Non-formality of planar configuration spaces in characteristic two

0.21 MB·

by  Paolo Salvatore

NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN
CHARACTERISTIC TWO
7
1 PAOLOSALVATORE
0
2

b
e
F Abstract. LetFk(R2)betheorderedconfigurationspaceofkdistinctpoints
in the plane. We prove that its singular cochain algebra with coefficients in
3 Z2 isnotformalasadifferentialgradedalgebrafork≥4.

]
T 1. Introduction
A
The notion of formality of a topological space X is usually introduced in the
.
h senseofrationalhomotopytheoryviasomecommutativedifferentialalgebramodel
t in characteristic 0 of X, like the Sullivan-deRham algebra A (X), or the alge-
a PL
m bra of differential forms if X is a manifold. However one can define formality
over any coefficient ring R in the non-commutative context, by requiring the al-
[
gebra of R-valued cochains C∗(X,R) to be quasi-isomorphic to the cohomology

2 H∗(X,R). This means that C∗(X,R) is connectedto its cohomologyH∗(X,R) by
v
a zig-zag of homomorphisms of differential graded associative R-algebras inducing
6
isomorphisms in cohomology. If R is a field then this property depends only on
1
8 the characteristic of R. If R is a field of characteristic 0 then the formality in the
6 associativesenseisequivalenttotheusualcommutativeformalitybyarecentresult
0 of Saleh [15].
.
1 Let us consider the euclidean configuration spaces
0
F (Rn)={(x ,...,x )∈(Rn)k|x 6=x for i6=j}.
7 k 1 k i i
1 KontsevichandLambrechts-Volichaveprovedtheirformalityincharacteristiczero.
:
v
Theorem 1.1. [10, 11] The configuration space F (Rn) is formal over R for any
i k
X k,n.
r
a The case of the configurationspaces in the plane Fk(R2) had been provedmuch
earlier by Arnold [1].
Whatcanbesaidaboutformalityovertheintegers,orinpositivecharacteristic?
It is not difficult to prove the following result.
Theorem 1.2. The configuration space F (Rn) is intrinsically formal over Z if
k
n≥k.
This means that any space with the same cohomology ring as the configuration
space is formal over Z. The case n < k is open in general. A special case is
F (R2)≃S1×(S1∨S1) that is formal over Z.
3
Geoffroy Horel has announced a proof of Z -formality of F (Rn) for any n > 2
p k
even, using ´etale cohomology.
One might think that formality over Z holds for all configuration spaces as in
the rational case.
1
2 PAOLOSALVATORE
However we found the following surprising result:
Theorem1.3. TheconfigurationspaceF (R2)isnotformaloverZ foranyk ≥4.
k 2
This immediately implies non-formality over the integers.
Corollary 1.4. F (R2) is not formal over Z for any k ≥4.
k
WeapproachthisquestionbyobstructiontheoryfollowingtheworkbyHalperin-
Stasheff [9] for cdga (commutative differential graded algebras) in characteristic 0
andits extensionto the non-commutativecaseby ElHaouari[7]. We implemented
the computation of the non-trivial obstruction in MATLAB. The relevant library
is attached.
In a follow up paper we will deal with configuration spaces in R3.
The paper is organized as follows: in section 2 we recall the presentation of
the cohomology rings of the configuration spaces and of their Koszul dual, the
Yang Baxter algebras. In section 3 we define formality of an algebra and recall
the obstructiontheory to it by Halperin-Stasheff via bigradedmodels. In section 4
we describe the Barratt-Eccles-Smith simplicial model for euclidean configuration
spaces. In section 5 we apply the obstruction theory and prove that F (R2) is
k
not formal over Z for k ≥ 4. In section 6 we interpret the obstruction class in
2
Hochschildcohomology. Finally in the appendix we describe the attachedsoftware
in MATLAB.
2. Cohomology of configuration spaces and Yang-Baxter algebras
We recall the presentation of the cohomology ring of the configuration spaces
F (Rn). Then we describe its Koszul dual, the Yang-Baxter algebra, and its geo-
k
metric interpretation.
