THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE Abstract.The paper introduces the notion of a truncating twisting functi* *on from a cubical set to a permutahedral set and the corresponding notion of twisted Cartesian product of these sets. The latter becomes a permutocub* *ical set that models in particular the path space fibration on a loop space. * *The chain complex of this twisted Cartesian product in fact is a comultiplic* *ative twisted tensor product of cubical chains of base and permutahedral chain* *s of fibre. This construction is formalized as a theory of twisted tensor pro* *ducts for Hirsch algebras. 1.introduction The paper continues [12] in which a combinatorial model for a fibration was constructed based on the notion of a truncating twisting function from a simpli* *cial set to a cubical set and on the corresponding notion of twisted Cartesian prod- uct of these sets being a cubical set. Applying the cochain functor we obtain a multiplicative twisted tensor product modeling the corresponding fibration. There arises a need to iterate this construction for fibrations over loop or * *path spaces the bases of which are modeled by cubical sets. A cubical base naturally requires a permutahedral fibre; this really agrees with the first usage of the * *permu- tahedra (the Zilchgons) as modeling polytopes for loops on the standard cube due to R.J. Milgram [15] (see also [7]). For this here we introduce the notion of a truncating twisting function # : Q* ** ! P*-1 from a 1-reduced cubical set Q to a monoidal permutahedral set P ([17]). For a permutahedral set L with a given P -action, # defines the corresponding twisted Cartesian product Q x# L. The latter becomes a permutocubical set. The permutocube is defined as a polytope which is obtained from the standard cube by the specific truncation procedure due to N. Berikashvili [5], see also bellow. * *The permutocube can be thought of as a modeling polytope for paths on the cube. The general theory of the truncating twisting functions here goes almost para* *llel to that of [12]. Namely, we construct a functor assigning to a cubical set Q a monoidal permutahedral set Q together with the canonical inclusion #U : Q ! Q of degree -1 being an universal example of a truncating twisting function: a* *ny ____________ 1991 Mathematics Subject Classification. Primary 55R05, 55P35, 55U05, 52B05,* * 05A18, 05A19 ; Secondary 55P10 . Key words and phrases. Cubical set, permutahedral set, permutocubical set, t* *runcating twist- ing function, twisted Cartesian product, double cobar construction, Hirsch alge* *bra. This research described in this publication was made possible in part by Awa* *rd No. GM1- 2083 of the U.S. Civilian Research and Development Foundation for the Independe* *nt States of the Former Soviet Union (CRDF) and by Award No. 99-00817 of INTAS. 1 2 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE # : Q* ! P*-1 factors as # : Q -#U-! Q -f#!P where f# is a map of monoidal permutahedral sets. The twisted Cartesian product PQ = Q x#U Q is a permutocubical set functo- rially depending on Q. Note that Q models the loop space |Q| and PQ models the path space fibration on |Q|. The chain complex C}*( Q) coincides with the cobar construction C* (Q). Furthermore, Cfi*(Q x#U Q) coincides with the acyclic cobar construction (C* (Q); C* (Q)). Moreover, applying the chain functor to # : Q* ! P*-1 we obtain a twisting cochain #* = C*(#) : C* (Q) ! C}*-1(P ) and then Cfi*(Q x# L) coincides with the twisted tensor product C* (Q) #* C}*(L). We construct the explicit diagonal for the permutocube Bn which agrees with that of Pn [17] by means of the natural embedding Pn ! Bn. The equalities C}*( Q) = C* (Q) and Cfi*(Q x# L) = C* (Q) #* C}*(L) allow us to transport these diagonals to the cobar construction C* (Q) and the twisted tensor product C* (Q) #*C}*(L) respectively. Thus, finally, we obtain comultiplicative models* * for the loop space |Q| and the twisted Cartesian product Q x# L. In fact the diagonal C* (Q) ! C* (Q) C* (Q) is determined by higher order chain operations {Ep,q: C* (Q) ! C* (Q) p C* (Q) q}p+q>0; in particular, the cooperation E1,1is the dual operation of the cubical version of Steenrod's cochain ^1-operation and all operations {Ep,q} define on C* (Q) the structure which we call a Hirsch coalgrebra. This structure together with t* *he action C}*(P ) C}*(L) ! C}*(L) and the twisting cochain #* describes the above mentioned comultiplication on the twisted tensor product C* (Q) #* C}*(L). Dually, the permutahedral ^-product of C*}( Q) induces a product on BC* (Q) C*}( Q) which, in fact, is determined by higher order cochain operations (1) {Ep,q: C* (Q) p C* (Q) q ! C* (Q)}p+q>0; in particular, the operation E1,1is the cubical version of Steenrod's cochain ^* *1- operation and all operations {Ep,q} define on C* (Q) the structure which we cal* *l a Hirsch algrebra. Again, this structure together with the coaction C*}(L) ! C*}(* *P ) C*}(L) and the twisting cochain #* : C*}(P ) ! C*+1(Q) describes the multiplica* *tion on the twisted tensor product C* (Q) #* C*}(L) induced by the permutocubical multiplication of C*fi(Qx#L). Note that this multiplication is not strictly ass* *ociative but could be extended to an A1 -algebra structure. We formalize this construction by developing the general theory of multiplica* *tive twisted tensor products for Hirsch algebras instead of dga's. A Hirsch algebra * *we define as an object (A, d, ., {Ep,q: A p A q ! A}p+q>0), i.e., (A, d, .) is an associative dga and the sequence of operations {Ep,q} determines a product on t* *he bar construction BA turning it into a dg Hopf algebra (this multiplication can * *be viewed as a perturbation of the shuffle product and is not necessarily associat* *ive). In particular E1,1has properties similar to ^1 product, so a Hirsch algebra can* * be considered as to have a structure measuring the lack of commutativity of A. Let C be a dg Hopf algebra and M be a dga and a dg C-comodule simultaneously. A twisting element OE : C ! A wa call multiplicative if the induced map C ! BA is* * a dg Hopf algebra map. In this case we introduce on A OEM a twisted multiplication THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 3 ~OEin terms of OE and the Hirsch algebra structure of A by the same formulas as* * in the case A = C* (Q), C = C*}(P ) and M = C*}(L) where OE = #* : C*}(P ) ! C*+1(Q) is automatically multiplicative. Furthermore, we apply the above machinery for a fibration F ! E ! Z on 1-connected space Z associated with a principal G-fibration G ! E0 ! Z by an action G x F ! F to obtain the following combinatorial model. Let Q = Sing1IZ SingIZ be the Eilenberg 1-subcomplex generated by singular cubes sending the 1-skeleton of the standard n-cube In into the base point of Z, and * *let P = SingPG and L = SingPF, where SingIand SingP denote the singular cubical and the permutahedral complex of a space respectively (see [17] and Section 2).