A CUBICAL MODEL FOR A FIBRATION
TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
Abstract.In the paper the notion of a truncating twisting function from a*
* simplicial set to a
cubical set and the corresponding notion of twisted Cartesian product of *
*these sets are introduced.
The latter becomes a cubical set. This construction together with the the*
*ory of twisted tensor
products for homotopy G-algebras allows to obtain multiplicative models f*
*or fibrations.
1.Introduction
For a fibration F ! E ! Y on the tensor product C*(Y ) C*(F) E. Brown [8] h*
*as introduced
a twisted differential dø such that the homology of the cochain complex (C*(Y )*
* C*(F), dø) is
isomorphic to the cohomology H*(E) but just additively. So there arises the pro*
*blem of introducing
of an associative multiplication on that complex to describe H*(E) as an algebr*
*a too. In this way
various multiplicative models were constructed in which either the associativit*
*y was abolished or
a differential was not a derivation (see, for example, L. Lambe and J. Stasheff*
* [21] for references).
The standard notion of a twisting function ø : X* ! G*-1from a simplicial set*
* to a simplicial
group and the notion of corresponding twisted Cartesian product X xøG does not *
*allow to intro-
duce directly a desired comultiplication on the twisted tensor product C*(X) C*
**(ø)C*(G) of the
simplicial chain complexes since it does not coincide with the simplicial chain*
* complex C*(X xøG).
The situation radically changes if we replace simplicial group G by a monoidal *
*cubical set and suit-
ably modify the notion of a twisting function.
This idea comes from recent results of N. Berikashvili: In [5] a multiplicati*
*ve model with asso-
ciative multiplication in the case when the fiber F is the cubical version of t*
*he Eilenberg-MacLane
space is constructed; in the next paper [6] a multiplicative model C*(Y ) OEC**
*2(F), OE : C*2(G) !
C*+1(Y ) is constructed where C*(Y ) is the singular simplicial cochain complex*
* of the base and
C*2(G) and C*2(F) are the singular cubical cochain complexes of the structure g*
*roup and the fiber
respectively.
In this paper we begin to develop the general theory of twisting functions to*
* form twisted
Cartesian products of abstract sets of different kind. The continuation will fo*
*llow in [19] where
twisted functions from cubical sets to permutahedral sets will be considered.
Here we introduce the notion of a truncating twisting function ø : X* ! Q*-1w*
*here X is
a 1-reduced simplicial setSand Q is a monoidal cubical set (that is there is gi*
*ven a cubical map
Q x Q ! Q with (Q x Q)n = p+q=nQpx Qq). For a cubical set L with a given Q-act*
*ion such a
twisting function ø determines the twisted Cartesian product X xøL being a cubi*
*cal set.
To present an universal example of a truncating twisting function we construc*
*t a functor as-
signing to a simplicial set X a monoidal cubical set X together with the canon*
*ical truncating
twisting function ø : X ! X in such a way that any truncating function ø0: X*!*
* Q*-1factors
through it, that is, ø0: X ø-! X ! Q where the second map is a monoidal cubical*
* map.
__________
1991 Mathematics Subject Classification. Primary 55R05, 55P35, 55U05, 52B05, *
*05A18, 05A19 ; Secondary
55P10.
Key words and phrases. Simplical set, cubical set, truncating twisting functi*
*on, twisted Cartesian product, cobar
construction, homotopy G-algebra.
This research described in this publication was made possible in part by Awar*
*d No. GM1-2083 of the U.S.
Civilian Research and Development Foundation for the Independent States of the *
*Former Soviet Union (CRDF)
and by Award No. 99-00817 of INTAS.
1
2 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
The twisted Cartesian product PX = X xø X is a cubical set functorially depe*
*nding on
X. Note that X models the loop space |X| and PX models the path space fibrat*
*ion on
|X|. Moreover, the normalized chain complex C2*( X) coincides with Adams' cobar*
* construction
C*(X) and C2*(PX) coincides with the acyclic cobar construction (C*(X); C*(X)*
*).
Applying the chain functor to ø : X* ! Q*-1we obtain a twisting cochain ø* = *
*C*(ø) :
C*(X) ! C2*-1(Q) and then C2*(X xø L) coincides with the twisted tensor product*
* C*(X) ø*
C2*(L). Furthermore, the cubical structure of X xø L evidently determines on C2*
**(X xø L) =
C*(X) ø*C2*(L) an associative comultiplication. Dually on C*(X) ø*C*2(L) C**
*2(X xøL) (we
have here the equality too provided the graded sets in question are of finite t*
*ype) one immediately
obtains a desired multiplication.
