Iterated integral and the loop product
Iterated integrals and the loop product
Koichi Fujii
1 Introduction
The purpose of this paper is to describe string topology from the viewpoint of
Chen’s iterated integrals. Let M be a compact closed oriented d-manifold and
LM be the free loop space ofM , the set of unbased smooth maps from S1 toM .
Let H∗(LM) be the homology of the free loop space shifted by the dimension of
the manifold i.e. H∗(LM) = H∗+d(LM). Chas and Sullivan found the product
on H∗(LM) which they called loop product [1]:
Hp(LM)⊗Hq(LM)→ Hp+q(LM).
They showed that this product makes H∗(LM) an associative, commutative
algebra.
Merkulov constructed a model for this product based on the theory of iter-
ated integrals, especially of the formal power series connection [10]. He showed
that there is an isomorphism of algebras
H∗(LM) ∼= H∗(ΛM ⊗ R
where ΛM is the de Rham differential graded algebra of M and R
the formal completion of the free graded associative algebra generated by some
noncommutative indeterminates.
On the other hand, Chen showed that the cohomology of the free loop space
of the simply-connected manifold is isomorphic to the cohomology of the cyclic
bar complex of differential forms via Chen’s iterated integrals (see [5] or [8]):
H∗(LM) ∼= H
∗(C(ΛM)).
In this paper, we construct a model for the loop product based on the the-
ory of the cyclic bar complex. We define a complex Hom(B(ΛM),ΛM) and
its subcomplex Hom(B(ΛM),ΛM) so that the Poincaré duality induces the
isomorphism of vector spaces
H∗(Hom(C(ΛM),R)) ∼= H∗−d(Hom(B(ΛM),ΛM)).
We can define a product on Hom(B(ΛM),ΛM) which realizes the loop product.
http://arxiv.org/abs/0704.0014v1
Theorem 1.1. Let M be a compact closed oriented simply-connected manifold.
Assume that H∗(M) is of finite type. Let A be a differential graded subalge-
bra of ΛM such that H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an
isomorphism of associative, commutative algebras
H∗(LM) ∼= H∗(Hom(B(A), A)).
The product defined on H∗(Hom(B(A), A)) corresponds to the loop product un-
der the isomorphism.
The paper is organized in the following way. In section 2, we briefly review
Chen’s iterated integrals. In section 3, we give a construction of a complex
Hom(B(A), A), and discuss its properties. In section 4, we give a proof of
theorem 1.1. In section 5, we study the iterated integrals on the free loop space
of the non-simply-connected manifolds. In section 6, we describe a relation
between the product on Hom(B(A), A) and the Goldman bracket. In this paper,
all the homologies have their coefficients in the field of real numbers.
Acknowledgement: The author would like to thank Professor Toshitake Kohno
much for helpful comments and gentle support.
2 Chen’s iterated integrals
We briefly review Chen’s iterated integrals (see [5], or [8]). Let M be a finite
dimensional smooth manifold and let LM be the free loop space of M , that is
the space of all smooth maps from S1 to M . Let ∆k be the k-simplex
{(t1, · · · , tk) ∈ R
k | 0 ≤ t1 ≤ · · · ≤ tk ≤ 1}.
We have an evaluation map
Φk : ∆k × LM →M
defined by
Φk(t1, · · · , tk; γ) = (γ(t1), · · · , γ(tk)).
Then define Pk to be the composition
(Λ∗M)⊗k → Λ∗Mk
→ Λ∗(∆k × LM)
→ Λ∗−kLM
where p∗ is the integration along the fiber of the projection p : ∆k×LM → LM .
Given ω1, · · ·ωk ∈ Λ
∗M , the iterated integral
ω1 · · ·ωk
is a differential form on LM of total degree |ω1| + · · · |ωk| − k, defined by the
formula
ω1 · · ·ωk = (−1)
(k−1)|ω1|+(k−2)|ω2|+···+|ωk−1|+k(k−1)/2Pk(ω1, · · · , ωk).