Consider the direction map π : F (Rn) → Sn−1 from the i-th to the j-th
i,j k
particle given by π (x ,...,x )=(x −x )/|x −x |.
i,j 1 k j i j i
Let ι∈Hn−1(Sn−1) be the standard fundamental class. We write
A =π∗ (ι)∈Hn−1(F (Rn)).
ij i,j k
The following computation of the cohomology ring is due to Arnold in the case
n=2 and Fred Cohen for n>2.
Theorem 2.1. (Arnold, Cohen) [1,5] The cohomology ring H∗(F (Rn)) for n≥2
k
has a presentation with (n−1)-dimensional generators A for 1 ≤ i 6= j ≤ n and
ij
relations
(1) A =(−1)nA
ij ji
(2) A2 =0
ij
(3) (Arnold) A A +A A +A A =0 for i6=j 6=k 6=i
ij jk jk ki ki ij
Thus up to the grading this ring depends only on the parity of n.
Corollary 2.2. The cohomology groups H∗(F (Rn)) are torsion free, and have a
k
graded basis provided by the set
{A ···A |j <···<j , i <j ∀t}
i1j1 iljl 1 l t t
NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 3
Corollary 2.3. The Poincare series of the configuration space is
k−1
P(F (Rn))= (1+mxn−1)
k
m=1
Y
Notice that F (Rn) is the total space of a tower of fibrations with fibers homo-
k
topicto∨ Sn−1 form=1,...,k−1. Thusthe additivestructuredoesnotseethe
m
twisting of the fibrations, but the cohomology ring does.
AremarkablefactisthatthecohomologyoftheconfigurationspacesisaKoszul
algebra ( see example 32 in [3]) . The following holds with coefficients in Z.
Proposition 2.4. The Koszul dual algebra of H∗(F (Rn)) for n≥2 is the Yang-
k
Baxter algebra YB(n) generated by classes B of degree n−2, for 1 ≤ i 6= j ≤ k
k ij
under the relations
(1) B =(−1)nB
ij ji
(2) (Yang-Baxter)
[B ,B ]=[B ,B ]=[B ,B ] for i6=j 6=k 6=i
ij jk jk ki ki ij
(3) [B ,B ]=0 when {i,j}∩{r,t}=∅
ij rt
Clearly this algebra up to the grading depends only on the parity of n.
Proposition 2.5. [6] The Yang-Baxter algebra is torsion free and has a graded
basis
{B ···B |j ≤···≤j , i <j ∀t}
i1j1 iljl 1 l t t
The Yang-Baxter algebra has the following geometric meaning for n > 2: it is
the Pontrjagin ring of the loop space of the configuration space F (Rn). Notice
k
that under this hypothesis F (Rn) is (n−2)-connected.
k
Let A∗ ∈ H (F (Rn)) be the homology basis dual to A . By the Hurewicz
ij n−1 k ij
theorem and adjointness we have isomorphisms
Hn−2(ΩFk(Rn))∼=πn−2(ΩFk(Rn))∼=πn−1(Fk(Rn))∼=Hn−1(Fk(Rn))
Let
B′ ∈H (ΩF (Rn))
ij n−2 k
be the class corresponding to A∗ under the composite isomorphism.
ij
Theorem 2.6. (Cohen-Gitler, Fadell-Husseini) [6, 8] There is an isomorphism of
algebras
H∗(ΩFk(Rn))∼=YBk(n)
for n>2 sending B′ to B .
ij ij
Corollary 2.7. The Poincare series of ΩF (Rn) for n>2 is
k
k−1
P(ΩF (Rn))= (1−mxn−1)−1
k
m=1
Y
This follows also from the homotopy equivalence
k−1
ΩF (Rn)≃ Ω(∨mSn−1)
k 1
m=1
Y
that is not multiplicative in general.
4 PAOLOSALVATORE
For n = 2 the (ungraded) algebra YB = YB(2) has also a similar geometric
k k
meaning that we explain. Let us start from the following well known result.
Proposition 2.8. The configuration space F (R2) is a classfying space of the pure
k
braid group on k strands Pβ .
k
Now consider the descending central series of Pβ defined by
k
G =Pβ and G =[G ,G ].