From computational point of view the next question is how to express this (co*
*)multiplication
in terms of tensor factors. It appears that the translation of the cubical diag*
*onal of C2*(X xøL)
describes the comultiplication on C*(X) ø*C2*(L) by canonical higher order cha*
*in operations of
degree k
Ek,1: C*(X) -!C*(X) k C*(X), k 0,
on C*(X), which agree with operations constructed by Baues [2], [3] on the norm*
*alized complex
CN*(X), and by the action C2*(Q) C2*(L) -!C2*(L) involving also ø*. Interesti*
*ngly, ø* takes
place in both definition of a twisted differential and of a twisted comultiplic*
*ation !
Moreover, the cooperation E1,1is the dual operation of Steenrod's cochain ^1-*
*operation and
all {Ek,1} operations introduce on C*(X) the structure of a homotopy G-coalgebr*
*a just being the
dual notion of a homotopy G-algebra in the sense of [12].
Of course, algebraically a reason of difficulty in obtaining the associative *
*multiplication on
twisted tensor products was hidden in the non-commutativity of the Alexander-Wh*
*itney diagonal.
The classical tool which measures the lack of commutativity, Steenrod's ^1-prod*
*uct, as we now
can clearly see, is not enough to compensate it. The above fact justifies the s*
*tructure of a homotopy
G-algebra as a good notion for a dga with ^1. Note also that the specification *
*of this structure
on a dg (co)algebra A is equivalent to a dg Hopf algebra structure on the (co)b*
*ar construction
( A)BA.
We develop the theory of multiplicative twisted tensor products for homotopy *
*G-algebras in the
following sense. In [24] it is shown that if a twisting element OE : C ! A from*
* a dg Hopf algebra
C to a commutative dga A is coprimitive, that is if the induced map C ! BA is a*
* map of dg
Hopf algebras (where on BA the standard shuffle multiplication is meant), and M*
* is dga being
also a comodule over C then the twisted tensor product A OEM is a dga with res*
*pect to standard
multiplication of the tensor product A M of dga's.
Now we replace the commutative A by a homotopy G-algebra. By definition this*
* structure
determines on BA a strictly associative multiplication converting BA into a dg *
*Hopf algebra (this
multiplication can be viewed as a perturbation of the shuffle one ). 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 associative multiplication ~OEin terms of OE and h*
*omotopy G-algebra
structure of A by the same formulas as in the case A = C*(X), C = C*2(Q) and M *
*= C*2(L). It
remains only to remark that ø* : C*2(Q) -!C*(X) provides a basic example of a m*
*ultiplicative
twisting element.
Thus, the above theory unifies commutative and homotopically commutative case*
*s, in particular,
the cases of singular and Sullivan de-Ram cochain complexes for topological spa*
*ces.
Now for a fibration F ! E ! Y on a 1- connected space Y associated with a pri*
*ncipal G-
fibration G ! E0! Y by an action G x F -!F we obtain a combinatorial model, bei*
*ng a cubical
set, in the following way. Let X = Sing1Y SingY be the Eilenberg 1-subcomplex*
* generated by
singular simplices sending the 1-skeleton of the standard n-simplex n to the b*
*ase point of Y , and
let Q = SingIG and M = SingIF. We have that Adams' map !*: C*(Y ) = C*( X) ! C*
*2*( Y )
is in fact realized by a monoidal cubical map ! : X ! SingI Y. Composing this *
*map with the
map of monoidal cubical sets SingI Y ! SingIG = Q induced by the canonical map *
* Y ! G of
monoids we immediately obtain a truncating twisting function ø : X ! Q.
A CUBICAL MODEL FOR A FIBRATION 3
The resulting twisted Cartesian product Sing1Y xøSingIF just provides the req*
*uired cubical
model of E: there exists a cubical weak equivalence Sing1Y xø SingIF ! SingIE. *
*Applying
cochain functor we obtain the above multiplicative twisted tensor product [6]. *
*Note also that in
this multiplicative model it could be introduced Steenrod's (co)chain operation*
*s as they are defined
for cubical sets [16].
By using the standard triangulation of the cubes one obtains a map of dg Hopf*
* algebras
C*N(G) ! C*2(G) and then it is possible to obtain again multiplicative twisted *
*tensor product
C*(Y ) ø0*C*N(F) but now with respect to the multiplicative twisting cochain ø*
*0*: C*N(G) !
*
C*2(G) ø-!C*(Y ). In other words, for a special twisting cochain one introduces*
* on Brown's model
a required multiplication.