3 Preliminaries
In this section, we give a construction of some complexes. Let A be an arbitrary
differential graded algebra in this section. Let A∨ denote the dual of A. The
bar complex of A, (B(A), dB), is defined by
B(A) = ⊕r≥0 ⊗
r sA,
dB(ω1, · · · , ωr) = −(−1)
(ω1, · · · , ωi−1, dωi, ωi+1, · · · , ωr)
−(−1)εi
(ω1, · · · , ωi−1, ωi ∧ ωi+1, ωi+2, · · · , ωr).
Here (sA)q = Aq+1 or Aq according as 0 ≤ q or 0 < q, and εi = deg(ω1, · · · , ωi).
We denote the totality of degree n elements by B(A)n. The coproductH
∗(B(A))
→ H∗(B(A)) ⊗H∗(B(A)) is defined by
(ω1, · · · , ωn) 7→
(ω1, · · · , ωi)⊗ (ωi+1, · · · , ωn).
Chen proved the following theorem.
Theorem 3.1 (Chen [5]). Let M be a simply-connected manifold and H∗(M)
be of finite type. Let A be a differential graded algebra of ΛM such that A0 =
R and H∗(A) ∼= H∗(ΛM) by the inclusion. Then there is an isomorphism of
coalgebras
H∗(B(A)) ∼= H
∗(ΩM)
given by
(ω1, · · · , ωn) 7→
ω1 · · ·ωn.
Let F pB(A) be a filtration of B(A) such that
F pB(A) = ⊕0≤r≤p ⊗
r sA.
Let Hom(B(A), A∨)n =
p+q=n Hom(B(A)p, A
q∨) and Hom(B(A), A∨) =
n Hom(B(A), A
∨)n. Its boundary is defined by
δϕ(ω1, · · · , ωr)(ω)
= ϕ(ω1, · · · , ωr)(dω) + (−1)
|ω|ϕ(dB(ω1, · · · , ωr))(ω)
− (−1)|ω|ϕ(ω2, · · · , ωr)(ω ∧ ω1)
+(−1)|ω|+εr−1(|ωr|+1)ϕ(ω1, · · · , ωr−1)(ω ∧ ωr).
Let us define the subcomplex of Hom(B(A), A∨), Hom(B(A), A∨), according
to the Chen’s normalization of the cyclic bar complex (see [4] or [8]). We define
Hom(B(A), A∨) to be the set of elements in Hom(B(A), A∨) which satisfy the
following equations for any ω, ωi ∈ A
>0 and f ∈ A0:
−ϕ(· · ·ωi−2, fωi−1, ωi, · · · )(ω) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )(ω)
+ϕ(· · · , ωi−1, df, ωi, · · · )(ω) = 0, 1 ≤ i ≤ r − 1,
−ϕ(ω1, · · · , ωr)(fω) + ϕ(fω1, · · · , ωr)(ω) + ϕ(df, ω1, · · · , ωr)(ω) = 0,
−ϕ(ω1, · · · , fwr)(ω) + ϕ(ω1, · · · , ωr)(fω) + ϕ(ω1, · · · , ωr, df)(ω) = 0.
It can be easily seen that it is isomorphic to the dual of the normalized cyclic
bar complex of A:
Hom(B(A), A∨) ∼= C(A)
Similarly, let Hom(B(A), A)n =
p−q=n Hom(B(A)p, A
q) and Hom(B(A), A)
n Hom(B(A), A)n. Its boundary is defined by
δϕ(ω1, · · · , ωr)
= (−1)|ϕ|−εrdϕ(ω1, · · · , ωr)− (−1)
|ϕ|−εrϕ(dB(ω1, · · · , ωr))
+(−1)|ϕ|−εrω1 ∧ ϕ(ω2, · · · , ωr)
−(−1)(|ωr|+1)(|ϕ|+1)ϕ(ω1 · · · , ωr−1) ∧ ωr.