* * We construct the Adams-Milgram map !* : C* (Q) ! C}*( Z) which in fact is realized by a monoidal permutahedral map ! : Q ! SingP Z. On the other hand, one has a map of monoidal permutahedral sets SingP Z ! SingPG = P induced by the canonical map Z ! G of monoids. The composition of these two maps immediately yields a truncating twisting function # : Q ! P . The resulting twisted Cartesian product Sing1IZ x# SingPF , being a permutocu- bical set, just provides the required model of E: there exists a permutocubical weak equivalence Sing1IZ x# SingPF ! SingBE, where SingB denotes the sin- gular permutocubical complex of a space. Applying the cochain functor we obtain a certain multiplicative twisted tensor product for the fibration. In particular, we can obtain a combinatorial model for the path space fibrati* *on 2Y ! P Y ! Y in the following way. Taking for the base Z = Y the cubical model Q = Sing2Y from [12] the above theory yields the twisted Cartesian model Sing2Y x#U Sing2Y being a permutocubical set. Consequently, we introduce on the acyclic bar construction B(BC*(Y ); BC*(Y )) the multiplication whose restriction to the double bar construction BBC*(Y ) is* * just the one constructed in [17]. To summarize we observe the following. In [12] it is indicated the homotopy G-algebra structure on C*(Y ) consisting of cochain operations {Ek,1: C*(Y ) k C*(Y ) ! C*(Y )}k 1, defining a multiplication on BC*(Y ). Here we extend this multiplication to the structure of Hirsch algebra on BC*(Y ), i.e., to operations (1) {Ep,q: (BC*(Y )) p (BC*(Y )) q ! BC*(Y )}p+q>0, which actually are cochain operations of type C*(Y ) m ! C*(Y ) n . This two se* *ts of operations including in particular ^, ^1 and ^2 operations, allow us to con- struct multiplicative models for Y, 2Y and multiplicative twisted tensor prod- ucts for path space fibrations on Y and Y as well as for fibrations associated* * with them. Finally, we mention that the geometric realization | Sing2Y | of Sing2Y is homeomorphic to the cellular model for the double loop space due to G. Carls* *son and R. J. Milgram [7] and is homotopically equivalent to the cellular model due* * to H.-J. Baues [3]. The paper is organized as follows. We adopt the notions and the terminology from [12]; note that here a (co)algebra need not have a (co)associative (co)mul* *ti- plication if it is not specially emphasized. In Section 2 we construct the fun* *ctor 4 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE from the category of cubical sets to the category of permutahedral sets; Sect* *ion 3 introduces the permutocubes; in Section 4 we introduce the notion of a permu- tocubical set; Section 5 introduces the notion of a truncating twisting functio* *n and the resulting twisted Cartesian product; in Section 6 we define an explicit dia* *g- onal on the permutocubes; in Section 7 we build the permutocubical set model for the double path space fibration; in Section 8 a permutocubical model and the corresponding multiplicative twisted tensor product for a fibration are constru* *cted, and, finally, in Section 9 the twisted tensor product theory for Hirsch algebra* *s is developed. 2. The permutahedral set functor Q For completeness we first recall some basic facts about permutahedral sets fr* *om [17] (compare, [13]). 2.1. Permutahedral sets. Permutahedral sets are combinatorial objects generated by permutahedra and equipped with the appropriate face and degeneracy operators. Naturally occurring examples include the double cobar construction, i.e., the cobar construction on Adams' cobar construction [1] with coassociative coproduct [3], [7], [12] . Per* *mu- tahedral sets are similar in many ways to simplicial or cubical sets with one c* *rucial difference: Permutahedral sets have higher order structure relations, whereas s* *truc- ture relations in simplicial or cubical sets are strictly quadratic. We note th* *at the exposition on polyhedral sets by D.W. Jones [11] makes no mention of structure relations. Let Sn+1 denote the symmetric group on n_+_1_= {1, 2, . .,.n +a1}nd recall that the permutahedron (the Zilchgon) Pn+1 is the convex hull of (n + 1)! verti* *ces (oe(1), . .,.oe(n + 1)) 2 Rn+1, oe 2 Sn+1 [8], [15]. As a cellular complex, Pn+* *1 is an n-dimensional convex polytope whose (n - k)-faces are indexed by all (ordere* *d) partitions M1| . .|.Mk+1 of n_+_1_. For 1 j k, let M2j-1|M2j be a partition of n_-_j_+_2; then each (n - k)-face corresponds to a composition of face opera* *tors dM2k-1|M2k. .d.M1|M2acting on Pn+1, where M2j-1|M2j is a special partition of n_-_j_+_2for 1 j k (see Theorem 2.1). Since a partition A|B of n_+_1_denotes the same (n - 1)-face as dA|B, we use the two symbols interchangeably (see figu* *re 1). THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 5 | | | | o|____________|_oo | | | | | | | d4|123 | | | | | | |(1, 1, 1) o|_____________||____|______||___________|______|______|oooooo | |d24|1|3 | d34|12 | |d14|23| o| d o|_____|_od |____________|_ood |______|oo | 124|3 | | 234||1 | 13|4|2 | |_____________| |______| |______| | o| o| |o |o |o |o o| | |d2|13|4 | d3|124 | |d1|234| | d12|34 | |d23|14| |d13|24| | | | | | | | | o|_____________|_____|______|____________|______|______|oooooo | | (0, 0, 0) | | | | | d123|4 | | | | | | | | | o|____________|_oo Figure 1: P4 as a subdivision of P3 x I. Labels A|B for general (n - 1)-faces of Pn+1 can be obtained in purely set- theoretic terms. For ffl = 0, 1 and 1 i n, let en-1i,ffldenote the (n - 1)* *-face (x1, . .,.xi-1, ffl, xi+1, . .,.xn) In. For 0 i j 1, let Ii,j= 1 - 2-i* *, 1 - 2-j I, where 2-1 is defined to be 0, and for M a non-empty set, let @M denote its cardinality and define @; = 0. When n = 1, label the vertices of P2 by e01,0$ 1* *|2 and e01,1$ 2|1. Inductively, if Pn has been constructed, n 1, obtain Pn+1 by subdividing and labeling the (n - 1)-faces of Pn x I as indicated below: _Face_of_Pn+1_|Label_or_subscript_ | en-1n,0 |n_|n + 1 en-1n,1 |n + 1|n_ A|B x I0,@B |A|B [ {n + 1} A|B x I@B,1 |A [ {n + 1}|B. Interestingly, some (but not all) compositions dC|DdA|B act on Pn+1. This sit* *u- ation is quite different from the simplicial or cubical cases in which all comp* *ositions @i@j or dfflidffljact on the standard n-simplex n or the standard n-cube In, r* *espec- tively. The conditions under which dC|DdA|B acts on Pn+1 can be stated in terms of set operations defined as follows. Given a non-empty ordered set A = {a1 < . .