Finally, we mention that the geometric realization | Sing1Y | of Sing1Y is h*
*omeomorphic to
the cellular model for the loop space observed by G. Carlsson and R. J. Milgram*
* [9]. In [2], [3]
H.-J. Baues has defined a geometric coassociative and homotopy cocommutative di*
*agonal on the
cobar construction CN*(Y ) by means of a certain cellular model for the loop s*
*pace (homotopically
equivalent to | Sing1Y |) the cellular chains of which coincide with the CN*(Y*
* ); consequently,
one obtains a homotopy G-coalgebra structure on CN*(Y ). Another modification o*
*f Adams' cobar
construction is considered by Y. Felix, S. Halperin and J.-C. Thomas [10].
The paper is organized as follows: Section 2 contains some background materia*
*l; in Section 3 we
construct two functors and P from the category of simplicial sets to the cate*
*gory of cubical sets,
so in fact here we construct the universal twisted Cartesian product PX = X xø *
*X; based on this
universal example we introduce the notion of a truncating twisting function and*
* the corresponding
twisted Cartesian product in the next Section 4; in Section 5 we build the cubi*
*cal set model for
the path space fibration; in Section 6 a cubical model and the corresponding mu*
*ltiplicative twisted
tensor product for a fibration are constructed, and, finally, in the Section 7 *
*the twisted tensor
product theory for homotopy G-algebras is developed containing an application t*
*o Brown's model
too.
2.Notation and preliminaries
Let R be a commutative ring with unit 1. A differential graded algebra (dga*
*) is a graded
R-module C = {Ci}, i 2 Z, with an associative multiplication ~ : Ci Cj -! Ci+j*
*and a
homomorphism (a differential ) d : Ci -! Ci+1with d2 = 0 and satisfying the Lei*
*bniz rule
d(xy) = d(x)y + (-1)|x|xd(y), where xy 2 Ci+jis the element ~(x y), x 2 Ci, y*
* 2 Cj, |x| = i.
We assume that a dga contains a unit 1 2 C0. A non-negatively graded dga C is c*
*onnected if
C0 = R. A connected dga C is n-reduced if Ci = 0, 1 i n. A dga is commutati*
*ve if ~ = ~T,
where T(x y) = (-1)|x||y|(y x).
A differential graded coalgebra (dgc) is a graded R-module C = {Ci}, i 2 Z, w*
*ith an coassocia-
tive comultiplication : C -! C C and a homomorphism (a differential ) d : C*
*i -!Ci-1
with d2 = 0 and satisfying d = (d 1 + 1 d) . A dgc C is assumed to have a *
*counit
ffl : C -!R, (ffl 1) = (1 ffl) = 1. A non-negatively graded dgc C is con*
*nected if C0 = R. A
connected dgc C is n-reduced if Ci= 0, 1 i n. A dgc is cocommutative if =*
* T.
A (connected) differential graded Hopf algebra (dgha) (C, ~, ) is a connecte*
*d dga (C, ~) and
a connected dgc (C, ) simultaneously such that : C -!C C is an algebra map.
A dga M is a (left) comodule over a dgha C if : M -!C M is a dga map.
2.1. Cobar and Bar constructions. For an R-module M let T(M) = 1i=0Ti(M),
Ti(M) = M i where T0(M) = R. Denote the tensor product of elements aj2 M by [a1*
*| . .|.an] 2
Tn(M). By s-1M we denote the desuspension of M, i.e. (s-1M)i= Mi+1.
Let (C*, dC, ) be a 1-reduced dgc. Let 0: C -! C C pr-!C>0 C>0. The (red*
*uced) cobar
construction C on C is the tensor algebra T(C~), ~C= s-1(C>0), with differenti*
*al d = d1+ d2
defined for ~c2 ~C>0by ____
d1[~c] = -[dC(c)]
4 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
and X 0 X
d2[~c] = (-1)|c[|~c0|c~00] for 0(c) = c0 c00,
and extended as a derivation.
The acyclic cobar construction (C; C) is the twisted tensor product C C in *
*which the tensor
differential is twisted by the canonical twisting map C -! C being an inclusion*
* of degree -1.
Let (A, dA, ~) be a 1-reduced dga. The (reduced) bar construction BA on A is*
* the tensor
coalgebra T(A~), A~= s-1(A>0), with differential d = d1+ d2 given for [~a1| . .*
*|.~an] 2 Tn(A~) by
nX _____
d1[~a1| . .|.~an] = - (-1)"i[~a1| . .|.dA(ai)| . .|.~an],
i=1
and n
X _____
d2[~a1| . .|.~an] = - (-1)"i[~a1| . .|.ai-1ai| . .|.~an],
P i=2
"i= j*0(*).
