, and among the remaining onesthose with ℓ ( p ) < p end up in a lower filtration degree. This proves the first part.The transpose ψ ∗ : C ∗ ( E ) → M ∗ of ψ is a quasi-isomorphism of dgas. Wefilter M ∗ by the dual filtration, which leads to a spectral sequence E r convergingto H ∗ ( E ). Since F is of finite type, we have E pq = (cid:0) C p ( X ) ⊗ C q ( F ) (cid:1) ∗ , (12.7) E pq = C p ( X ) ⊗ H q ( F )(12.8)by the cohomological Künneth theorem [4, Prop. VI.10.24, case II], hence E pq = H p ( X ) ⊗ H q ( F )(12.9)by its homological counterpart [4, Thm. VI.9.13] and the assumption that t ( x )annihilates H ∗ ( F ) for any x ∈ X . By the same argument as before, the productson (12.8) and (12.9) are componentwise. This concludes the proof. (cid:3) Appendix A. Comparison with Baues’ diagonal
In this section we compare the diagonal on Ω C ( X ) defined by Baues [1, Sec. 1]for a 1-reduced simplicial set with the diagonal (3.17) induced by the hgc structureof C ( X ). Proposition A.1.
For a -reduced simplicial set X the diagonal (3.17) on Ω C ( X ) is the same as Baues’. This implies that the diagonal (3.17) is also equal to the one constructed byHess–Parent–Scott–Tonks via homological perturbation theory [13, Secs. 4 & 5].
Proof.
Let x ∈ X be an n -simplex. The terms in Baues’ formula for ∆( x ) [1, p. 334]are indexed by the subsets b ⊂ n −
1. It not difficult to see, cf. the proof of [18,Prop. 5.4], that in analogy with formula (3.7) Baues’ diagonal is of the form(A.1) h x i 7→ h x i ⊗ ∞ X k =0 ˜ E k ( x )for certain functions(A.2) ˜ E k : C ( X ) → Ω k C ( X ) ⊗ Ω C ( X ) . Moreover, each non-zero summand appearing in ˜ E k ( x ) can be written as(A.3) ± (cid:10) x p (cid:12)(cid:12) · · · (cid:12)(cid:12) x p k (cid:11) ⊗ h x p k +1 i for the unique interval cut p of [ n ] associated to e k such that x p k +1 contains thevertices indexed by b plus 0 and n . Hence, up to sign, we get the same termsin s ⊗ ( k +1) ˜ E k ( x ) as in s ⊗ ( k +1) E k ( x ) = ( − k AW e k ( x ).It remains to verify the sign in the claimed identity(A.4) s ⊗ ( k +1) ˜ E k ( x ) = ( − k AW e k ( x ) . We proceed by induction on k . The case k = 0 is trivial because AW (1) the identitymap and ˜ E = s − the inverse of the suspension map s .For k > p : p k +1 p p k +1 · · · k +1 p k − k p k k +1 p k +1 for the surjection e k with those for the interval cut(A.6) p ′ : p k p p k · · · k p k − for e k − . We compute the exponents of all the signs involved, always modulo 2.The exponents of the permutation signs differ byperm( p ) − perm( p ′ ) ≡ ( p k − p k − ) p + 1 + k − X i =1 ( p i +1 − p i + 1) ! (A.7) ≡ ( p k − p k − ) k − X i =1 p i + k ! since we have to move the interval corresponding to e k (2 k ) = k before all preceding(inner) intervals corresponding to e k (1) = e k (3) = · · · = e k (2 k −
1) = k + 1. Theexponents of the position signs change by p k − because of the additional innerinterval for e k (2 k −
1) = k + 1.The sign for the summand (A.3) is the sign of the shuffle ( n − r b, b ). Hence,by passing from k − k , the exponent of this sign changes by(A.8) ( p k − p k − − p + k − X i =1 ( p i +1 − p i + 1) ! ≡ ( p k − p k − − k − X i =1 p i + k + 1 ! because we have to move all elements in the interior of the k -th interval beforeall previous values occurring in b , that is, all vertices in x p k +1 with indices strictlybetween 0 and p k − .Still modulo 2, the changes in the exponents add up to(A.9) k X i =1 p i + k + 1 ≡ k X i =1 ( p i − p i − + 1) + 1 ≡ (cid:12)(cid:12) (cid:10) x p | . . . | x p k (cid:11) (cid:12)(cid:12) + 1 . This is exactly the exponent of the sign change we get when we pass from k − k in (A.4). The sign exponent | h x p | . . . | x p k i | arises because we have to move theadditional suspension operator past the element h x p | . . . | x p k i . Another minus signcomes from the increased exponent on the right-hand side of (A.4). This completesthe proof. (cid:3) Strictly speaking, this is not a shuffle in the sense of Section 2.4 as 0 / ∈ n − { , . . . , n − } . ZCZARBA’S TWISTING COCHAIN IS COMULTIPLICATIVE 23