We define Hom(B(A), A) to be the set of elements in Hom(B(A), A) which
satisfy the following equations for any ω, ωi ∈ A
>0 and f ∈ A0:
−ϕ(· · ·ωi−2, fωi−1, ωi, · · · ) + ϕ(· · · , ωi−1, fωi, ωi+1, · · · )
+ϕ(· · · , ωi−1, df, ωi, · · · ) = 0, 1 ≤ i ≤ r − 1,
−f ∧ ϕ(ω1, · · · , ωr) + ϕ(fω1, · · · , ωr) + ϕ(df, ω1, · · · , ωr) = 0,
−ϕ(ω1, · · · , fwr) + ϕ(ω1, · · · , ωr) ∧ f + ϕ(ω1, · · · , ωr, df) = 0.
The cup product on Hom(B(A), A) is defined by
ϕ1 ∪ ϕ2(ω1, · · · , ωr)
0≤i≤r
(−1)|ϕ1|(|ϕ2|+εr−εi)ϕ1(ω1, · · · , ωi) ∧ ϕ2(ωi+1, · · · , ωr).
Since δ(ϕ1 ∪ ϕ2) = δϕ1 ∪ ϕ2 + (−1)
|ϕ1|ϕ1 ∪ δϕ2, H∗(Hom(B(A), A)) becomes
an algebra. This product can be induced on H∗(Hom(B(A), A)).
The E1-term of their spectral sequences associated with the filtration F
pB(A)
can be calculated from the cohomology of A.
Proposition 3.2. There is an isomorphism of vector spaces
H∗(Hom(F
pB(A)/F p−1B(A), A∨)) ∼= Hom(⊗
psH(A), H(A)∨)
Proof. Let A be a differential graded subalgebra of A such that A
= Ap for p
> 1, A
= R and
A1 = dA0 ⊕A
There is an isomorphism of vector spaces
Hom(F qB(A)/F q−1B(A), A∨) ∼= Hom(F
qB(A)/F q−1B(A), A
Since A
= R, there is an isomorphism
H0(Hom(F
qB(A)/F q−1B(A), A
)) ∼= Hom(⊗sH(A), H(A)
Therefore we obtain the proposition.
4 Proof of Theorem 1.1
We give the proof of theorem 1.1 in this section. There is a differential graded
subalgebra of A, A, such that A
= R and H(A) ∼= H(A) by the inclusion. Then
we obtain the isomorphism of algebras
H∗(Hom(B(A), A)) ∼= H∗(Hom(B(A), A))
by proposition 3.2. Therefore it suffices to verify the theorem in the case A0 = R.
The following result is due to Chen.
Theorem 4.1 (Chen [5]). H∗(LM) ∼= H∗(Hom(B(A), A
Proof. We define ψ : C∗(LM)→ Hom(B(A), A
∨) by
ψ(σ)(ω1, · · · , ωn)(ω) =
π∗ω ∧
ω1 · · ·ωn.
Let FpC∗(LM) be a filtration of C∗(LM) such that
FpCr(LM) = { σ : ∆
r → LM | π ◦ σ = σ′ ◦ π′ for some σ′ ∈ Cq(M),
q ≤ p, π′ : ∆r → ∆q } .
Let {Erp,q} be the associated spectral sequence. Define a filtration of Hom(B(A), A
FpHom(B(A), A) = {f ∈ Hom(B(A), A
∨) | f(ω1, · · · , ωn)(ω) = 0, ∀ω ∈ A
≥p+1}.
It can be easily shown that ψ preserves the filtrations of C∗(LM) and Hom(B(A), A
On E2-level, the map
ψ : Hp(M)⊗Hq(ΩM)→ Hp(A
∨)⊗Hq(B(A)
is given by
σ1 ⊗ σ2 7−→
(ω1, · · · , ωn 7→
ω1 · · ·ωn)
Theorem 3.1 asserts that this is an isomorphism. Therefore we obtain the
theorem.