Clearly it is a 1-reduced dgc with respect to the AW diagonal. Let SingY be the*
* singular simplicial
complex of a topological space Y and X = Sing1Y SingY be the (Eilenberg) 1-su*
*bcomplex
generated by those singular simplices which send the 1-skeleton of the standard*
* simplex n, n
0, at the base point of Y. Let define the dgc C*(Y ) as C*(Y ) = C*(X). Then Ad*
*ams' cobar
construction C*(Y ) is the cobar construction of the dgc C*(Y ).
2.3. Cubical sets. A cubical set is a sequence of sets Q = {Qn}n 0 with boundar*
*y operators
dffli: Qn ! Qn-1, ffl = 0, 1, 1 i n, and degeneracy operators ji: Qn ! Qn+1*
*, 1 i n + 1,
satisfying the standard conditions [15].
An example of a cubical set is the singular cubical set SingIY = {SingInY }n *
*0of a space Y,
where SingInY is the set of all continuous maps In -!Y [22].
Analogously to a simplicial set for a cubical set Q and an R-module A its cha*
*in complex in
coefficients A is defined which will be denoted by (C~2*(Q; A), d). The normali*
*zed chain complex
(C2*(Q; A), d) of Q is defined as the quotient C2*(Q; A) = ~C2*(Q; A)=D*(Q), wh*
*ere D*(Q) is the
subcomplex of (C~2*(Q; A), d) generated by the degenerate elements of Q. Note *
*also that both
C~2*(Q) and C2*(Q) are DG-coalgebras with respect to the canonical comultiplica*
*tion determined
by the Cartesian product decomposition of the n-cube In = I x . .x.I[27]. For a*
* space Y we will
denote C2*(SingIY ; Z) by C2*(Y ).
The (tensor) product of two cubical sets Q and Q0is
[
Q x Q0= {(Q x Q0)n = Qpx Q0q}= ~
p+q=n
where (jp+1(a), b) ~ (a, j1(b)), (a, b) 2 QpxQ0q, and endowed with the obvious *
*face and degeneracy
operators [15].
A graded monoidal cubical set we define as a cubical set Q with a cubical map*
* ~ : Q x Q ! Q,
which is associative and has the unit e 2 Q0. Clearly, for a monoidal cubical s*
*et its chain complex
C2*(Q; R) is a dg Hopf algebra as well as C*2(Q; R).
For a graded monoidal cubical set Q a Q-module we define as a cubical set L t*
*ogether with
associative action Q x L ! L and the unit of Q acting on it as identity. In thi*
*s case C*2(L; R) is a
dga comodule over dg Hopf algebra (C*2(Q; R), d).
A CUBICAL MODEL FOR A FIBRATION 5
3.The cubical set functors X and PX
3.1. The cubical set functor X. First, for a simplicial set X = {Xn, @i, si}n *
*0, let define the
graded set 0X as follows. Let Xc be the graded set of formal expressions
Xcn+k= {jik. .j.i1ji0(x)| x 2 Xn}n 0;k 0,
where
i1 . . .ik, 1 ij n + j - 1, 1 j k, ji0= 1,
and let ~Xc= s-1(Xc>0) denote the desuspension of Xc. Then define 00X as the f*
*ree graded monoid
(without unit) generated by ~Xc. Elements of 00X we denote by ~x1.~.x.kfor xj2*
* Xcj, 1 j k.
The product of two elements ~x1.~.x.kand ~y1.~.y.`is defined by concatenation ~*
*x1.~.x.k~y1.~.y.`.
This only subject to the associativity relation and no other relations whatsoev*
*er between the
expressions. The total degree of an element ~x1.~.x.kis the sum m(k)= m1+ . .+.*
*mk, mj= |~xj|,
and we write ~x1.~.x.k2 ( 0X)m(k).
Let 0X be the monoid obtained from 00X via
0X = 00X/~ ,
______ ____
where jp+1(x). ~y~ ~x. j1(y)for x, y 2 Xc, |x| = p + 1. Clearly, we have the in*
*clusion MX 0X
of graded monoids where MX denotes the free monoid generated by ~X= s-1(X>0).