<.am } Z, let IA : A ! @A_be the index map ai 7! i; for z 2 Z let A + z = {a1 + z < . .<.am + z} with the understanding_that addition takes preference over set operations. For 1 p n* *, let pdenote the set containing the last p elements of n_, i.e., _p= {n - p + 1 < . * *.<.n}; in particular, _p= {q < . .<.n}when p + q = n + 1. Definition 2.1. Given non-empty disjoint subsets A, B n_, define the lower and upper disjoint unions 6 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE æ At_B = In_ØA(B)I+ @A - 1 [ @A_,ifminB = min(n_ØA) n_ØA(B) + @A -i1,fminB > min(n_ØA); and æ ___ A__tB = In_ØB(A)I[ @B - 1,ifmax A = max (n_ØB) n_ØB(A), ifmax A < max (n_ØB). If either A or B is empty, define At_B = A__tB = A [ B. In particular, if A|B is a partition of n_, then At_B = A__tB = n_-_1_. Given a partition A1| . .|.Ak+1 of n_+_1_, define A(0)= A[k+2]= ;; inductively,* * given A(j), 0 j k, let A(j+1)= A(j)t_Aj+1; and given A[j], 2 j k + 2, let A[j-1]= Aj-1__tA[j]. And finally, for 1 j k + 1, let A(j)= A1 [ . .[.Aj. Now to a given (n - k)-face A1| . .|.Ak+1 of Pn+1, assign the compositions of f* *ace operators dA(k)|A(k-1)t_(n+1_\A(k)).d.A.(1)|A(0)t_(n+1_\A(1)) (2) = dA(1)_tA[3]|A[2].d.A.(k)_tA[k+2]|A[k+1] and denote either composition by dA1|...|Ak+1. Note that both sides of relation (2) are identical when k = 1, reflecting the* * fact that each (n - 1)-face is a boundary component of exactly one higher dimensional face (the top cell of Pn+1). On the other hand, each (n - 2)-face A|B|C is a bo* *und- ary component shared by exactly two (n - 1)-faces. Consequently, A|B|C can be realized as a quadratic composition of face operators in two different ways giv* *en by (2) with k = 2: (3) dAt_B|At_CdA|B[C = dA__tC|B__tCdA[B|C (see Figure 2). Relation (3) reminds us of the quadratic relation @i@j = @j-1@i (i < j)for face operators in a simplicial set. Example 2.1. In P8, the 5-face A|B|C_=_12|345|678 = 12|345678_\ 12345|678. Since At_B = {1234}, At_C = {567}, At C = {12} and Bt C = {34567}, we obtain the following quadratic relation on 12|345|678 : d1234|567d12|345678= d12|34567d12345|678; similarly, on 345|12|678 we have d1234|567d345|12678= d34567|12d12345|678. Similar relations on the six vertices of P3 appear in Figure 2 below. d2|1d13|2= d1|2d3|12 d3|12 d2|1d23|1= d2|1d3|12 |_____________|oo d13|2| | d23|1 | | | | d1|2d13|2= d2|1d1|2|o3 123 |o d2|1d2|13= d1|2d23|1 | | | | d1|23| | d2|13 |____________|_oo d1|2d12|3= d1|2d1|23 d12|3 d1|2d2|13= d2|1d12|3 THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 7 Figure 2: Codimension 2 relations on P3. For 1 p < n, let Qp(n) = partitionsA|B ofn_| p_ A orp_ B, Qp(n) = {partitionsA|B ofn_| _p A or_p B}, Qqp(n)= Qp(n) [ Qq(n), where p + q = n + 1. GivenPa sequence of (not necessarily distinct) positive integers {nj}1 j ksuch * *that n = nj, let Pn1,...,nk(n) = {partitionsA1| . .|.Ak ofn_| @Aj = nj}. Theorem 2.1. Let A|B 2 Pp,q(n + 1)and C|D 2 P**(n). Then dC|DdA|B denotes an (n - 2)-face of Pn+1 if and only if C|D 2 Qqp(n). Proof.If dC|DdA|B denotes an (n - 2)-face, say X|Y |Z, then according to relati* *on (3) we have either A|B = X| Y [Z and C|D = Xt_Y |Xt_Z or A|B = X [ Y |Zand C|D = X__tZ| Y __tZ. Hence there are two cases. Case_1:_A|B = X|Y [ Z. If minY = minY [ Z, then p_ Xt_Y ; otherwise minY [ Z = minZ and p_ Xt_Z. In either case, C|D= Xt_Y |Xt_Z 2 Qp(n). Case_2:_A|B = X [ Y |Z. If max_X_= max X [ Y, then _q X__tZ;_otherwise_ max(X [ Y ) = max Y and _q Y tZ. In either case, C|D= Xt Z|Y tZ 2 Qq(n). Conversely, given A|B 2 Pp,q(n + 1) and C|D 2 Qqp(n), let 8 < A|S (C)|S (D), C|D 2 Qp(n) [A|B; C|D] = : T (C)|T (D)|B, C|D 2 Qq(n), where S (X)= I-1B q_\ X - p + 1and T (X) = I-1A p_\ X . A straightforward calculation shows that [X|Y [ Z; Xt_Y |Xt_Z]= X|Y |Z = [X [ Y |Z; X__tZ| Y __tZ]. Consequently, if X|Y |Z = [A|B; C|D], either A|B = X| Y [ Z andC|D = Xt_Y |Xt_Z when C|D 2 Qp(n) or A|B = X [ Y |Zand C|D = X__tZ| Y __tZ when C|D 2 Qq(n). 8 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE On the other hand, if C|D 62 Qqp(n), higher order structure relations involving* * both face and degeneracy operators appear. This rich structure distinguishes "permu- tahedral sets" from simplicial or cubical sets whose structure relations are st* *rictly quadratic. To motivate the definition of an abstract permutahedral set,_we first constru* *ct the universal example-singular permutahedral sets. Define 0_= 0 = ;. For 1 r n and r + s = n + 1, define canonical projections r,s: Pn ! Pr x Ps, mapping each face A|B 2 Qsr(n) homeomorphically onto the (n - 2)-product cell æ _____ _____ _ A \ s - 1| B \ s - 1x sA|B 2 Qs(n), r_x A \ r_-_1_| B \ r_-_1_A|B 2 Qr(n), and each face A|B 62 Qsr(n) onto the (n - 3)-product cell _____ _____ A \ s - 1| B \ s - 1x A \ r_-_1_| B \ r_-_1_, _____ _____ where A \ s - 1| B \ s - 1is a particular partition of r_and A \ r_-_1_| B \ r_* *-_1_is a particular partition of _s(see Figure 3). THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 9 3|12 12 x 3|2 _____________||oo o_____________||o 13|2 | | 23|1 | | | | | | | | 2,2 | | |o 123 |o ________- 1|2x 23 | 12 x 23 | 2|1x 23 | | | | | | | | 1|23 | | 2|13 | | _____________||oo o____________|_|o 12|3 12 x 2|3 |o________________o |o________________o |Q Q |Q Q |Q Q | | Q | QQ | Q |o___ Q|___________|ooQo | Q | Q | Q | |Q | Q | Q | | Q|o________________|o||QQ | Q |o_______________|_o|QQ |o |oQ | o oQ | |o________ |______o | |Q | Q | |Q | Q | | | | | | Q | Q|o | Q | Qo| | | | | | Q |o | | Q |o | | | | | | | | | | | | | | | | | |o_______________o|_|| | |o_______________|o_| |o___ |____| ______|o | |o________ |______|o | Q | | Q | | Q | Q | Q | | Q | | Q | Q | Q |o | Q|o | Q | Q | Q Q | Q Q | Q Q | Q Q | Q|o________________|oQ Q |o________________|oQ 2,3 1234 _______________- 12 x 234 | | 3,2 || ||1 x 2,2 | | |? |? 123 x 34 _______________- 12 x 23 x 34 2,2x 1 |o________________o |o________________o |Q Q |Q Q |Q Q | | Q | Q | Q | Q|o | Qo | Q | Q | Q | |Q | Q | Q | | Q|o________________|o||QQ | QQ |o_______________|_o|QQ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |o___ |____| ______|o | | |o________ |______|o | Q | | Q | | Q | Q | Q | | Q | | Q | Q | Q |o | Q|o | Q | Q | Q Q | Q Q | Q Q | Q Q | Q|o________________|oQ Q |o________________|oQ Figure 3: Some canonical projections on P3 and P4. 