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 1reduced 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. 9900817 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 ^1operation 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 Ccomodule 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
1connected space Z associated with a principal Gfibration G ! E0 ! Z by
an action G x F ! F to obtain the following combinatorial model. Let Q =
Sing1IZ SingIZ be the Eilenberg 1subcomplex generated by singular cubes
sending the 1skeleton of the standard ncube 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 AdamsMilgram 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
Galgebra 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 ndimensional convex polytope whose (n  k)faces are indexed by all (ordere*
*d)
partitions M1 . ..Mk+1 of n_+_1_. For 1 j k, let M2j1M2j be a partition
of n__j_+_2; then each (n  k)face corresponds to a composition of face opera*
*tors
dM2k1M2k. .d.M1M2acting on Pn+1, where M2j1M2j is a special partition of
n__j_+_2for 1 j k (see Theorem 2.1). Since a partition AB of n_+_1_denotes
the same (n  1)face as dAB, we use the two symbols interchangeably (see figu*
*re
1).
THE TWISTED CARTESIAN MODEL FOR THE DOUBLE PATH SPACE FIBRATION 5
 
 
o_____________oo
 
 
 
 d4123 
 
 
 (1, 1, 1)
o______________________________________________oooooo
 d2413  d3412  d1423
o d o______od _____________ood ______oo
 1243   2341  1342 
_____________ ______ ______ 
o o o o o o o
 d2134  d3124  d1234
 d1234  d2314 d1324 
      
o________________________________________________oooooo
 
(0, 0, 0)  
 
 d1234 
 
 
 
 
o_____________oo
Figure 1: P4 as a subdivision of P3 x I.
Labels AB 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 en1i,ffldenote the (n  1)*
*face
(x1, . .,.xi1, ffl, xi+1, . .,.xn) In. For 0 i j 1, let Ii,j= 1  2i*
*, 1  2j
I, where 21 is defined to be 0, and for M a nonempty 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$ 21. 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_

en1n,0 n_n + 1
en1n,1 n + 1n_
AB x I0,@B AB [ {n + 1}
AB x I@B,1 A [ {n + 1}B.
Interestingly, some (but not all) compositions dCDdAB 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 nsimplex n or the standard ncube In, r*
*espec
tively. The conditions under which dCDdAB acts on Pn+1 can be stated in terms
of set operations defined as follows.
Given a nonempty 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 nonempty 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 AB 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[j1]= Aj1__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(k1)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 ABC is a bo*
*und
ary component shared by exactly two (n  1)faces. Consequently, ABC can be
realized as a quadratic composition of face operators in two different ways giv*
*en by
(2) with k = 2:
(3) dAt_BAt_CdAB[C = dA__tCB__tCdA[BC
(see Figure 2). Relation (3) reminds us of the quadratic relation @i@j = @j1@i
(i < j)for face operators in a simplicial set.
Example 2.1. In P8, the 5face ABC_=_12345678 = 12345678_\ 12345678.
Since At_B = {1234}, At_C = {567}, At C = {12} and Bt C = {34567}, we
obtain the following quadratic relation on 12345678 :
d1234567d12345678= d1234567d12345678;
similarly, on 34512678 we have
d1234567d34512678= d3456712d12345678.
Similar relations on the six vertices of P3 appear in Figure 2 below.
d21d132= d12d312 d312 d21d231= d21d312
_____________oo
d132  d231
 