It appears that 0X canonically admits the structure of a cubical set. Namely*
*, denote by
i: Xn -!Xix Xn-i, i(x) = @i+1. .@.n(x) x @0. .@.i-1(x), 0 i n,
the components of AW diagonal, and let xn denote an n-simplex, i.e. xn 2 Xn. Th*
*en
i(xn) = ((x0)i, (x00)n-i) 2 Xix Xn-i
for_all n 0. Define_the face operators d0i, d1i: ( 0X)n-1-! ( 0X)n-2on a (mon*
*oidal) generator
xn 2 (X~)n-1 (Xc)n-1by
__ ____ ______
d0i(xn) = (x0)i. (x00)n-i,i = 1, ..., n - 1,
__ _____
d1i(xn) = @i(xn), i = 1, ..., n - 1,
and extend to elements ~x1.~.x.k2 MX by
_____ _________
d0i(~x1.x.~.k) = ~x1.(.x.0q)jq. (x00q)mq+1-jq.~.x.k,
______
d1i(~x1.x.~.k) = ~x1.@.j.q(xq).~.x.k,
where m(q-1)< i m(q), jq= i - m(q-1), 1 q k, 1 i n - 1.
Then for d0i, d1ithe defining identities of a cubical set can be easily check*
*ed on MX. In particular,
the simplicial relations between @i's imply the cubical relations between d1i's*
*; the associativity
relations between i's imply the cubical relations between d0i's, and the commu*
*tativity relations
between @i's and j's imply the cubical relations between d1i's and d0j's.
Before we extend the face operators on the whole__0X define a degeneracy oper*
*ator ji :
( 0X)n-1-! ( 0X)n on a (monoidal) generator _x2 (Xc)n-1by
____
ji(_x) = ji(x)
and extend to elements ~x1.~.x.k2 0X by
ji(~x1.x.~.k) = ~x1.j.j.q(__xq) . .~.xk,
jn(~x1.x.~.k) = ~x1.~.x.mk-1. jmk+1(__xk),
where m(q-1)< i m(q), jq= i - m(q-1), 1 q k, 1 i n - 1.
Finally, inductively extend the face operators on these degenerate elements s*
*o that the defining
identities of a cubical set are satisfied. It is easy to see that the cubical s*
*et { 0X, d0i, d1i, ji} depends
functorially on X.
It is convenient to verify the above cubical relations by the following combi*
*natorics of the
standard cube (compare, [4]).
6 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
Remark 3.1. Motivated by the combinatorial description of the standard (n+1)-si*
*mplex n+1we
denote the set {0, 1, ..., n + 1} by [0, 1, ..., n + 1] to assign to whole In. *
*Next for the face operators
d0i$ x1, ..., xi-1, 0, xi+1,i...,=xn,1, ..., n
d0i$ x1, ..., xi-1, 1, xi+1,i...,=xn,1, ..., n
(here faces of In are written in the Euclidean coordinates) put the corresponde*
*nce
d0i$ [0, 1, ..., i][i,i...,=n1+,1],..., n
d1i$ [0, 1, ...,^-, ...,in=+11],, ..., n.
In general, any q-dimensional face of In is expressed as a sequence of blocks
[0, i1..., ik1][ik1, ..., ik2][ik2, ..., ik3]...[ikp-1, ..., ikp, n+1], 0 < i1*
*< . .<.ikp< n+1, q = kp-p+1.
In this combinatorics a cubical degeneracy operator jiis thought of as to add*
* a formal element
* to the set [0, 1, ..., n + 1] at the i + 1-th place: ji[0, 1, ..., n + 1] = [*
*0, 1, ..., i - 1, *, i, ..., n + 1] with
the following convention: [0, 1, ..., i-1, *][*, i, ..., n+1] = [0, 1, ..., n+1*
*] that guarantees the equality
d0iji= 1 = d1iji. Geometrically this just agrees with the standard projections *
*In+1! In.
____
Now suppose X has a fixed vertex *. Then declare s0(*)as a unit_of_ 0X and de*
*note it by e.
Moreover, for each x 2 Xn, n > 0, we put the relation jn(~x) = sn(x). Let ( X, *
*d0i, d1i, ji) be the
resulting (unital) graded monoidal cubical set.
In particular, for a 1-reduced simplicial set X we will have the following id*
*entities
__ ____ ______ ______ ______ _____
d01(xn) = (x0)1. (x00)n-1= e . (x00)n-1= (x00)n-1= @0(xn),
__ ______ ____ ______ ______ _____n
d0n-1(xn) = (x0)n-1. (x00)1= (x0)n-1. e = (x0)n-1=x@n(xn),2 Xn.
Thus, all the face operators @iof X are included in the definition of X.