1 k i 1 i−1
The direct sum of the subquotients forms a Lie algebra
L =⊕ (G /G )
k i i i+1
under the bracket induced by taking commutators. Let U denote the universal
enveloping algebra functor.
Theorem 2.9. [6] There is an isomorphism YBk ∼=ULk sending Bij to the Artin
generator A of the pure braid group.
ij
3. Formality and bigraded models
We present the definition of formality in rational homotopy theory and in the
non-commutativesense. Wethenproceedtodescribethenon-commutativeversions
of the bigraded models by Halperin and Stasheff, due to El Haouari.
Definition 3.1. [9] A topological space X is rationally formal if there is a zig-
zag of quasi-isomorphisms of commutative differential graded algebras connecting
theSullivan-deRhamalgebraAPL(X)toitscohomologyH∗(APL(X))∼=H∗(X,Q)
equipped with the trivial differential.
If X is a manifold this is equivalent to the existence of a zig-zag of quasi-
isomorphismsbetweenthealgebraofde-RhamformsΩ∗(X)anditsrealcohomology
H∗(X,R). IfX isacomplexmanifoldwemightusetheringofcomplexdifferential
formsandcohomologywithcomplexcoefficients. Namelythenotionofformalityin
characteristic zero does not depend on the field for connected spaces of finite type
(Theorem6.8in [9]). The same is true in positive characteristic,by a similar proof
following the obstruction theory in [7].
Kontsevich [10] and Lambrechts-Volic [11] have proved that the configuration
space F (Rn) is formal over R for any k and n (Theorem 1.1 ).
k
This approach uses the commutative algebra of real PA forms and proves also
the formality of the little discs operads.
Arnold had easily proved the formality of F (C) [1]. His quasi-isomorphism
k
embeds H∗(F (C),C) into the algebra of complex differential forms Ω∗(F (C))
k C k
sending A to d(z −z )/(z −z ).
ij i j i j
We turn now to the non-commutative case.
Definition 3.2. LetR be acommutativering. AtopologicalspaceX is R-formal,
or formal over R, if the algebra of singular cochains C∗(X,R) is connected to its
cohomology H∗(X,R) by a zig-zag of quasi-isomorphisms of differential graded
R-algebras.
For spaces of finite type with torsion free homology Z-formality is universal as
it implies R-formality for any ring R.
NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 5
Here is an application of formality to computations: let ΩX and LX be respec-
tivelythe basedandthe unbasedloopspaceofX. LetHH denote the Hochschild
∗
homology functor from differential graded R-algebras to graded R-modules.
Proposition3.3. Let X bea simply connectedR-formal spaceof finitetype. Then
there are isomorphisms of graded R-modules
TorH∗(X,R)(R,R)∼=H∗(ΩX,R)
HH∗(H∗(X,R))∼=H∗(LX,R)
ThisfollowsfromthecollapseoftherelevantEilenberg-Moorespectralsequences.
ThetheoryofR-formalityisstudiedbyElHaouari[7],whoextendstheobstruc-
tion theory by Halperin-Stasheff to the non-commutative case. The key point is
the existence of a bigraded model that is then deformed to an actual model. A
bigraded module V = ⊕Vn has an upper dimensional grading, the degree, and a
k
lower grading, the resolution level. The tensor algebra TV on a bigraded module
inherits a bigrading by additivity. We work in the category of cochain differential
graded algebras over a commutative ring R with upper grading, such that the dif-
ferentialishomogeneousofdegree1. AbigradedR-algebrahasadifferentialthatis
alsohomogeneouswithrespecttothelowergrading,andlowersitby1. Thetensor
algebraon a bigradedmodule inherits a bigradingas well, and its cohomologytoo.
We saythataR-DGAAisconnectedifH0(A)=R,generatedbythe unitofA.
Proposition 3.4. (2.1.1.of [7]) Let R be a field. Given a connected graded R-
algebra H, there exists a bigraded R-module V and a differential d on the tensor
algebra T(V) together with a quasi-isomorphism
ρ:(T(V),d)→(H,0)
such that the homology in positive resolution level vanishes:
H (T(V),d)=0.
+
Moreover V = H and ρ| is the identity. The algebra (T(V),d) is unique up to
0 V0
isomorphism and is called the bigraded model of H.