 
d12d132= d21d12o3 123 o d21d213= d12d231
 
 
d123  d213
_____________oo
d12d123= d12d123 d123 d12d213= d21d123
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) = partitionsAB ofn_ p_ A orp_ B,
Qp(n) = {partitionsAB 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 AB 2 Pp,q(n + 1)and CD 2 P**(n). Then dCDdAB denotes
an (n  2)face of Pn+1 if and only if CD 2 Qqp(n).
Proof.If dCDdAB denotes an (n  2)face, say XY Z, then according to relati*
*on
(3) we have either
AB = X Y [Z and CD = Xt_Y Xt_Z
or
AB = X [ Y Zand CD = X__tZ Y __tZ.
Hence there are two cases.
Case_1:_AB = XY [ Z. If minY = minY [ Z, then p_ Xt_Y ; otherwise minY [
Z = minZ and p_ Xt_Z. In either case, CD= Xt_Y Xt_Z 2 Qp(n).
Case_2:_AB = 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, CD= Xt ZY tZ 2 Qq(n).
Conversely, given AB 2 Pp,q(n + 1) and CD 2 Qqp(n), let
8
< AS (C)S (D), CD 2 Qp(n)
[AB; CD] = :
T (C)T (D)B, CD 2 Qq(n),
where
S (X)= I1B q_\ X  p + 1and T (X) = I1A p_\ X .
A straightforward calculation shows that
[XY [ Z; Xt_Y Xt_Z]= XY Z = [X [ Y Z; X__tZ Y __tZ].
Consequently, if XY Z = [AB; CD], either
AB = X Y [ Z andCD = Xt_Y Xt_Z
when CD 2 Qp(n) or
AB = X [ Y Zand CD = X__tZ Y __tZ
when CD 2 Qq(n).
8 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
On the other hand, if CD 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 examplesingular 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 AB 2 Qsr(n) homeomorphically onto the (n  2)product cell
æ _____ _____ _
A \ s  1 B \ s  1x sAB 2 Qs(n),
r_x A \ r__1_ B \ r__1_AB 2 Qr(n),
and each face AB 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
312 12 x 32
_____________oo o_____________o
132   231  
   
  2,2  
o 123 o ________ 12x 23  12 x 23  21x 23
   
   
123   213  
_____________oo o_____________o
123 12 x 23
o________________o o________________o
Q Q Q Q
Q Q   Q  QQ  Q
o___ Q___________ooQo  Q  Q
 Q  Q  Q  Q
  Qo________________oQQ  Q o________________oQQ
o oQ  o oQ  o________ ______o 
Q  Q  Q  Q     
 Q  Qo  Q  Qo    
 Q o   Q o     
         
  o_______________o_  o_______________o_
o___ ____ ______o  o________ ______o 
Q   Q   Q  Q 
Q   Q   Q  Q 
Q o  Qo  Q  Q 
Q Q  Q Q  Q Q  Q Q 
Qo________________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
 Qo  Qo  Q  Q
 Q  Q  Q  Q
  Qo________________oQQ  QQ o________________oQQ
        
         
         
         
         