Remark 3.2. Note that in the definition of X we have to add formally the degen*
*eracies, since
simplicial degeneracies are not applicable unless the last one. This is also ju*
*stified by the geometrical
fact that in the path space fibration a degenerate singular n-simplex of base l*
*ifts to a singular (n-1)-
cube of the fibre which need not to be degenerate in general (cf. the proof of *
*Theorem 5.1).
3.2. The cubical set functor PX. First, for a simplicial set X weSdefine the cu*
*bical set P0X as
follows. Consider the Cartresian product Xcx 0X = {(Xcx 0X)n = p+q=nXcpx ( 0X*
*)q} of
the graded sets Xc and 0X (ignoring for the moment the underlying structures).*
* Let
Xcex 0X = Xcx 0X= ~,
where (jp+1(x), y) ~ (x, j1(y)), (x, y) 2 Xcpx ( 0X)q. Then introduce on Xcex 0*
*X the face oper-
ators d0i, d1iand the degeneracy operators jias follows. For an element (x, y) *
*2 Xpx ( 0X)q
Xcpx ( 0X)q, p + q = n, let
( 0 i-1 ________
00)p+1-i.1y), i p,
d0i(x, y) = ((x ) , (x
(x, d0i-p(y)), p < i n,
(
d1i(x, y) = (@i-1(x), y), 1 i p,
(x, d1i-p(y)), p < i n,
ji(x, y) = (ji(x),1y), i p,
ji(x, y) = (x, ji-p(y)),p < i n + 1.
It is easy to check that the face operators satisfy the canonical cubical ident*
*ities. These data
uniquely extend to the structure of a cubical set on whole Xcex 0X. The resulti*
*ng cubical set is
denoted by P0X, and then obtain the cubical set PX from P0X by replacing 0X by*
* X.
A CUBICAL MODEL FOR A FIBRATION 7
It is convenient to verify the cubical relations in P0X by the following comb*
*inatorics of the
standard cube.
Remark 3.3. To P0X it corresponds the following combinatorial model of the stan*
*dard cube
(compare Remark 3.1). Now denote the set {0, 1, ..., n + 1} by 0, 1, ..., n + 1*
*] and assign to whole
In+1, while assign to its any proper q-face a sequence of blocks
j1...js1][js1...js2][js2...js3]...[jst-1...jst, n + 1], 0 j1< . .<.jst<*
* n + 1, q = st- t.
In particular the dimension of the block 0...j] is greater by one than the di*
*mension of the block
[0...j]. The face and degeneracy operators are acting on blocks as in Remark 3.*
*1, but for the first
block we have the following correspondence
d0i$ j1...ji-1][ji-1...js1],1 i < s1,
d1i$ j1...bji...js1],1 i < s1.
Obviously one has the inclusion of graded sets X -!PX defined by y -!(*, y),*
* * 2 X0, and
the projection PX -,!X defined by (x, y) -!x. On the other hand, the canonical *
*cellular map
_ : In+1-! n+1([27]) admits the following combinatorial description
j1...js1][js1...js2][js2...js3]...[jst-1...jst, n + 1] -!{j1...j*
*s1}
(see, Fig. 1); in particular, the face 0][0, 1, ..., n + 1] of In+1, i.e. d01(I*
*n+1), is mapped onto the
minimal vertex (the base point) 0 2 n+1. So that _ can be thought of as a comb*
*inatorial model
for the projection ,.
|s0 0____________|ss3
| || |
| | |
0 || | ||
|s | 0 ____________|||ss1|
| | ________- | | | | 0123]
[0123] | | | | | |
| |s0 | 0 |____|_____|ss2
| | |
| | |
| | |
| | |
|s @@I |___________|ss
0 @ 0 1
@ ø |
@ |
|_
@ |
@ |?
@ s3
@ jjQ|Q|
j j || Q Q
0 j_j_____s||_____QQj_s2
QQQ | j j
Q Q |jj| 0123
Q j|s
1
Figure 1: The universal truncating twisting function ø.
8 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
4.Truncating twisting functions and twisted Cartesian products
It appears that the cubical set PX can be viewed as a twisted Cartesian produ*
*ct determined
by the inclusion function ø : X -! X, x -!~x, being of degree -1. It will be re*
*ferred to as the
universal truncating function. Here we give the general formalism for descripti*
*on such functions.