10 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE Now each A|B 2 Pr,s(n + 1)is an (n - 1)-face of Pn+1 homeomorphic to PrxPs, so choose a homeomorphism ffiA|B : PrxPs ! A|B. In singular permutahedral sets, ffiA|B face operators pullback along the cellular projection Pn -r,s!Pr x Ps -! A|B and degeneracy operators pullback along the cellular projections ffi, fij : Pn ! Pn-1, where ffi identifies the faces i_|n_\ i_and n_\ i_|i_, 1 i n - 1, and fij i* *dentifies the faces j|n_\ j and n_\ j|j, 1 j n. Note that ff1 = fi1 and ffn-1 = fin; * *the projections fij were first defined by R.J. Milgram in [15] and denoted by Dj. Example 2.2. Let Y be a topological space. The singular permutahedral set of Y is a tuple (SingP*Y, dA|B, %i, &j), where SingPn+1Y = {continuous mapsPn+1 ! Y }, n 0, face operators dA|B : SingPn+1Y ! SingPnY are defined by dA|B(f) = f O ffiA|B O r,s for each A|B 2 Pr,s(n + 1) and degeneracy operators %i, &j, : SingPnY ! SingPn+1Y are defined by %i(f) = f O fii and&j(f) = f O ffj for each 1 i n and 1 j n - 1. It is easy to check that singular permutahedral sets are in fact permutahedral * *sets per Definition 2.2 below. For example, for the presence of a higher order struc* *ture relation see Figure 4. P2 x P1 i ii1 P P P 2,1i i i P PfPfi13|2 i i i P P P i i i 1,1 ffi2|1 1,2 PPq P2 _____- P1 _____- P1 x P1 _____-P2 _____-P1 x P2 _____-P3P3 ff1 ffi1|23 || H H J | | H H d2|1d1|23d12|34(f)Jd1|23d12|34(f) | 2,2 | H H J |? || H H H J P2 x P2 ||&1d2|1d1|23d12|34(f) = H H J | | H H JJ^ d12|34(f) |ffi12|34 | d13|2d12|34(f) H HHj |? ___________________________________________-|Y __________oeP4 f Figure 4: The quartic relation &1d2|1d1|23d12|34= d13|2d12|34. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 11 Now SingP Y determines the singular (co)homology of Y in the following way: Form the "chain complex" (C*(SingP Y ), d) of SingP Y with X d = (-1)rsgn(A; B) dA|B, A|B2Pr,s(n+1) where sgn(A; B) denotes the sign of the shuffle. Note that if f 2 C*(SingP4Y ) and d13|2d12|34(f)6= 0, the component d13|2d12|34(f)of d2(f) 2 C*(SingP2Y ) is not cancelled and d2 6= 0 (see Figure 4). Thus d is not a differential and (C*(SingP Y ), d) is not a complex in the classical sense. So form the quotient P C}*(Y ) = C* Sing Y =D, where D is the submodule generated by the degeneracies; then C}*(Y ), dis the complex of singular permutahedral chains on Y. The sequence of cellular project* *ions Pn+1 Ø-!In _-! n, Ø = (1 x 2,2) . .(.1 x 2,n-1) 2,n, _ is defined in [18](see also [12]), induc* *es a sequence of homomorphisms C*(SingY ) ! C*(SingIY ) ! C*(SingP Y ) ! C}*(Y ) whose composition is a chain map that induces a natural isomorphism H*(Y ) H}*(Y ) = H*(C}*(Y ), d). Although the first two terms in the sequence above are non-normalized chain com- plexes of singular simplicial and cubical sets, the map between them is not a c* *hain map. In general, a cellular projection between polytopes induces a chain map be- tween corresponding singular complexes if one uses normalized chains in the tar* *get. Finally, we note that SingP Y also determines the singular cohomology ring of Y since the diagonal on the permutahedra and the Alexander-Whitney diagonal on the standard simplex commute with projections. We are ready to define the notion of an abstract permutahedral set. For purpo* *ses of applications, only relation (4) in the definition below is essential; the ot* *her relations may be assumed modulo degeneracies. Definition 2.2. Let P = {Pn+1}n 0 be a graded set together with face operators dA|B : Pn+1 ! Pn for each A|B 2 P**(n + 1)and degeneracy operators %i, &j : Pn ! Pn+1 for each 1 i n + 1, 1 j n such that %1 = &1 and %n+1 = &n. Then P, dA|B, %i,i&js a permutahedral set if the following structure relations hold: For all A|B|C 2 P***(n + 1) (4) dAt_B|At_CdA|B[C = dA__tC|B__tCdA[B|C. For all A|B 2 Pr,s(n + 1)and C|D 2 P**(n)\ Qsr(n) (5) dC|DdA|B = &jdM|N dK|LdA|B where 12 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE 8 >>>K|L = n_Ø (r_\ D)|r_\ D, >>>M|N = C__t(r_\ D)| (DØ (r_\ D))_t(r_\ D), >> or >>>K|L = r_\ C|n_Ø (r_\ C), >>> >:M|N = (r_\ C)t_(CØ (r_\ C))| (r_\ C)t_D, j = @ (r_\ C)when r 2 D. For all A|B 2 P**(n + 1)and 1 < j < n (for j = 1, n see (7) below) (6) 8 >><1, ifA = j_orB = j_, &jdj|n_Øj, ifA|B 2 Qj(n + 1), A 6= j_orB 6= j_, dA|B&j = > _ _ n+1-j >:&j-1dj-1_|n_Øj-1_,ifA|B 2 Q (n + 1), A 6= j_orB 6= j_, &j&jdM|N dK|L, ifA|B =2Qn+1-jj(n + 1)where 8 __ __ >>>K|L = At j_\ B | BØ j_\ B t j_\ B , >>>M|N = @ j \ A |n_-_1_Ø@ j \ A , >> or >>>K|L = j\ A t j\ B | j \ A tB, >>> _ __ _ _ __ >:M|N = j_-_1_|n_-_1_Øj_-_1_, when j 2 B. For all A|B 2 P**(n + 1)and 1 i n + 1 æ (7) dA|B%i= 1,% ifA = {i}or B = {i}, jdC|D,where 8 >>>C|D = In+2_Øi(AØi)|In+2_Øi(B), > or >>>C|D = In+2Øi(A) |In+2Øi(BØi) , : j = I ___ ___ B (i)+ @A when {i}$ B. For all i j %i%j = %j+1%i, (8) &i&j&= &j+1&i, i%j = %j+1&i, %i&j = &j+1%i. 2.2. The Cartesian product of permutahedral sets. Let P 0= {Pr0, d0A|B, &0i, %0j} and P 00= {Ps00, d00A|B, &00i, %00j} be permu* *tahedral sets and let ( ), [ P 0x P 00= (P 0x P 00)n = Pr0x Ps00 ~ , r+s=n+1 n 1 where (a, b)~ (c, d)if and only if a = &0r(c) and d = &001(b), i.e., (&0r(c), b) = (c, &001(b)) for all(c, b) 2 Pr0x Ps00. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 13 Definition 2.3. The product of P 0and P 00, denoted by P 0xP 00, is the permuta* *hedral set 0 00 P x P , dA|B, &i, %j with face and degeneracy operators defined by 8 i j >>< d0r (a), b, ifA|B 2 Qs(n), i _\A|r_\B j (9) dA|B(a, b) = > a, d00 (b),ifA|B 2 Qr(n), >: s_\(A-n+s)| s_\(B-n+s) &idM|N dK|L (a, b), otherwise, where 8 >>>K|L = r_\ A| (r_\ B)[ s_-_1_+ r >>>M|N = (r_\ A)t_(BØ (r_\ B))| (r_\ A)t_B >> or >>>K|L = A [ (BØ (r_\ B))|r_\ B >>> __ __ >:M|N = At (r_\ B)| (BØ (r_\ B))t(r_\ B) i = @A + @ (r_\ B)- 1 when r 2 A; æ 0 (10) &i(a, b)= (&i(a),ab),, &010 i < r, , i-r+1(b)r i n; æ 0 (11) %j(a, b)= %j(a),ab,, %001 j r, j-r+1(b),r < j n + 1. Remark 2.1. Note that the right-hand side of the third equality in (9) reduces * *to the first two; indeed, if r 2 B, then K|L 2 Qs(n) and M|N 2 Qr(n); if r 2 A, K|L 2 Qs(n) and M|N 2 Qr(n) if m2_+ r - 1 A \ (r_-_1_\ A), m2 = @(r_\ B), while for m2_+ r - 1 6 A \ (r_-_1_\ A) one has K|L 2 Qs(n), M|N 62 Qr(n) and r - 1 2 L. Example 2.3. The canonical map ' : SingP X xSingP Y ! SingP (X xY ) defined for (f, g) 2 SingPrX x SingPsY by '(f, g) = (f x g) O r,s is a map of permutahedral sets. Consequently, if X is a topological monoid, the singular permutahedral complex SingP X inherits a canonical monoidal structure. Definition 2.4. A monoidal permutahedral set is a permutahedral set P with a map ~ : P x P ! P of permutahedral sets which is associative and has the unit e 2 P1. Clearly, for a monoidal permutahedral set P its chain complex (C}*(P ; R), d)* * is a dg Hopf algebra. For a permutahedral set L a P -module structure on it we define as a permutah* *e- dral map P xL ! L being associative and with the unit of P acting on L as ident* *ity. In this case C*}(L; R) is a dga comodule over dg Hopf algebra (C*}(P ; R), d). 