Lemma 4.2. H∗(Hom(B(A), A)) ∼= H∗−d(Hom(B(A), A
Proof. We define a chain map P : Hom(B(A), A)→ Hom(B(A), A∨) by
P (ϕ)(ω1, · · · , ωn)(ω) =
ω ∧ ϕ(ω1, · · · , ωn).
Define a filtration of Hom(B(A), A) by
FpHom(B(A), A) = {ϕ ∈ Hom(B(A), A) | ϕ(ω1, · · · , ωn) ∈ A
≥d−p}.
The map P preserves those filtrations. On E2-level, the map
P : Hd−p(A)⊗Hq(B(A)
∨)→ Hp(A
∨)⊗Hq(B(A)
is given by
ω ⊗ ϕ 7−→
ω ∧ τ
This is isomorphic and we obtain the lemma.
Proof of theorem 1.1. We can verify that H∗(LM) is isomorphic to
H∗(Hom(B(A), A)) as vector spaces by composing the maps in theorem 4.1 and
lemma 4.2. We can also verify that there is an isomorphism of associative,
commutative algebras. Indeed, the cup product of Hom(B(A), A) on E2-level
Hd−p(A)⊗Hq(B(A)
∨)⊗Hd−s(A)⊗Ht(B(A)
∨)→ H2d−p−s(A)⊗Hq+t(B(A)
is given by
a⊗ g ⊗ b⊗ h 7→ (−1)(d−p+q)(d−s)a ∧ b⊗ g · h,
where g · h satisfies
g · h(ω1, · · · , ωn) =
g(ω1, · · · , ωi)h(ωi+1, · · · , ωn).
Then the following theorem asserts that the loop product and the cup product
coincide on E2-level.
Theorem 4.3 (Cohen-Jones-Yan [6]). Let M be a simply-connected manifold.
Then {Erp,q} becomes an algebra and converges to H∗(LM) as algebras. On
E2-level, the product
µ : Hp(M ;Hq(LM))⊗Hs(M ;Ht(LM))→ Hp+q−d(M ;Hs+t(LM))
is given by
µ((a⊗ g)⊗ (b ⊗ h)) = (−1)(d−s)(p+q−d)(a · b)⊗ (gh)
where a ∈ Hp(M), b ∈ Hs(M), g ∈ Hq(ΩM), h ∈ Ht(ΩM), a · b is the intersec-
tion product and gh is the Pontryagin product.
Therefore we obtain the theorem.
5 The conjugacy classes of fundamental groups
Let π denote a fundamental group of a smooth manifold M and J denote an
augmentation ideal of the group ring of π, Rπ. Chen showed that the completion
of the fundamental group with respect to the powers of its augmentation ideal is
isomorphic to the dual of the 0-th cohomology of the bar complex of differential
forms via iterated integrals [3]:
Rπ/Jp ∼= H
0(B(A))∨
where A is a differential graded subalgebra of ΛM such that A0 = R and
H∗(A) ∼= H∗(M).
Based on this work, we study iterated integrals on the free loop space of the
non-simply-connected manifold. Let π̃ denote the set of conjugacy classes of π
and J̃p denote pr(Jp) where pr is the projection of Rπ onto Rπ̃.
Theorem 5.1. Let M be a smooth manifold and H∗(M) is of finite type. Let A
be a differential graded subalgebra of ΛM such that the map Hq(A)→ Hq(ΛM)
induced by the inclusion is isomorphic if q = 0, 1 and injective if q = 2. Then
there is an isomorphism of vector spaces
Rπ̃/J̃p ∼= H0(Hom(B(A), A
We give the proof of this theorem in this section. Let ∗ be a fixed point
in S1. In this section, let LM be a set of smooth maps from S1 to M which
are constant maps near ∗. Let ΩxM be a subspace of LM whose elements
send ∗ to x ∈ M . Let Diff(S1, ∗) denote diffeomorphisms of S1 which coincide
with identity map near ∗. We define α, β : ∆q → LM to be equivalent by a
reparameterization iff there is a smooth map τ : ∆q → Diff(S1, ∗) such that
β(ξ)(t) = α(ξ)(τ(t, ξ)), ∀(t, ξ) ∈ S1 ×∆q.