Definition 4.1. Let X be a 1-reduced simplicial set and Q be a monoidal cubical*
* set. A sequence
ø = {øn}, øn : Xn -!Qn-1, n 1, of degree -1 functions is a truncating twistin*
*g function if it
satisfies:
ø(x) = e, x 2 X1,
d0iø(x) = ø@i+1. .@.n(x) . ø@0.1.@.i-1(x), xi2 Xnn-,1,n 1
d1iø(x) = ø@i(x), 1 i n - 1,x 2 Xn, n 1
jnø(x) = øsn(x), x 2 Xn, n 1.
Remark 4.1. Note that by definition a truncating twisting function commutes onl*
*y with the last
degeneracy operators (compare [27]), since it is so for the universal truncatin*
*g function.
Proposition 4.1. Let X be a 1-reduced simplicial set and Q be a monoidal cubica*
*l set. A sequence
ø = {øn}n 1of degree -1 functions øn : Xn -!Qn-1is a truncating twisting functi*
*on if and only
if the monoidal map f : X -!Q defined by f(~x1.x.~.k) = ø(x1) . .ø.(xk) is a m*
*ap of cubical sets.
Given a truncating twisting function ø and a cubical set L with cubical map Q*
* x L -!L the
corresponding twisted Cartesian product X xøL is defined by replacing X by L i*
*n the definition
of PX. Namely, we get the following
Definition 4.2. Let X be a 1-reduced simplicial set, Q be a monoidal cubical se*
*t and L be a
Q-module Q x L -!L. Let ø = {øn}n 1, øn : Xn -!Qn-1, be a truncating twisting f*
*unction. The
twisted Cartesian product X xøL is the graded set
X xøL = XcxL= ~,
where (jp+1(x), y) ~ (x, j1(y)), (x, y) 2 Xcpx Lq, and endowed with face and de*
*generacy operators
d0i, d1i, and jidefined for (x, y) 2 Xpx Lq Xcpx Lq by
8
><(@1. .@.p(x), ø(x) . y),i = 1,
d0i(x, y) = >(@i. .@.p(x), ø@0. .@.i-2(x)1. 0,
with the following properties:
(i) Ep,qis of degree 1 - p - q;
(ii) Ep,q= 0 except E1,0= 1 = E0,1and Ek,1, k 1;
(iii) The homomorphism E : BA BA ! A defined by
E([~a1| . .|.~ap] [~b1| . .|.~bq]) = Ep,q(a1, ..., ap; b1, ..*
*., bq)
is a twisting element in the dga (Hom(BA BA, A), r), i.e. satisfies rE = -E *
*^ E (this
condition implies that the comultiplicative coextension ~E : BA BA ! BA is a *
*chain map);
(iv) The multiplication ~E is associative, i.e. it turns BA into a dg Hopf alge*
*bra.
Entirely dually one can formulate the notion of a homotopy G-coalgebra.
The conditions (iii) and (iv) can be rewritten in terms of components Ep,q(se*
*e [12]). In particular
the operation E1,1satisfies the conditions similar to that of Steenrod's ^1 pro*
*duct: the condition
(iii) gives
dE1,1(a; b) - E1,1(da; b) + (-1)|a|E1,1(a; db) = (-1)|a|ab - (-1)|a|(|*
*b|+1)ba,
so it measures the non-commutativity of the product of A (thus, a hga with Ek,1*
*= 0 for k 1
is just a commutative dga). We denote E1,1(a, b) = a ^1b. This notation is just*
*ified also by the
other condition which follows from (iii):
c ^1ab = (c ^1a)b + (-1)|a|(|c|-1)a(c ^1b). (8)
This formula means that the map a ^1 - : A ! A is a derivation which in the cas*
*e of C*(X) is
called as the Hirsch formula. As for the map - ^1 c : A ! A, it follows from (i*
*ii) that it is a
derivation only up to homotopy and for the suitable homotopy it just serves the*
* operation E2,1:
dE2,1(a, b; c)-E2,1(da, b; c) - (-1)|a|E2,1(a, db; c) - (-1)|a|+|b|E2,1(a*
*,(b;9dc))=
(-1)|a|+|b|ab ^1c - (-1)|a|+|b||c|(a ^1c)b - (-1)|a|+|b|a(b ^*
*1c).
Main examples of hga's are: C*(X) (see [2], [3],[13] and previous section) an*
*d the Hochschild
cochain complex of an associative algebra where E1,1and E2,1were defined by Ger*
*stenhaber in [11]
and the higher operations were described in ([17], [13]). One more example is t*
*he cobar construction
of a dg Hopf algebra [18]. Note also that certain algebras (including polynomia*
*l ones), which are
realized as the cohomology of topological spaces, admit a non-trivial hga struc*
*ture too [26].