14 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE 2.3. The permutahedral set functor Q. Let Q = (Qn, d0i, d1i, ji)n 0 be a cubical set. Recall that the diagonal : C* (Q) ! C* (Q) C* (Q) of Q is defined on a 2 Qn by X (a) = sgn(A; B) d0B(a) d1A(a), where d0B= d0j1...d0jq, d1A= d1i1...d1ip, the summation is over all shuffles {A* *, B} = {i1 < ... < iq, j1 < ... < jp} of the set n_. In particular the extreme cases A* * = ; and B = ; give the primitive part of the diagonal with sgn(;; B) = sgn(A; ;) = +. First, for Q let define the graded set 0Q as follows. Let Qc*be the graded s* *et of formal expressions Qcn+k= {&ik. .&.i1&i0(a)| a 2 Qn}n 0;k 0, where i1 . . .ik, 1 ij n + j - 1, 1 j k, &i0= 1, and let ~Qc= s-1(Qc>0) denote the desuspension of Qc. Then define 00Q as the f* *ree graded monoid (without unit) generated by ~Qc. Let 0Q be the monoid obtained from 00Q via 0Q = 00Q/ ~ , _______ ____ where &p+1(a). ~b~ ~a. &1(b)for a, b 2 Qc, |a| = p + 1. Clearly, we have the in* *clusion MQ 0Q of graded monoids where MQ denotes the free monoid generated by ~Q= s-1(Q>0). Then we introduce the canonical structure of a permutahedral_set_on 0Q as follows. First define the degeneracy operator &i by &i(~a)_= &i(a)for a monoid* *al generator ~a2 ~Q; next, for ~a2 ~Q ~Qcdefine %j(~a) = jj(a); and finally, if ~* *ais any other element of Q~cdefine its degeneracy accordingly to (8). Use formulas (10) and (11) to extend both degeneracy operators on decomposables. Now for ~a2 ~Qn+1 ~Qcn+1, define the face operator dM1|M2 by ______ ______ dM1|M2(~a) = d0M2(a). d1M1(a), M1|M2 2 P*,*(n + 1), while for other elements of Q~cand for decomposables in 0Q use formulas (5)-(7) and (9) to define dM1|M2 by induction on grading._ Now suppose Q has a fixed vertex *. Then j1(*)is declared as a unit, e, of 0Q. This relation converts 0Q into a (unital) graded monoidal permutahedral set denoted by ( Q, dM1|M2, &i, %j). In particular, we have the following identities: _____ di|n+1_\i(_a)= d1i(a),1 i n, _____ dn+1_\i|i(_a)= d0i(a),1 i n. Thus, for a 1-reduced cubical set Q all its face operators are involved in the definition of Q. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 15 Remark 2.2. Note that the definition of Q uses all cubical degeneracies. This is justified geometrically by the fact that a degenerate singular n-cube in the* * base of a path space fibration lifts to a singular (n - 1)-permutahedron in the fibre, * *which is degenerate with respect to Milgram's projections. On the other hand, we must formally adjoin the other degeneracies to achieve relations (5) (c.f., the defi* *nition of the cubical set X on a simplicial set X [12]). 3. The permutocubes The pertmutocube Bn is an n-dimensional polytope discovered by N. Berikashvili which can be thought of as a "twisted Cartesian productö f the cube and the permutahedron. Originally the permutocube Bn has been obtained from In by the following truncation procedure: First the n-cube is truncated at the minimal vertex a0 = (0, ..., 0), then it is truncated along those n - 1-faces that cont* *ained a0, and continuing so the last truncation is along those 1-faces (edges) of the n-c* *ube that contained a0. Hence, B2 is a pentagon (Figure 6 ), for B3 see Figure 8. In particular at a0 one obtains the permutahedron Pn. So that we get the natural cellular embedding (see Figures 5,7) (12) ffi;]n_: Pn ! Bn. The notation for the above inclusion map is motivated by the following com- binatorial description of Bn. First remark that the faces of Bn are in one-to-o* *ne correspondence with partitions A]M1|...|Mm of all subsets of the set n_in which* * only A is allowed to be the empty set ;. Since faces of Pn correspond to all (non-em* *pty) partitions of n_the canonical bijection n_-! ;]n_is thought of as a combinatori* *al analog of ffi;]n_. Let A(n) be the set of all (ordered) subsets of n_including the empty set ; t* *oo. In particular, @A(n) = 2n. For ~ 2 A(n) let A~ denote its corresponding subset * *in n_. First we introduce a face operator diwhich is thought of as deliting i-th e* *lement of n_; so that it resembles the simplicial operator @i-1. We have the one-to-one correspondence between the set A(n) and the set of formal compositions of di's defined by A~ = {1, ...,^ik, ...,^i1, ..., n} ! dik. .d.i1. Then to a face of Bn corresponding to the subset A~ n_we assign the compositi* *on of face operators dik. .d.i1. Now for a set A~ let P0r,m1,...,mq(A~ ) = {partitionsA0]M~1|...|M~q ofA~ |@A0 = r 0, @M~j= mj 1* *}, 1 j q, 1 q @A~ . For example, q = @A~ if and only if A0 = ; and each M~j consists of a single element. Such partitions just correspond to the vertices o* *f Bn. For A~ = m_we simply denote P0(m_) by P0(m). Next introduce the second type of a face operator dA]M for those (n - 1)-faces of Bn which correspond to partitions A0]M~ 2 P0r,m(A~ ) where A = IA~(A0) and M = IA~(M~); in particular the face operator d;]n_just denotes the single (n - * *1)- permutahedral face ffi;]n_(Pn) Bn. Then any (n - k - q)-face u of Bn corresponding to a partition A0]M~1|...|M~q* * 2 P0r,m1,...,mq(A~ ) can be expressed as the composition of face operators dAq]Mq. .d.A1]M1dik. .d.i1, 16 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE with Aj = IBj(Bj+1), Mj = IBj(M~q-j+1), Bj = A~ \ (M~q-j+2[ . .[.~Mq), B1 = A~, 1 j q, and let denote this composition by dA0]M~1|...|M~qor by du. For example, for n = 9 if {i2 < i1} = {2 < 5}, then A~ = {1, ^2, 3, 4, ^5, 6,* * 7, 8, 9}, and for the 4-face u of B9 corresponding to 38]14|6|79 2 P02,2,1,2(A~), one gets u = d38]14|6|79(B9) = d24]13d1235]4d12346]57d2d5(B9). We have that Bn also admits a realization as a subdivision of the standard n- cube In compatible with inclusion (12) (see, Figures 6,8). Indeed, let B0 = * a* *nd label the endpoints of B1 = [0, 1] via e01,0$ d;]1and e01,1$ d1. Inductively, i* *f Bn-1 has been constructed, obtain Bn as a subdivision of Bn-1 x I in the following w* *ay: _Face_of_Bn____|Label__________ | en-1n,0 |dn-1_]n | en-1i,1 |di, i 2 n_ | dA]M x I0,@M |dA]M[{n} | dA]M x I@M,1 |dA[{n}]M. From this we evidently see that that each proper m-cell em of Bn has the form em = ep x eq+1, m = p + q, where ep and eq+1 are top cells of Bp and Pq+1 respectively. Consequently, on proper cells of the permutocube we have the acti* *on of a permutahedral face operator dM1|M2 as dM1|M2(em ) = ep x dM1|M2(eq+1). These operators are connected together with di and dA]M by the canonical re- lations. Namely, combinatorially the relations between dA]M and dM1|M2 reflect the associativity of the partition procedure, while the relations between di an* *d ei- ther dA]M or dM1|M2 reflect the commutativity of the deleting and the partition procedures. These relations together with those involving degeneracies incorporated in the singular permutocubes (see Example 4.1) motivates the notion of a permutocubical set given in the next section. 