         
o___ ____ ______o   o________ ______o 
Q   Q   Q  Q 
Q   Q   Q  Q 
Q o  Qo  Q  Q 
Q Q  Q Q  Q Q  Q Q 
Qo________________oQ Q o________________oQ
Figure 3: Some canonical projections on P3 and P4.
10 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
Now each AB 2 Pr,s(n + 1)is an (n  1)face of Pn+1 homeomorphic to PrxPs,
so choose a homeomorphism ffiAB : PrxPs ! AB. In singular permutahedral sets,
ffiAB
face operators pullback along the cellular projection Pn r,s!Pr x Ps ! AB and
degeneracy operators pullback along the cellular projections
ffi, fij : Pn ! Pn1,
where ffi identifies the faces i_n_\ i_and n_\ i_i_, 1 i n  1, and fij i*
*dentifies
the faces jn_\ j and n_\ jj, 1 j n. Note that ff1 = fi1 and ffn1 = 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, dAB, %i, &j), where
SingPn+1Y = {continuous mapsPn+1 ! Y }, n 0,
face operators
dAB : SingPn+1Y ! SingPnY
are defined by
dAB(f) = f O ffiAB O r,s
for each AB 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 PfPfi132
i i i P P P
i i i 1,1 ffi21 1,2 PPq
P2 _____ P1 _____ P1 x P1 _____P2 _____P1 x P2 _____P3P3
ff1 ffi123
 H H J 
 H H d21d123d1234(f)Jd123d1234(f)  2,2
 H H J ?
 H H H J P2 x P2
&1d21d123d1234(f) = H H J 
 H H JJ^ d1234(f) ffi1234
 d132d1234(f) H HHj ?
___________________________________________Y __________oeP4
f
Figure 4: The quartic relation &1d21d123d1234= d132d1234.
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) dAB,
AB2Pr,s(n+1)
where sgn(A; B) denotes the sign of the shuffle. Note that if f 2 C*(SingP4Y )
and d132d1234(f)6= 0, the component d132d1234(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,n1) 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 nonnormalized 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 AlexanderWhitney 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
dAB : Pn+1 ! Pn
for each AB 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, dAB, %i,i&js a permutahedral set if the following structure relations hold:
For all ABC 2 P***(n + 1)
(4) dAt_BAt_CdAB[C = dA__tCB__tCdA[BC.
For all AB 2 Pr,s(n + 1)and CD 2 P**(n)\ Qsr(n)
(5) dCDdAB = &jdMN dKLdAB where
12 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
8
>>>KL = n_Ø (r_\ D)r_\ D,
>>>MN = C__t(r_\ D) (DØ (r_\ D))_t(r_\ D),
>> or
>>>KL = r_\ Cn_Ø (r_\ C),
>>>
>:MN = (r_\ C)t_(CØ (r_\ C)) (r_\ C)t_D,
j = @ (r_\ C)when r 2 D.
For all AB 2 P**(n + 1)and 1 < j < n (for j = 1, n see (7) below)
(6) 8
>><1, ifA = j_orB = j_,
&jdjn_Øj, ifAB 2 Qj(n + 1), A 6= j_orB 6= j_,
dAB&j = > _ _ n+1j
>:&j1dj1_n_Øj1_,ifAB 2 Q (n + 1), A 6= j_orB 6= j_,
&j&jdMN dKL, ifAB =2Qn+1jj(n + 1)where
8 __ __
>>>KL = At j_\ B  BØ j_\ B t j_\ B ,
>>>MN = @ j \ A n__1_Ø@ j \ A ,
>> or
>>>KL = j\ A t j\ B  j \ A tB,
>>> _ __ _ _ __
>:MN = j__1_n__1_Øj__1_,
when j 2 B.
For all AB 2 P**(n + 1)and 1 i n + 1
æ
(7) dAB%i= 1,% ifA = {i}or B = {i},
jdCD,where
8
>>>CD = In+2_Øi(AØi)In+2_Øi(B),
> or
>>>CD = 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, d0AB, &0i, %0j} and P 00= {Ps00, d00AB, &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 , dAB, &i, %j
with face and degeneracy operators defined by
8 i j
>>< d0r (a), b, ifAB 2 Qs(n),
i _\Ar_\B j
(9) dAB(a, b) = > a, d00 (b),ifAB 2 Qr(n),
>: s_\(An+s) s_\(Bn+s)
&idMN dKL (a, b), otherwise, where
8
>>>KL = r_\ A (r_\ B)[ s__1_+ r
>>>MN = (r_\ A)t_(BØ (r_\ B)) (r_\ A)t_B
>> or
>>>KL = A [ (BØ (r_\ B))r_\ B
>>> __ __
>:MN = 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,
, ir+1(b)r i n;
æ 0
(11) %j(a, b)= %j(a),ab,, %001 j r,
jr+1(b),r < j n + 1.
Remark 2.1. Note that the righthand side of the third equality in (9) reduces *
*to
the first two; indeed, if r 2 B, then KL 2 Qs(n) and MN 2 Qr(n); if r 2 A,
KL 2 Qs(n) and MN 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 KL 2 Qs(n), MN 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= s1(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= s1(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 dM1M2 by
______ ______
dM1M2(~a) = d0M2(a). d1M1(a), M1M2 2 P*,*(n + 1),
while for other elements of Q~cand for decomposables in 0Q use formulas (5)(7)
and (9) to define dM1M2 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, dM1M2, &i, %j).
In particular, we have the following identities:
_____
din+1_\i(_a)= d1i(a),1 i n,
_____
dn+1_\ii(_a)= d0i(a),1 i n.
Thus, for a 1reduced 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 ncube 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 ndimensional 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 ncube is truncated at the minimal
vertex a0 = (0, ..., 0), then it is truncated along those n  1faces that cont*
*ained a0,
and continuing so the last truncation is along those 1faces (edges) of the nc*
*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 onetoo*
*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 (nonem*
*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 ith e*
*lement
of n_; so that it resembles the simplicial operator @i1. We have the onetoone
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~qj+1), Bj = A~ \ (M~qj+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 4face u of B9 corresponding to 38]14679 2 P02,2,1,2(A~), one gets
u = d38]14679(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 Bn1
has been constructed, obtain Bn as a subdivision of Bn1 x I in the following w*
*ay:
_Face_of_Bn____Label__________