Remark 3.5. Proposition 3.4 works also over a commutative ring R when Hi and
Tori (R,R) are free R-modules for each i. In particular it holds for R = Z and
H
H =H∗(F (Rn)).
k
Example3.6. InthecaseoftheconfigurationspacecohomologyH =H∗(F (Rn)),
k
the lower grading of the Yang-Baxter algebra YB(n) = H! is the length of words
k
in the standard generators. The suspended filtered dual V = s(YB(n))∗ generates
k
the bigradedmodel (T(V),d)=B(YB(n))∗ thatis the filtered dualofthe barcon-
k
struction. Inotherwordsthedifferentialisthederivationinducedbythecoproduct
d : V −µ→∗ V ⊗V ⊂ T(V) dual to the multiplication µ of YB(n). The resolution
k
map Ψ:(T(V),d)→H is then chosen by
Ψ(B∗)=A ,
ij ij
Ψ((B ...B )∗)=0 ∀l >1
i1j1 iljl
We do not need to consider filtered duals for n > 2 since in that case YB(n) is
k
of finite type, i.e. finite dimensional in each degree.
6 PAOLOSALVATORE
A bigraded module A has an induced filtration level by F (A)=⊕ A∗ . We
k i≤k i
say that the differential lowers filtration level by h if d(F (A))⊂F (A).
k k−h
The key idea of Halperin-Stasheff is to deform the bigraded model of the coho-
mology H(A) of a DGA A in order to get a model for the algebra itself.
Theorem 3.7. (2.2.2 of [7]) Let A be a connected DGA over a field R . Let
(T(V),d) be the bigraded model of H∗(A). Then there is a differential D on T(V)
and a quasi-isomorphism
π :(T(V),D)−→A
such that D −d lowers filtration level by 2. If π′ : (T(V),D′) → A is another
solution to this problem, then there exists an isomorphism
φ:(T(V),D)−→(T(V),D′)
such that φ−id lowers filtration level by 1, and π′◦φ ≃π. Any solution is called
a model of A.
The following result completes the theory, compare Theorem 5.3 in [9].
Proposition 3.8. Let A,B be connected DGA’s over a field R with an isomor-
phism φ¯ : H(A) ∼= H(B). Let (T(V),DA),(T(V),DB) be the respective models.
Then A and B are quasi-isomorphic if and only if there exists an isomorphism
φ:(T(V),D )→(T(V),D ) such that φ−id lowers filtration.
A B
We give a first immediate application.
Definition3.9. AspaceX isintrinsicallyR-formalifitisR-formal,andanyother
space with the same cohomology R-algebra is also R-formal.
Theorem 3.10. The configuration space F (Rn) is intrinsically Z-formal if n≥k.
k
Proof. We can assume n >2 since intrinsic formality of S1 ≃F (R2) is easy. The
2
generators of the bigraded model (T(V),d) for H =H∗(F (Rn),R) are in degree
k
n−1,2n−3,...,1+i(n−2),...
since V ∼=s(YBn)∗, and respective level
0,1,2,3,...,i−1,...
Considerthe model (T(V),D) for the singular cochainalgebraC∗(Y,R) of a space
Y withH∗(Y,R)∼=H. WehavethatD−dlowersthefiltrationby2. Allgenerators
are in degree 1 mod n−2, andso their differentials are in degree 2 mod n−2. For
a generator v of level i−1, and thus degree 1+i(n−2), a homogeneous non-zero
monomial can be a term of (D −d)(v) only if it has degree 2 mod n−2. But
quadratic monomials v v , with v ∈ V and b ∈ V , have to be ruled out
a b a a−1 b−1
because they would have degree 2+i(n−2)=[1+a(n−2)]+[1+b(n−2)], thus
filtration level (a−1)+(b−1)= i−2, contradicting that D−d lowers filtration
by two. The monomial is then the product of at least n generators, that is in
degree at least n+n(n −2) = 2+ (n+1)(n −2), so i > n. Then d = D on
V ≤ n. The top cohomology of the configuration space (and Y) is in dimension
(k−1)(n−1) < n(n−1), so there is no obstruction to extending the restriction
(T(V ),D)→H inductively to a quasi-isomorphism (T(V),D)→H.