Let C∗(LM) be a chain complex having as a basis the totality of equiva-
lence classes of smooth simplexes of LM . Let C∗(ΩxM) be a chain complex
having as a basis the totality of equivalence classes of smooth simplexes of ΩxM .
C∗(ΩxM) becomes a noncommutative associative algebra as follows. The prod-
uct of σ1 and σ2 in C∗(ΩxM) is defined to be the path product or 0 according
as degσ1+degσ2 ≤ 1 or > 1. The augmentation ε : C∗(ΩxM) → R is given by
εσ = 1 or 0 according as degσ = 0 or > 0.
Let σ be a smooth simplex of M . Define for each σ
Cq(LM)(σ) = {
niτi ∈ Cq(LM) | π♯τi = σ}.
Cq(LM)(σ) becomes a noncommutative associative algebra. Let ε(σ) denote
the augmentation of Cq(LM)(σ), given by
niτi 7→
ni. Define a filtration
of Cq(LM)(σ) by
FpCq(LM) = (kerε)
p ⊕ (⊕σ:∆q→M (kerε(σ))
Proposition 5.2. The map ψp : FpCq(LM) → Hom(F
p−1B(A), A∨) given by
(ω1, · · · , ωp) 7→
π∗ω ∧
ω1 · · ·ωp
is well-defined, chain map and FpCq(LM) ⊂ kerψp.
Proof. The well-definedness can be verified by the following lemma which can
be verified as in proposition 1.5, proposition 4.1.1 [2], and in proposition 1.5.3
Lemma 5.3 (Chen). (1) If α and β ∈ C∗(LM) are equivalent by a reparame-
terization, then
ω1 · · ·ωn = β
ω1 · · ·ωn.
(2) If τ1, τ2 ∈ Cq(LM)(σ), then
(τ1 · τ2)
ω1 · · ·ωn =
ω1 · · ·ωi ∧ τ
ωi+1 · · ·ωn.
(3) If f ∈ Λ0M , then for any i
ω1 · · · fωi−1 · · ·ωn +
ω1 · · · fωi · · ·ωn +
ω1 · · ·ωi−1df ωi · · ·ωn = 0.
To verify FpCq(LM) ⊂ kerψp, it suffices to show (kerε(σ))
p ⊂ kerψp. Let
s denote the section of π, which sends points of M to the constant map. Take
(σ1 − s♯σ) · (σ2 − s♯σ) · · · · ·(σp − s♯σ) ∈ (kerε(σ))
p, where σ ∈ Cq(M) and
σi ∈ Cq(LM)(σ). Then
(σ1 − s♯σ) · (σ2 − sσ) · · · · · (σp − s♯σ)
π∗ω ∧
ω1 · · ·ωp−1
σ∗ω ∧ (σ1 − s♯σ)
ω1 · · · (σk − s♯σ)
∗1 · · · ∧ (σp − s♯σ)
Therefore we obtain the proposition.
Let C∗(M,x) denote a set of smooth simplexes ofM neighborhood of whose
vertices are at x in M . We define
C ⊗ sC⊗p = C∗(M,x)⊗ sC∗(M,x)
Here (sC∗(M,x))q = Cq+1(M,x) or 0 according as q > 0 or q ≤ 0. Its boundary
is given by the sum of the boundary on each complex. Let us construct a chain
map Φ : C ⊗ sC⊗p → FpC∗(LM)/Fp+1C∗(LM) considering the following three
cases:
case 1: If (σ1, · · · , σp) ∈
sC(M,x)⊗p
, then
Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · · · · (σp − x)
where x is regarded as a constant map.