Appendix B. Szczarba operators and degeneracy maps
Apparently, neither in Szczarba’s paper [21] nor elsewhere in the literature onecan find a proof that Szczarba’s twisting cochain (5.3) and his twisted shufflemap (5.16) are actually well-defined on normalized chain complexes. The purposeof this appendix is to close this gap.Recall from [21, eq. (3.1)] and [14, eq. (6)] that the simplicial operators(B.1) D i ,k : X m → X m + k for 0 ≤ k ≤ n , i ∈ S n and m ≥ n − k are recursively defined by(B.2) D ∅ , = id and D i ,k = D ′ i ′ ,k s ∂ i − k if k < i , D ′ i ′ ,k if k = i , D ′ i ′ ,k − s if k > i for n ≥ i ′ = ( i , . . . , i n ). Here D ′ denotes the derived operator of asimplicial operator D , compare [21, p. 199] or [14, p. 1863].For n ≥ S n × [ n ] → S n − × [ n − , ( i , p ) ( j , q )recursively via(B.4) j = ( i − , j ′ ) , q = q ′ + 1 if p < i , ( j ′ , q ′ ) := Φ( i ′ , p ) , j = i ′ , q = 0 if p = i or i + 1, j = ( i , j ′ ) , q = q ′ + 1 if p > i + 1 , ( j ′ , q ′ ) := Φ( i ′ , p − i ′ = ( i , . . . , i n ). Note that the base case n = 1 is completely coveredby the second line above since i = 0 in that case. Lemma B.1.
Let n ≥ , i ∈ S n and p ∈ [ n ] , and set ( j , q ) = Φ( i , p ) . (i) For any ≤ k < p and any simplex x of dimension m ≥ n − k − we have D i ,k s p − − k x = s q D j ,k x. (ii) For any p < k ≤ n and any simplex x of dimension m ≥ n − k we have D i ,k x = s q D j ,k − x. Proof.
These are direct verifications by induction on n , based on the definitionsof D i ,k and Φ. The base cases are i = (0), k = 0, p = 1 and i = (0), k = 1, p = 0, respectively. In the induction step of the first formula, one distinguishes thecases k < i (with the subcases i < p − i ∈ { p − , p } and i > p ), k = i (withthe subcases i < p − i = p −
1) and k > i . For the second formula onehas the cases k < i , k = i and k > i (with the subcases p < i , p ∈ { i , i + 1 } and p > i + 1).For instance, for n > k < i and i > p we haveSz i s p x = D ′ i ′ ,k s ∂ i − k s p − − k x = D ′ i ′ ,k s s p − − k ∂ i − k − x (B.5) = D ′ i ′ ,k s p − k s ∂ i − k − x = (cid:0) D i ′ ,k s p − − k (cid:1) ′ s ∂ i − k − x = (cid:0) s q ′ D j ′ ,k (cid:1) ′ s ∂ i − k − x by induction, where ( j ′ , q ′ ) = Φ( i ′ , p ). Then j = ( i − , j ′ ) and q = q ′ + 1, hence= s q ′ +1 D ′ j ′ ,k s ∂ i − − k x = s q D ′ j ,k x since k < p ≤ i − (cid:3) Proposition B.2.
Let ≤ p ≤ n , and let x be an n -simplex. (i) For i ∈ S n and ( j , q ) = Φ( i , p ) we have Sz i s p x = s q Sz j x. (ii) For i ∈ S n +1 and ( j , q ) = Φ( i , p ) we have b Sz i s p x = s q b Sz j x. In particular, Szczarba’s twisting cochain t and the twisted shuffle map ψ descendto the normalized chain complexes.Proof. These formulas follow from Lemma B.1 and the identities (2.17) and (2.18).For example, we haveSz i s p x = D i , σ ( s p x ) D i , σ ( ∂ s p x ) · · · D i ,n σ (( ∂ ) n s p x )(B.6) = D i , s p − σ ( x ) · · · D i ,p − s σ (( ∂ ) p − x ) D i ,p σ ( s ( ∂ ) p x ) · D i ,p +1 σ (( ∂ ) p x ) · · · D i ,n σ (( ∂ ) n − x )= s q D j , σ ( x ) · · · s q D j ,p − σ (( ∂ ) p − x ) · · s q D j ,p σ (( ∂ ) p x ) · · · s q D j ,n − σ (( ∂ ) n − x )= s q Sz j x. The last claim is a consequence of the formulas just established and, for the twistingcochain t , the identity t ( s x ) = σ ( s x ) − x . (cid:3) References [1] H.-J. Baues, The double bar and cobar constructions,
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