≤n
(cid:3)
NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 7
A special case not covered by the theorem is that of F (R2) ≃ S1×(S1∨S1),
3
which is Z-formal , since wedges and products preserve formality [9].
In order to prove the non-formality of F (R2) (Theorem 1.3) over Z for k > 3
k 2
we introduce simplicial versions of these configuration spaces in the next section.
4. The Barratt-Eccles complex
We recall the Smith simplicial model for planar configuration spaces, obtained
by filtering the Barratt-Eccles complex [2]. For simplicity we write F := F (R2),
k k
and we adopt Z -coefficients throughout although we do not state it explicitly.
2
Definition 4.1. The full Barratt-Eccles complex on k elements is the geometric
bar construction of the symmetric group on k letters WΣ .
k
This is a simplicial set with (WΣ ) = (Σ )l+1, so any element is a string of
k l k
permutations. Thefaceoperatorsdeleteasinglepermutationinthestring,andthe
degeneracies double a permutation in the string.
The symmetric group Σ acts diagonally levelwise.
k
In particular WΣ has 2 non-degenerate simplices in each degree that are
2
(12|21|12|..), (21|12|21|..).
Consider the sub-simplicial set F (WΣ )⊂ WΣ spanned by non-degenerate sim-
t 2 2
plices of dimension at most t−1.
Thegeometricrealization|F (WΣ )|hasthehomotopytypeofthe(t−1)-sphere.
t 2
We can extend the filtration to any k >2 as follows.
There are simplicial versions of the projections seen in section 2, that we still
denote
π :WΣ →WΣ , for 1≤i6=j ≤k.
ij k 2
Levelwise these maps are powers of the function π :Σ →Σ defined by
ij k 2
(12) if σ−1(i)<σ−1(j)
π (σ)=
ij ((21) if σ−1(i)>σ−1(j).
Definition 4.2. The t-th stage F (WΣ ) of the filtration of the Barratt-Eccles
t k
complex is defined by
F (WΣ ) = π−1(F (WΣ ) )
t k l ij t 2 l
i,j
\
Proposition 4.3. [2] The geometric realization of the simplicial set F (WΣ ) has
t k
the homotopy type of the configuration space F (Rt) , and the realization of π can
k ij
be identified up to homotopy to the projection π : F (Rt) → St−1 considered in
ij k
section 2.
LetN∗ denotethenormalizedcochainfunctorfromsimplicialsetstodifferential
graded algebras (over Z in our case).
2
Definition 4.4. The Barratt-Eccles DG-algebra with complexity t and arity k is
E∗(k):=N∗(F (WΣ )).
t t k
8 PAOLOSALVATORE
Here k is the arity of the Barratt-Eccles operad, that is equal to the number of
pointsintheconfiguration. Theupperindexindicatesthedegree. Weareinterested
mainly in t=2.
Observe that each Ei(k) is a free module over Z [Σ ], so that k! divides its di-
t 2 k
mensionoverZ (thisholdsforanycoefficientring). Forfixedtthetopdimensional
2
non-trivial vector space is in degree (t−1) k .
2
To get a flavour we present some cases of the generating polynomial
(cid:0) (cid:1)
Ptk(x)= dimZ2(Eti(k))xi
i
X
P2(x)=2(1+x)
2
P3(x)=6(1+5x+6x2+2x3)
2
P4(x)=24(1+23x+104x2+196x3+184x4+86x5+16x6)
2
P2(x)=2(1+x+x2)
3
P3(x)=6(1+5x+25x2+60x3+70x4+38x5+8x6)
3
P4(x)=24(1+23x+529x2+5550x3+30214x4+97048x5+...)
3
Let us denote the elements of the symmetric group Σ by the following letters:
3
A=(123),B =(132),C =(213),D=(231),E =(312),F =(321).
Then E2(3) is the free Z [Σ ]-module on the 6 simplices
2 2 3
(A|D|F)∗, (A|E|F)∗, (A|C|F)∗, (A|C|D)∗, (A|B|F)∗, (A|B|E)∗,
hence it is 36-dimensional over Z .