en1n,0 dn1_]n

en1i,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 mcell 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 dM1M2 as dM1M2(em ) = ep x dM1M2(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 dM1M2 reflect
the associativity of the partition procedure, while the relations between di an*
*d ei
ther dA]M or dM1M2 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! Bp1,qn1,i 2 p_,
dA]M : Bp,qn! Bpr,q+r1n1,A]M 2 P0pr,r(p),
dM1M2 : Bp,qn! Bp,q1n1,M1M2 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; dM1M2, &i, %j}q 0; p+q=n
is a permutahedral set and
didj = dj1di, i < j,
didA]M = dA\j]Mdj, j = I1A(i), i 2 p__r_,
didM1M2 = dM1M2di,
dM1M2dA]M = dA]M dM3M4, M1M2 2 Qr(q + r),
M3M4 = M1 + 1  r \ q_+_1_M2 + 1  r \ q_+_1_,
dM1M2dA]M = dA2]L2dA1]L1,A1L1 = A [ I1M(M1 \ r_) I1M(M2 \ r_),
A2L2 = A I1M(M1 \ r_), M1M2 62 Qr(q + r),
dijj = jjdi, i < j;
dijj = 1, i = j;
dijj = jjdi1, i > j;
dA]M jj = jidA1]M1, A1M1 = Ip+1_\j(A \ j) Ip+1_\j(M),
i = IA (j), j 2 A,
dA]M jj = %idA1]M1, A1M1 = 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_\ jj,
dM1M2jj = jjdM1M2,
diij = ijdi, i = &, %,
dA]M ij = ij+r1dA]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 : Bp1 x Pq+1 ! Bp x Pq+1, 1 i p,
f~fiA]M : Bpr x Pq+r 1xr,q+1!Bpr x Pr x Pq+1 ffiA]Mx1!Bp x P*
*q+1,
1 x ffiM1M2: 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_\jjand *
*dj.
Then for f 2 (SingB X)p,qndefine
di : (SingB Y )p,qn! (SingB Y )p1,qn1,
dA]M : (SingB Y )p,qn! (SingB Y )pr,q+r1n1,
dM1M2 : (SingB Y )p,qn! (SingB Y )p,q1n1,
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,
dM1M2(f) = f O (1 x ffiM1M2),
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 , dM1M2, 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 dM1M2,
where the summation is over all i 2 n_, A]M 2 P0**(p) and M1M2 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 nonnormalized 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 1reduced cubical set and P =
(Pn+1, dM1M2, &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,
dM1M2#(a) = #d0M2(a) . #d1M1(a),M1M2 2 P*,*(n),a 2 Qn,
%i#(a) = #ji(a), i 2 n_.
Note that since the first condition above we in particular get
din_\i#(a)= #d1i(a),i 2 n_,
dn_\ii#(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 1reduced 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 1reduced cubical set and P =
(Pn+1, dM1M2, &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 , dM1M2, 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),
dM1M2(a, b)= (a, dM1M2(b)), M1M2 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 , dM1M2, 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 ! Mn1, n 1. Then the twisted Cartesian product Qx#M
can be thought of as a permutocubical resolution of a 1reduced 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 1reduced 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
 