4.Permutocubical sets Definition 4.1. A permutocubical set is a graded set B = {Bp,qn| p, q 0; p + q = n}n 0 together with face and degeneracy operators di : Bp,qn! Bp-1,qn-1,i 2 p_, dA]M : Bp,qn! Bp-r,q+r-1n-1,A]M 2 P0p-r,r(p), dM1|M2 : Bp,qn! Bp,q-1n-1,M1|M2 2 P*,*(q + 1), jj : Bp,qn! Bp+1,qn+1,j 2 p_+_1_, &i, %j : Bp,qn! Bp,q+1n+1,i 2 q_+_1_, j 2 q_+_2_, THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 17 that satisfy the following relations: For each p 0 the graded set {Bp,qn; dM1|M2, &i, %j}q 0; p+q=n is a permutahedral set and didj = dj-1di, i < j, didA]M = dA\j]Mdj, j = I-1A(i), i 2 p_-_r_, didM1|M2 = dM1|M2di, dM1|M2dA]M = dA]M dM3|M4, M1|M2 2 Qr(q + r), M3|M4 = M1 + 1 - r \ q_+_1_|M2 + 1 - r \ q_+_1_, dM1|M2dA]M = dA2]L2dA1]L1,A1|L1 = A [ I-1M(M1 \ r_)| I-1M(M2 \ r_), A2|L2 = A| I-1M(M1 \ r_), M1|M2 62 Qr(q + r), dijj = jjdi, i < j; dijj = 1, i = j; dijj = jjdi-1, i > j; dA]M jj = jidA1]M1, A1|M1 = Ip+1_\j(A \ j)| Ip+1_\j(M), i = IA (j), j 2 A, dA]M jj = %idA1]M1, A1|M1 = Ip+1_\j(A)| Ip+1_\j(M \ {j}), i = IM (j), j 2 M, r > 1, dA]M jj = 1, A]M = p_+_1_\ j|j, dM1|M2jj = jjdM1|M2, diij = ijdi, i = &, %, dA]M ij = ij+r-1dA]M ,i = &, %, jijj = jj+1ji, i j, iijj = jjii, i = &, %. Example 4.1. For a topological space Y define the singular permutocubical compl* *ex SingB Y as follows: Let (SingB Y )p,qn= {continuous mapsBp x Pq+1 ! Y }p,q 0; p+q=n, Bp x Pq+1 is a Cartesian product of the permutocube Bp and the permutohedron Pq+1. Let ffiix 1 : Bp-1 x Pq+1 ! Bp x Pq+1, 1 i p, f~fiA]M : Bp-r x Pq+r 1x-r,q+1-----!Bp-r x Pr x Pq+1 ffiA]Mx1-----!Bp x P* *q+1, 1 x ffiM1|M2: Bp x Pq ! Bp x Pq+1, 18 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE be the maps in which ffii and ffiA]M are the canonical inclusions, while ffiM1|* *M2 is defined in Example 2.2. Consider also the maps fljx 1 : Bp+1 x Pq+1 ! Bp x Pq+1,j 2 p_+_1_, 1 x ffj : Bp x Pq+2 ! Bp x Pq+1,j 2 q_+_1_, 1 x fij : Bp x Pq+2 ! Bp x Pq+1,j 2 q_+_2_, where flj : Bp+1 ! Bp is the projection that identifies the faces dp+1_\j|jand * *dj. Then for f 2 (SingB X)p,qndefine di : (SingB Y )p,qn! (SingB Y )p-1,qn-1, dA]M : (SingB Y )p,qn! (SingB Y )p-r,q+r-1n-1, dM1|M2 : (SingB Y )p,qn! (SingB Y )p,q-1n-1, and jj : (SingB Y )p,qn! (SingB Y )p+1,qn+1, &i, %j: (SingB Y )p,qn! (SingB Y )p,q+1n+1, as compositions di(f) = f O (ffiix 1), dA]M (f) = f O ~ffiA]M, dM1|M2(f) = f O (1 x ffiM1|M2), ji(f) = f O (flix 1), &i(f) = f O (1 x ffi), %i(f) = f O (1 x fii). It is easy to check that (SingB Y, di, dA]M , dM1|M2, ji, &i, %i) is a permutoc* *ubical set. The singular permutocubical complex SingB Y determines the singular (co)ho- mology of Y in the following way: Form the "chain complex" (C*(SingB Y ), d) of SingB Y with X d = (-1)i+1di- sgn(A; M)(-1)@AdA]M + sgn(M1; M2)(-1)@M1 dM1|M2, where the summation is over all i 2 n_, A]M 2 P0**(p) and M1|M2 2 P**(q + 1). Then consider the quotient being a chain complex in the classical sense (i.e., d2 = 0) Cfi*(Y ) = C*(SingB Y )=D, where D is the submodule of C*(SingB Y ) generated by the degenerate elements of SingB Y. Now let ' : Bn ! In be the cellular projection defined by the property that it maps homeomorphically the faces dn_\i]i(Bn) and di(Bn) onto the faces d0i(In) a* *nd d1i(In) respectively, 1 i n. Then the composition of maps Bp x Pq+1 OE-!Ip x Iq = Ip+q _-! p+q, OE = ' x Ø, clearly induces a composition of maps of graded sets SingY -_!SingIY -OE!SingB Y THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 19 denoted by the same symbols. After the passage on the non-normalized chains (unless the last one) one gets a sequence of homomorphisms C*(SingY ) ! C*(SingIY ) ! C*(SingB Y ) ! Cfi*(Y ), whose composition is a chain map inducing a natural isomorphism H*(Y ) Hfi*(Y ) = H*(Cfi*(Y ), d). Since the diagonal on the permutocube constructed in Section 6 is compatible wi* *th the AW diagonal on the standard simplex under the above cellular projections, Hfi*(Y ) determines the singular cohomology ring of Y as well. Basic examples of a permutocubical set are provided in the next section. 5.Truncating twisting functions and twisted Cartesian products An universal example of truncating twisting function is just the canonical in* *clu- sion function #U : Q ! Q, x ! ~x, of degree -1, where Q is the permutahedral set for a cubical set Q constructed above. The geometrical interpretation of #U answers to the truncation procedure that converts In into Bn mentioned in Section 3. By this the permutocube is thought of as a "twisted Cartesian productö f the cube and the permutohedron (see Fig. 5,7). Motivated by this here we give the general formalism for such functions and t* *hen the corresponding notion of twisted Cartesian product. Definition 5.1. Let Q = (Qn, d0i, d1i, ji) be a 1-reduced cubical set and P = (Pn+1, dM1|M2, &i, %i) be a monoidal permutahedral set. A sequence # = {#n}n 1 of degree -1 functions #n : Qn ! Pn is called a truncating twisting function if #(a) = e, a 2 Q1, dM1|M2#(a) = #d0M2(a) . #d1M1(a),M1|M2 2 P*,*(n),a 2 Qn, %i#(a) = #ji(a), i 2 n_. Note that since the first condition above we in particular get di|n_\i#(a)= #d1i(a),i 2 n_, dn_\i|i#(a)= #d0i(a),i 2 n_, for any a 2 Qn>0. Remark 5.1. By definition a truncation twisting function involves only the perm* *u- tahedral degeneracy operator %i, since it is in fact arisen by the cubical dege* *neracy operator ji (cf. Remark 2.2). We have the following Proposition 5.1. Let Q be a 1-reduced cubical set and P be a monoidal permu- tahedral set. A sequence # = {#n}n 1 of degree -1 functions #n : Qn ! Pn is a truncating twisting function if and only if the monoidal map f : Q ! P defined by f(~a1. .~.ak) = #(a1) . .#.(ak) is a map of permutahedral sets. Proof.Obvious. 20 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE Definition 5.2. Let Q = (Qn, d0i, d1i, ji) be a 1-reduced cubical set and P = (Pn+1, dM1|M2, &i, %i) be a monoidal permutahedral set and L be a permutahedral set with P -module structure. Let # = {#n}n 1, #n : Qn ! Pn be a truncating twisting function. The twisted Cartesian product Q x# L is the Cartesian product of sets [ Q x L = {(Q x L)p,qn= Qp x Lq+1} n=p+q endowed with the face and degeneracy operators di, dA]M , dM1|M2, jj, &j, %j de* *fined for (a, b) 2 Qp x Lq+1 by : di(a, b) = (d1i(a), b), i 2 p_, dA]M (a, b) = (d0M(a), #d1A(a) . b),A]M 2 P0*,*(p), dM1|M2(a, b)= (a, dM1|M2(b)), M1|M2 2 P*,*(q + 1), jj(a, b) = (jj(a), b), j 2 p_+_1_, &j(a, b) = (a, &j(b)), j 2 q_+_1_, %j(a, b) = (a, %j(b)), j 2 q_+_2_. It is easy to check that (Q x# L, di, dA]M , dM1|M2, jj, &j, %j) is a permuto* *cubical set. Remark 5.2. Note that to a twisted Cartesian product Q x# L in fact corresponds the sequence of graded sets L '-!Q x# L ,-!Q with '(b) = (a0, b) and ,(a, b) = a, a0 2 Q0, a 2 Q, b 2 L. Example 5.1. Let M = {ek}k 0 be the free minoid on a single generator e1 2 M1 with trivial permutahedral set structure and let # : Q ! M be the sequence of constant maps #n : Qn ! Mn-1, n 1. Then the twisted Cartesian product Qx#M can be thought of as a permutocubical resolution of a 1-reduced cubical set Q. 5.1. The permutocubical set functor PQ. For the universal truncating twisting function #U the corresponding twisted Cartesian product implies the following Definition 5.3. A functor from the category of 1-reduced cubical sets to the ca* *te- gory of permutocubical sets defined by Q ! Q x#U Q is the universal permutocu- bical functor and is denoted by P. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 21 ;]1 ;] |________________|ss | | | | | | 2|1 | | s |s | | ffi ;]2||1 B2 | | ____________-;]2_ | | P | | | 2 | | | |s _________________||ss 1|2 @I ;]1|2 ;]2 @ | @ #U | | ' @ | @ ;]1 |? ;] @ |________________|ss @ | | @ | | | | @ | | | | | I2 | | | | | | | _________________||ss ;]1|2 ;]2 Figure 5: The universal truncating twisting function #U . d2 _____________rr | | d | | 2]1| | |r | | B2 |d1 | | B1 d;]1|2 | ______________rr ______________||rr d;]1 d1 d1]2 Figure 6: B2 as a subdivision of B1 x I. 22 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE ;] ;]|b____________QQrr1 | b pr;]2|1 | Q | b|b | Q 3|2|r1 |r;]3r|2br_____________Q|r;]2|1;]1|2| b| ;]3|bb|1b| | | | bbr3|1|2 | b r|bb|r;]3||1|2 | P3 2|3|r|1| | ;]2|3|1||| |;]3 || || ||r ;]1|b|r__3___|____|rQ_____|_rr;]3|2;]1|3|2||B3 2|1|r|3 |1|3|2 ;b]br2|1|||3| QQ || Q QQ||r_______________b-b_____________||rrQQ 1|2|3 ffi;]3_ ;]1|2|3 ;]2|3 @@I | @ #U | |' @ | @ |? ;] ;]1|b___________QQ_rr | b | Q | b b | Q | b _____________Q|rr;]2;]1|2 | | | | | | | | || |;]3 || I3 ;]1|b|_______|____|rrQ3 | b b || QQ || b b _____________||rrQQ ;]1|2|3 ;]2|3 Figure 7: The universal truncating twisting function #U . |b____________QQrrd2 | b pr | Q |d b|b d3 | d1Q |r23]r|dbr_____________Q|r1 |bb b| 3|]12 | | b r|bb|r | | |d2]1|3 || |d13]2 || b|r__|___|____|rQ_____|_rr|dB3;]123d b br| || dQQ1]23|| 12]3 b b_____________||rrQQ Figure 8: B3 as a subdivision of B2 x I. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 23 6.The diagonal of permutocubes Here we construct the explicit diagonal B : C*(Bn) ! C*(Bn) C*(Bn) for permutocubes which induces a diagonal for a permutocubical set too. 6.1. The orthogonal stream. Suppose that an n-dimensional polytope X is realized as a subdivision of the cube In so that each m-dimensional cell em X, 0 m n, is itself a subdivis* *ion of Im (Im need not to be a face of In, cf. Bn). In particular, we have an induced partial ordering on the set of all vertices* * of em defined by x y if there is an oriented polygonal line from x to y. Let em be a cell. For a cell ek em let Im(ek) Im be the face of Im of the minimal dimension m(ek) that contains ek. Then we introduce the following Definition 6.1. Let em X be a cell and x 2 em be a vertex. An orthogonal stream OSx(em ) of x with support em is a pair (Ux, Vx) of collections of those* * faces Ux = {u1, ..., ur} and Vx = {v1, ..., vs} of em which satisfy the following con* *ditions: 1. max ur = x = minv1 and dimur + dimv1 = m; 2. Im(ui)= Im(ur), dimui= dimur and max ui x, 1 i r; 3. Im(vj)= Im(v1), dimvj = dimv1 and minvj x, 1 j s. The union [x2emSOx(em ) is denoted by SO(em ). A pair (ui, vj) 2 OSx(e) is referred to as a complementary pair (CP), while t* *he pair (ur, v1) 2 OSx(e) to as a strong complementary pair (SCP) (compare, [17]) and will be denoted by (ux, vx). Clearly, any vertex x of em Bn uniquely defines (ux, vx) in OSx(em ), and, * *con- sequently, the whole OSx(em ) is uniquely determined by the vertex x. In partic* *ular, if x coincides with a vertex of Im then dim ux = m(ux) and dim vx = m(vx), so that Ux and Vx actually lay on orthogonal faces of Im at the vertex x. For Bn, an orthogonal stream OSx(Bn) admits the specific combinatorial descri* *p- tion. First, let B a linearly ordered (finite) set and let y = {b1, b2, ..., bm* * }, m 1, be any (non-ordered) sequence formed by its elements (i.e., corresponding to some * *ele- ment of Sm ). Then it corresponds two sequences with ordered blocks uy = A1|...* *|Ap and vy = C1|...|Cq defined as follows: A1 = {bj1< ... < b1} is the first maximal block of decreasing elements (i.e., bj1< bj1+1), A2 = {bj2< ... < bj1+1} is the* * next such a block, and so on, while C1 = {b1 < ... < bk1} is the first maximal block* * of increasing elements (i.e., bk1 > bk1+1), C2 = {bk1+1< ... < bk2} is the next su* *ch a block, and so on. For example, for B = 5_and y = {2, 1, 4, 3, 5} one gets uy = 12|34|5 and vy = 2|14|35. Now let x = ;]a1|...|an-k, 0 k n, be a vertex of Bn, i.e., the set {a1, .* *.., an-k} is the same as dik. .d.i1(n_) with {a1, ..., an-k} = n_\ A0, A0 = {ik < . .<.i1* *}. For the sequence x0 = {a1, ..., an-k} let (ux0, vx0) = (A1|...|Ap, C1|...|Cq) b* *e the corresponding pair determined above. Then for the SCP (ux, vx) we get the equal* *ity (ux, vx) = (A0]A1|...|Ap, C1]C2|...|Cq). For example, x = ;]2|1|3|6|5, then (ux, vx) = (4]12|3|56 , 2]136|5). Next for a partition a = A0]A1|...|A` of an ordered (finite) set we define the right-shift R and the left-shift L operators respectively as follows (compare, * *[17]): Let Mi Aiand Nj Aj, 0 i < `, 0 < j `, be proper subsets, while M0 = A0 24 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE is also allowed, as well as all the subsets to be the ;. Let RMi(a) = A0]A1| . .|.Ai\ Mi|Ai+1[ Mi| . .|.A`forminMi> max Ai+1, LNj(a) = A0]A1| . .|.Aj-1[ Nj|Aj\ Nj| . .|.A`forminNj > max Aj-1, where R; = Id = L;. Then each CP (u, v) 2 (Ux, Vx) can be obtained from the SCP (ux, vx) by successive application of the above operators as (u, v) = (RM`-1 . .R.M1RM0 (ux) , LN1 . .L.N`(vx)) for some {Mi}0 i<`and {Nj}00 and the induced comodule structure L : C*}(L) ! C*}(P ) C*}(L) by the action P xL ! L the permutocubical multiplication of the left side of (15) can be expr* *essed by the following formula. Let a1 m1, a2 m2 2 C* (Q) #* C*}(L) and kL: C*}(L) ! C*}(P ) kP C*}(L) be the iterated L with 0L=PId: C*}(L) ! C*}(L), and let pL(m1) = c11 . . .cp1 mp+11, q-1L(m2) = c12 . . .cq-12 mq2. Then (16) ~((a1 m1) (a2 m2)) = X p q-1 p* *+1 q (-1)ffla1Ep,q(#(c11), . .,.#(c1); a2, #(c12), ..., #(c2 )) m1* * m2, p 0; q 1 ffl = |mp+11|(|a2| + |c12| + . .+.