2
By Proposition4.3 we have quasi-isomorphic DGA’s E∗(k)≃C∗(F (R2)).
2 k
Proposition 4.5. Non-formality of E∗(k) for k≥4 is equivalent to Theorem 1.3.
2
Consider the 1-cochain
ω =π∗((12|21)∗)∈E1(k).
ij ij 2
For example for k = 3 ω is the sum of 9 generators, and for k = 4 of 144
ij
generators.
Consider the 1-cochain in E1(3)
2
Ar :=(F|E)∗+(E|F)∗+(E|D)∗+(E|C)∗+(E|A)∗+(E|B)∗+(D|E)∗+(C|E)∗+(A|E)∗.
Lemma 4.6. A resolution of the Arnold relation for 3 points on the cochain level
is given by
d(Ar)=ω ω +ω ω +ω ω
13 12 23 12 23 13
Proof. Let us compute the pairing < d(Ar),X >=< Ar,∂(X) > for basis genera-
tors X ∈ E2(3)∗. The cochain Ar is the sum of all generators of the form (E|x)∗
2
with E 6= x ∈ Σ , and of the form (x|E)∗ with x 6= B and x 6= E. Thus d(Ar)
3
vanishes on all generators not containing E, and also on chains (E|x|y), since in
complexity 2 there are no chains of the form (E|x|E), and
<Ar,∂(E|x|y)>=<Ar,(E|x)+(E|y)>=1+1=0 mod 2
If E is in the middle position, d(Ar) pairs non-trivially with chains of the form
(B|E|x). Since 1and3swapgoingfromB to E,only 2canmovewithrespectto 1
and 3, comparing E =(312) and x, hence the possibilities are x = D = (231) and
NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 9
x = F = (321). Finally d(Ar) pairs non-trivially on chains of the form (x|B|E),
and again only 2 can move from x to B so that x=A or x=C. This proves that
(4.1) d(Ar)=(B|E|D)∗+(B|E|F)∗+(A|B|E)∗+(C|B|E)∗.
The cochain ω ω pairs non trivially with chains (x|y|z) with (13) and (12) sub-
13 12
strings of x, (31) and (12) of y, and (31) and (21) of z. This forces x = (123) or
x=(132). In the first case y =(312) and z =(321). In the second y =(312), and
either z =(321) or z =(231), hence
(4.2) ω ω =(A|E|F)∗+(B|E|F)∗+(B|E|D)∗.
13 12
By similar considerations we obtain
(4.3) ω ω =(A|B|F)∗+(A|E|F)∗,
23 12
(4.4) ω ω =(A|B|E)∗+(A|B|F)∗+(C|B|E)∗.
23 13
Comparing the sum of equations 4.2, 4.3, 4.4 with equation 4.1 we conclude.
(cid:3)
5. The bigraded model of the Barratt Eccles complex
InthissectionwestartbuildingthebigradedmodeloftheBarratt-Ecclesalgebra
E∗(4) and verify its non-formality. Let (TW,D) be the model of E∗(4) of Theorem
2 2
3.7 , obtained by deforming the bigraded model (TW,d) ≃ H∗(F ), and equipped
4
with a quasi-isomorphismφ:(TW,D)→E∗(4).
2
By Theorem 3.7 d=D on W and W . We can characterize it explicitly.
0 1
Lemma 5.1. On W , D =d=0. On W , D =d is determined by
0 1
(B∗)2 if i=k andj =l
ij
B∗B∗ +B∗B∗ =[B∗,B∗] for j <l
d((B B )∗)= ij kl kl ij ij kl
ij kl Bi∗jBk∗l+Bi∗lBk∗i+Bk∗lBk∗i forj =landk <i
B∗B∗ +B∗B∗ +B∗ B∗ forj =landk >i
ij kl il ik kl ik
Proof. TheproductoftwogeneratorsBijBuv oftheYang-Baxteralgebraisastan-
dard basis generator if and only if j ≤ v. Instead, for j > v, the Yang Baxter
relations yield
B B if {i,j}∩{u,v}=∅
uv ij
B B =
ij uv
(BuvBij +BujBvj +BvjBuj otherwise
By dualizing the formula on basis generators we obtain the result.