 
 
21  
s s 
 ffi ;]21 B2 
 ____________;]2_  
P   
2   
s _________________ss
12 @I ;]12 ;]2
@ 
@ #U 
 '
@ 
@ ;]1 ? ;]
@ ________________ss
@  
@  
 
@  
 
 I2 
 
 
 
_________________ss
;]12 ;]2
Figure 5: The universal truncating twisting function #U .
d2
_____________rr
 
d  
2]1 
r 
 B2 d1
 
B1 d;]12 
______________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;]21  Q
 bb  Q
32r1 r;]3r2br_____________Qr;]21;]12
b ;]3bb1b  
 bbr312  b rbbr;]312 
P3 23r1  ;]231 ;]3 
 r ;]1br__3_______rQ______rr;]32;]132B3
21r3 132 ;b]br213 QQ 
Q QQr_______________bb_____________rrQQ
123 ffi;]3_ ;]123 ;]23
@@I 
@ #U 
'
@ 
@ ? ;]
;]1b___________QQ_rr
 b  Q
 b b  Q
 b _____________Qrr;]2;]12
  
   
  ;]3  I3
;]1b___________rrQ3 
b b  QQ 
b b _____________rrQQ
;]123 ;]23
Figure 7: The universal truncating twisting function #U .
b____________QQrrd2
 b pr  Q
d bb d3  d1Q
r23]rdbr_____________Qr1
bb b 3]12 
 b rbbr  
d2]13  d13]2 
br_________rQ______rrdB3;]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 ndimensional polytope X is realized as a subdivision of the
cube In so that each mdimensional 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 (nonordered) 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 = 12345 and vy =
21435.
Now let x = ;]a1...ank, 0 k n, be a vertex of Bn, i.e., the set {a1, .*
*.., ank}
is the same as dik. .d.i1(n_) with {a1, ..., ank} = n_\ A0, A0 = {ik < . .<.i1*
*}.
For the sequence x0 = {a1, ..., ank} 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 = ;]21365, then (ux, vx) = (4]12356 , 2]1365).
Next for a partition a = A0]A1...A` of an ordered (finite) set we define the
rightshift R and the leftshift 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\ MiAi+1[ Mi . ..A`forminMi> max Ai+1,
LNj(a) = A0]A1 . ..Aj1[ NjAj\ Nj . ..A`forminNj > max Aj1,
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, q1L(m2) = c12 . . .cq12 mq2.
Then
(16) ~((a1 m1) (a2 m2)) =
X p q1 p*
*+1 q
(1)ffla1Ep,q(#(c11), . .,.#(c1); a2, #(c12), ..., #(c2 )) m1*
* m2,
p 0; q 1
ffl = mp+11(a2 + c12 + . .+.cq12).
Corollary 8.1. Let F ! E i!Z be the fibration associated with Gfibration G !
E0ß!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, ..., bq1) [~ap+1 . ..~an] O*
* [~bq . ..~bm],
p 0; q 1
ffl = (~ap+1 + . .+.~an)(b0 + ~b1 + . .+.~bq1).
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 1reduced 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)aE1,1(a; db) = (1)aab  (1)a(b+1)b*
*a,
so it measures the noncommutativity 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 Galgebras [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
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A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze
st., 1, 380093 Tbilisi, Georgia
Email address: kade@@rmi.acnet.ge
A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze
st., 1, 380093 Tbilisi, Georgia
Email address: sane@@rmi.acnet.ge