case 2: If (σ1, · · · , σp) ∈
sC(M,x)⊗p
, then
Φ : (σ1, · · · , σp) 7−→ (σ1 − x) · (σ2 − x) · · ·σi · · · (σp − x)
where σi : ∆
1 ∋ ξ 7→ σi(ξ)(t) ∈ ΩxM is
σi(ξ)(t)
σi((1 − ξ)((1 − t)v0 + tv2) + ξ(1 − 2t)v0 + 2ξtv1), if 0 ≤ t ≤ 1/2
σi((1 − ξ)((1 − t)v0 + tv2) + ξ(2 − 2t)v1 + ξ(2t− 1)v2), if 1/2 ≤ t ≤ 1
Here v0, v1, v2 are the vertices of the standard simplex ∆
case 3: If (γ, σ1, · · · , σp) ∈ C1(M,x)⊗
sC(M,x)⊗p
, then
Φ : (γ, σ1, · · · , σp) 7−→ γ
t (σ1 − x)γt · · · γ
t (σp − x)γt
where γt : [0, 1] ∋ s 7→ γ(st) ∈ M , t ∈ ∆
Lemma 5.4. The following diagram commutes:
C ⊗ sC⊗p
−−−−→ FpC1(LM)/Fp+1C1(LM)
C ⊗ sC⊗p
−−−−→ FpC0(LM)/Fp+1C0(LM)
Proof. For case 2,
∂′Φ(σ1, · · · , σp)− Φ∂(σ1, · · · , σp)
= (σ1 − x) · · · (σ
i · σ
i − σ
i − σ
i + σ
i − σ
i + x) · · · (σp − x)
= (σ1 − x) · · · (σ
i − x) · (σ
i − x) · · · (σp − x) ∈ Fp+1C0(LM)
where σ
i , σ
i , σ
i are the faces of σi.
For case 3,
∂′Φ(γ, σ1, · · · , σp)− Φ∂
′(γ, σ1, · · · , σp)
= γ−1 · (σ1 − x) · γ · · · γ
−1 · (σp − x) · γ − (σ1 − x) · · · (σp − x)
∈ Fp+1C0(LM).
Therefore we obtain the lemma.
Proposition 5.2 gives the map
Hq(FpC(LM)/Fp−1C(LM))→ Hq(Hom(F
pB(A)/F p−1B(A), A∨)).
Lemma 5.5. For q = 0, the following map is isomorphic:
H0(FpC(LM)/Fp+1C(LM)) ∼= H0(Hom(F
pB(A)/F p−1B(A), A∨)).
Proof. We obtain the following surjection by lemma 5.4.
Φ : H0(C ⊗ sC
⊗p) ։ H0(FpC(LM)/Fp+1C(LM)).
Composing with the isomorphism ⊗pH1(M) ∼= H0(C ⊗ sC
⊗p), the map
⊗pH1(M) ։ H0(FpC(LM)/Fp+1C(LM))→ Hom(⊗
pH1(A),R)
is given by
(σ1, · · · , σn) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
This is isomorphic and we obtain the lemma.
Lemma 5.6. For q = 1, the following map surjective:
H1(FpC(LM)/Fp+1C(LM)) ։ H1(Hom(F
pB(A)/F p−1B(A), A∨)).
Proof. It suffices to show that the following map obtained by lemma 5.4 is
surjective.
ker∂ → H1(FpC(LM)/Fp+1C(LM))→ Hom(⊗
psH(A), H(A)∨)1
If (γ, σ1, · · · , σp) ∈ ker∂ ∩
C0(M,x)⊗
sC(M,x)⊗p
, then
(γ, σ1, · · · , σp) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
ωp, if deg ω = 0
0, otherwise
through the above map.
If (γ, σ1, · · · , σp) ∈ ker∂ ∩
C1(M,x)⊗
sC(M,x)⊗p
, then
(γ, σ1, · · · , σp) 7→
(ω1, · · · , ωp) 7→
ω1 · · ·
when deg ω = 1. Then we can verify the surjectivity and obtain the lemma.