(cid:3)
We shall show that d and D do not agree on W , and there is no “gauge equiv-
2
alence” transformation fixing the problem.
Following the construction ofthe bigradedmodel in [7, 9] we canassume that φ
sends each generator in W ∼=H1(F ) to a cocyle representative.
0 4
Definition 5.2. Let φ:W →E∗(4) be the linear map given by
0 2
φ(B∗)=ω .
ij ij
We move to the next set of generators in filtration 1
10 PAOLOSALVATORE
Definition 5.3. Let φ:W →E1(4) be the linear map given by
1 2
0 if (i,j)=(k,l)
ω ∪ ω if j <l
φ((B B )∗)= ij 1 kl
ij kl πk∗il(Ar) if i>k and j =l
π∗ (Ar)+ω ∪ ω if i<k and j =l.
ikl il 1 kl
We remark that the ∪ product of 1-cochains over Z has a simple form. Each
1 2
k-cochaincoverZ isdeterminedbyitssupportSupp(c),thesetofnon-degenerate
2
k-chains sent to 1 by c (rather than 0). Now for given 1-cochains c,c′ the relation
Supp(c∪ c′)=Supp(c)∩Supp(c′)completelydeterminesc∪ c′. Theconstruction
1 1
of ∪ goes back to Steenrod, and is presented by McClure-Smith in [12].
1
The homomorphism π : WΣ → WΣ is induced by the simplicial version of
kil 4 3
the forgetful projection from 4 to 3 configuration points that we describe below
(the projection to 2 points has been described in section 4).
We need a function π :Σ →Σ telling how a permutation twists the indices
kil 4 3
{k,i,l}. Consider the three symbols j = k,j = i,j = l. A permutation σ ∈ Σ
1 2 3 4
gives a sequence (σ(1),σ(2),σ(3),σ(4)) containing a unique ordered subsequence
(j ,j ,j ), where τ ∈ Σ . We set π (σ) = τ. The construction induces
τ(1) τ(2) τ(3) 3 kil
a map on the W-construction still denoted π : WΣ → WΣ , and then on
kil 4 3
normalized cochains
π∗ =N∗(π ):E∗(3)→E∗(4).
kil kil t t
Lemma5.4. Thehomomorphism ΦofDefinition5.2and5.3iscompatiblewiththe
differentials, and thus defines a homomorphism of DGA φ:(T(W ),d)→E∗(4).
≤1 2
Proof. We have that d((B2)∗)=(B∗)2 by lemma 5.1
ij ij
In the first case of 5.3
φ(d((B2)∗))=φ((B∗)2)=ω2 =0=d(φ((B2)∗)).
ij ij ij ij
Namely the top-dimensional 1-cocyle (12|21)∗ ∈ E1(2) squares to zero by dimen-
2
sionalreasons,andthenso does its pullback ω ∈E1(3). In the secondcaseof5.3
ij 2
by Lemma 5.1
d((B B )∗)=[B∗,B∗] if j <l, and
ij kl ij kl
φ(d((B B )∗))=φ([B∗,B∗])=ω ∪ω +ω ∪ω =d(ω ∪ ω )
ij kl ij kl ij kl kl ij ij 1 kl
by Steenrod’s formula [12]. By definition
d(ω ∪ ω )=d(φ((B B )∗)).
ij 1 kl ij kl
In the third case of 5.3 j =l and i>k we have from Lemma 5.1 that
d((B B )∗)=B∗B∗ +B∗B∗ +B∗ B∗
ij kl ij kl ij ki kj ki
and φ maps the right hand side to
ω ω +ω ω +ω ω =π∗ (d(Ar)) =d(π∗ (Ar))=dφ((B B )∗).
il kl il ki kl ki kil kil ij kl
In the last case of 5.3 for i<k we have the same formula
d((B B )∗)=B∗B∗ +B∗B∗ +B∗ B∗
ij kl ij kl ij ik kj ik
(we only swap i and k to have the term B in standard form), so
ik
φ(d((B B )∗))=ω ω +ω ω +ω ω =π∗ (d(Ar))+[ω ,ω ]=
ij kl il kl il ik kl ik ikl il kl