Remark 7.1. Note that the E2,1measures also the lack of associativity of E1,1=^*
*1; in particular,
condition (iv) yields
a ^1(b ^1c) - (a ^1b) ^1c = E2,1(a, b; c) + (-1)(|a|+1)(|b|+1)E2,1(b, a*
*; c).
14 TORNIKE KADEISHVILI AND SAMSON SANEBLIDZE
The last condition implies that the commutator [a, b] = a ^1 b - (-1)(|a|+1)(|b*
*|+1)b ^1 a satisfies
the Jacobi identity. Together with (8) it implies on the H(A) a Lie bracket of *
*degree -1. Besides (8)
and (9) imply that [a, -] : H(A) ! H(A) is a derivation, so H(A) becomes a Gers*
*tenhaber algebra
[11]. Note that this notion is not a particular case of hga. We have that the i*
*nduced Gerstenhaber
algebra structure on H(C*(X)) = H*(X) is trivial since of the existence of ^2 p*
*roduct.
7.1. Multiplicative twisted tensor products. Let C be a dgc, A be a dga and M b*
*e a dg
comodule over C. Brown's twisting element ø : C ! A determines the following ma*
*ps: a dga map
(the multiplicative extension of ø) fø : C ! A, a dgc map (the comultiplicativ*
*e coextension of
ø) gø : C ! BA and the twisted differential dø = d 1 + 1 d + ø\- : A M ! *
*A M.
Suppose now that C is a dg Hopf algebra and M is a dga in addition with M ! C*
* M beeng a
dga map. In general dø is not a derivation with respect to the multiplication o*
*f the tensor product
A M. But if A is a commutative dga (in this case BA is a dg Hopf algebra with*
* respect to
the shuffle product ~sh) and gø : C ! BA is a map of dg Hopf algebras, then the*
* twisted tensor
product A øC is a dga ([24]).
Suppose now that A is a homotopy G-algebra. In this case BA is again a dg Hop*
*f algebra with
respect to the multiplication ~E.
Definition 7.2. A twisting element ø : C -!A in Hom(C, A) we call multiplicativ*
*e if the comul-
tiplicative coextension C -!BA is an algebra map.
It is clear that if ø : C -!A is a multiplicative twisting element and if g :*
* B -!C is a map of
dg Hopf algebras then the composition øg : B -!A is again a multiplicative twis*
*ting element.
The canonical projection BA -!A provides an example of the universal multipli*
*cative element.
For a commutative dga A one has the equality ~E = ~sh, so Proute's twisting e*
*lement is
multiplicative (see, for example, [25]).
We have that the argument of the proof of formula (6) immediately yields
Theorem 7.1. Let ø* : C -!A be a multiplicative twisting element. Then the tens*
*or product A M
with the canonical twisting differential dø*= d 1 + 1 d + ø*\- becomes a dg*
*a (A M, dø*, ~ø*)
with the twisted multiplication ~ø*determined by formula (6).
Thus the above theorem includes the twisted tensor product theory for commuta*
*tive algebras
([24]).
Corollary 7.1. For a homotopy G-algebra A the acyclic barconstruction B(A; A) b*
*ecomes a dga
with the twisted multiplication determined by formula (7).
7.2. Brown's model as a dga. Now it is possible to replace in Corollary 6.1 the*
* cubical cochains
by the simplicial ones to introduce on Brown's model an associative multiplicat*
*ion. Indeed, let
C*N(F) denote the normalized singular simplicial cochain complex of F.
Corollary 7.2. Let F -!E i-!Y be a fibration as in Corollary 6.1. Then on the t*
*ensor product
C*(Y ) C*N(F) there are both twisted differential dø0and the multiplication ~ø0*
*, with ø0: C*N(G) !
C*(Y ) being multiplicative twisting element, such that (C*(Y ) C*N(F), dø0, *
*~ø0) is a dga with
cohomology algebra isomorphic to H*(E).
Proof.It is sufficient to observe that the map ' : C*N(G) -! C*2(G) induced by *
*triangulation
of the cubes (see, for example, [10]) is a map of dgha's, so that the compositi*
*on of ' 1 with
: C*N(F) -! C*N(G) C*N(F) defines on C*N(F) a comodule structure over the d*
*gha C*2(G).
Then we take ø0= ø*'.
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A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidz*
*e st., 1, 380093
Tbilisi, Georgia
E-mail address: kade@rmi.acnet.ge
A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidz*
*e st., 1, 380093
Tbilisi, Georgia
E-mail address: sane@rmi.acnet.ge
*