|cq-12|). Corollary 8.1. Let F ! E i-!Z be the fibration associated with G-fibration G ! E0-ß!Z by the action GxF ! F. Then the tensor product C* (Z) C*}(F ) becomes a dga (C* (Z) C*}(F ), d#, ~) with both twisted differential d# and the multipl* *ication ~. In particular, letting P = L = Q in (16) we deduce the following explicit formula for the multiplication on the acyclic bar construction B(C* (Z); C* (Z)) converting it into a dga. For a = a0 [~a1| . .|.~an], b = b0 [~b1| . .|.~b* *m], ai, bj 2 C* (Z), 0 i n, 0 j m, let (17) X ab = (-1)ffla0Ep,q(a1, ..., ap; b0, b1, ..., bq-1) [~ap+1| . .|.~an] O* * [~bq| . .|.~bm], p 0; q 1 ffl = (|~ap+1| + . .+.|~an|)(|b0| + |~b1| + . .+.|~bq-1|). Using the fact that BC*(Y ) has an associative multiplication [12] we canonic* *ally introduce on the acyclic bar construction B(BC*(Y ); BC*(Y )) the multiplication by (17) that agrees with the one on the double bar construction BBC*(Y ) [17]. 34 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE 9. Twisted tensor products for Hirsch algebras The notion of a Hirsch (co)algebra naturally generalizes the one of a homotopy G-(co)algebra. Again the structure such a (co)algebra on the cubical (co)chain complex of a topological space defined by the diagonal of permutahedra became the motivation for the material of this section and that formulas (16) and (17) established in the previous section are valid in a purely algebraic situation. Let for a dga A (Hom (BA BA, A), r) be the canonical dga with ^-product, where BA BA has the standard tensor coalgera structure. We have the following definition Definition 9.1. A Hirsch algebra is a 1-reduced associative dga A with multilin* *ear maps Ep,q: A p A q ! A, p, q 0, p + q > 0, satisfying the following conditions: (i) Ep,qis of degree 1 - p - q; (ii) E1,0= Id = E0,1and Ek>0,0= 0 = E0,k>0; (iii) The homomorphism E : BA BA ! A defined by E([~a1| . .|.~ap] [~b1| . .|.~bq]) = Ep,q(a1, ..., ap; b1, ..., b* *q) is a twisting element in the dga (Hom(BA BA, A), r), i.e., it satisfies rE = -E ^ E. Entirely dually one can formulate the notion of a Hirsch coalgebra. The condition (i) guarantees that the comultiplicative coextension ~E : BA BA ! BA is a map of degree 0, the condition (ii) guarantees that the empty bracket [ ] 2 BA is a unit for ~E , and the condition (iii) guarantees that ~E * *is a chain map; thus BA becomes a dg Hopf algebra with not necessarily associative multiplication ~E (cf. [9], [19]). The condition (iii) can be rewritten in terms of components Ep,q. In particul* *ar the operation E1,1satisfies the conditions similar to that of Steenrod's ^1 pro* *duct: dE1,1(a; b) - E1,1(da; b) + (-1)|a|E1,1(a; db) = (-1)|a|ab - (-1)|a|(|b|+1)b* *a, so it measures the non-commutativity of the product of A (thus, a Hirsch algebra with Ep,q= 0 for p, q 1 is just a commutative dga). Main examples of Hirsch (co)algebras are: C* (Q) (see previous section), in particular, Adams' cobar construction C*(X) ([17]), and the singular simplicial cochain complex C*(X): in [14] a twisting element E : BC*(X) BC*(X) ! C*(X) satisfying (i)-(iii) is constructed and these conditions determined E uni* *quely up to the standard equivalence of twisting elements. 9.1. Multiplicative twisted tensor products. Let A be a Hirsch algebra, C be a dg Hopf algebra, and M be a dga being a dg comodule over C. Definition 9.2. A twisting element # : C ! A in Hom(C, A) we call multiplicative if its comultiplicative coextension C ! BA is an algebra map. It is clear that if # : C ! A is a multiplicative twisting element and if g : B ! C is a map of dg Hopf algebras then the composition #g : B ! A is again a multiplicative twisting element. THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 35 The canonical projection BA ! A provides an example of the universal multi- plicative element. We have that the argument of the proof of formula (16) immediately yields Theorem 9.1. Let #* : C ! A be a multiplicative twisting element. Then the tensor product A M with the canonical twisting differential d#* = d 1 + 1 * * d + #*\- becomes a dga (A M, d#*, ~) with the twisted multiplication ~ determined by formula (16). Thus the above theorem includes the twisted tensor product theory both for homotopy G-algebras [12] and for commutative algebras ([16]). Corollary 9.1. For a Hirsch algebra A the acyclic bar construction B(A; A) cano* *n- ically becomes a dga with the twisted multiplication determined by formula (17). 36 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE References [1]J. F. Adams, On the cobar construction, Proc. Nat. Acad. Sci. (USA), 42* * (1956), 409- 412. [2]J. F. Adams and P. J. Hiltion, On the chain algebra of a loop space, 30* * (1955), 305-330. [3]H.-J. Baues, Geometry of loop spaces and the cobar construction, Memoir* *es of the AMS, 25 (1980), 1-170. [4]N. Berikashvili, On the differentials of spectral sequences (Russian), * *Proc. Tbilisi Mat. Inst., 51 (1976), 1-105. [5]_____, On the third obstruction, Bull. Georg. Acad. Sci., to appear. [6]E. Brown, Twisted tensor products, Ann. of Math., 69 (1959), 223-246. [7]G. Carlsson and R. J. Milgram, Stable homotopy and iterated loop spaces* *, Handbook of Algebraic Topology (Edited by I. M. James), North-Holland (1995), 505* *-583. [8]H.S.M. Coxeter and W.O.J. Moser, Generators and relations for discrete * *groups, Springer-Verlag, 1972. [9]E. Getzler and J.D. Jones, Operads, homotopy algebra, and iterated inte* *grals for double loop spaces, preprint, 1995. [10]V.K.A.M. Gugenheim, On the chain complex of a fibration, Ill. J. Math.,* * 16 (1972), 398-414. [11]D. W. Jones, A general theory of polyhedral sets and corresponding T-co* *mplexes, Dis- sertationes Mathematicae, CCLXYI, Warszava (1988). [12]T. Kadeishvili and S. Saneblidze, A cubical model for a fibration, prep* *rint, AT/0210006. [13]______-, Permutahedral complex modeling the double loop space, Proc. of the International Meeting, ISPM-98, Mathematical Methods in Modern Theor* *etical Physics, School and Workshop, Tbilisi, Georgia, September 5-18 (1998), 2* *31-236. [14]L. Khelaia, On the homology of the Whitney sum of fibre spaces, Proc. T* *bilisi Math. Inst., 83 (1986), 102-115. [15]R. J. Milgram, Iterated loop spaces, Ann. of Math., 84 (1966), 386-403. [16]A. Proute, A1 -structures, Modele minimal de Bauess-Lemaire des fibrati* *ons, preprint. [17]S. Saneblidze and R. Umble, Diagonals on the Permutahedra, Multiplihedr* *a and Asso- ciahedra, preprint, AT/0209109. [18]J.-P. Serre, Homologie singuliere des 'espaces fibr'es, applications, A* *nn. Math., 54 (1951), 429-505. [19]A.A. Voronov, Homotopy Gerstenhaber algebras, preprint, QA/9908040. A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze st., 1, 380093 Tbilisi, Georgia E-mail address: kade@@rmi.acnet.ge A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze st., 1, 380093 Tbilisi, Georgia E-mail address: sane@@rmi.acnet.ge