Proof of theorem 1.1. Consider the spectral sequences ofC(LM)/FpC(LM) and
Hom(F p−1B(A), A∨) associated with FqC(LM) and Hom(F
qB(A), A∨), re-
spectively. Lemma 5.5 asserts that ψp is isomorphic on E1-level at degree 0:
H0(FqC(LM)/Fq+1C(LM)) ∼= H0(Hom(F
qB(A)/F q−1B(A), A∨)).
Lemma 5.6 asserts that ψp is surjective on E1-level at degree 1:
H1(FqC(LM)/Fq+1C(LM)) ։ H1(Hom(F
qB(A)/F q−1B(A), A∨)).
Then there is an isomorphism on Er-level at degree 0 for r ≥ 1. We have
Rπ̃/J̃p ∼= H0(C(LM)/FpC(LM)) ∼= H0(Hom(F
pB(A), A∨)).
Therefore we obtain the theorem.
6 The Goldman bracket
This section is devoted to the proof of the following theorem.
Theorem 6.1. Let M be a compact closed oriented surface with genus g. Then
the Goldman bracket induces a Lie algebra structure on lim
Rπ̃/J̃pand there is
an isomorphism of Lie algebras
Rπ̃/J̃p ∼= H0(Hom(B(H
∗(M)), H∗(M)∨)).
Goldman showed that the vector space spanned by the free homotopy classes
of closed curves on a closed oriented surface has a Lie algebra structure [9]. This
work led Chas and Sullivan to the string topology. We would verify that this
structure makes lim
Rπ̃/J̃p a Lie algebra. On the other hand, we can construct
a bracket on H0(Hom(B(H
∗(M)), H∗(M)∨)) by the cup product defined in
section 3 and the Connes’s operator. Here we regard H∗(M) as a differential
graded algebra with a trivial differential. Theorem 6.1 asserts that those two
Lie algebras are isomorphic.
First we describe a relation between this bracket and the augmentation ideal
of the group ring of the surface group to induce a Lie algebra structure on
Rπ̃/J̃p. Then we construct a bracket on H0(Hom(B(A), A
∨)) and verify
the isomorphism of Lie algebras
Rπ̃/J̃p ∼= H0(Hom(B(A), A
Finally we verify the isomorphism
H0(Hom(B(A), A
∨)) ∼= H0(Hom(B(H
∗(M)), H∗(M)∨).
The following proposition makes lim
Rπ̃/J̃p a Lie algebra.
Proposition 6.2. (1) If p ≥ 1 and q ≥ 2, then [J̃p, J̃q] ⊂ J̃p+q−2.
(2) If p ≥ 2 , then [J̃p,Rπ̃] ⊂ J̃p−1.
Proof. We give a proof of (1). Take (σ1−x) · · · (σp−x) ∈ J̃p, (τ1−y) · · · (τq−y)
∈ J̃q, where σi ∈ ΩxM and τi ∈ ΩyM . Assume that all curves are immersions
and σi τj intersect transversally for any i, j. Let {σi♯τj} denote the set of
intersection points of σi and τj . Also assume that all the intersection points are
distinct i.e. {σi♯τj} ∩ {σk♯τl} = φ if i 6= k or j 6= l. Then,
[σ, τ ] =
s∈σi♯τj
{ε(s;σi, τj)γs,x · (σi − x) · · · (σp − x)(σ1 − x) · · ·
·(σi−1 − x) · γ
s,x · ·γs,y · (τj − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ
−γs,x · (σi+1 − x) · · · (σp − x)(σ1 − x) · · · (σi−1 − x) · γ
s,x ·
·γs,y · (τj+1 − y) · · · (τq − y)(τ1 − y) · · · (τj−1 − y) · γ
∈ J̃p+q−2.
Here γs,x is a path from s to x along σi and γs,y is a path from s to y along τj .
The proof of (2) can be verified in the same way.
Let A be a differential graded subalgebra of ΛM such thatH∗(A) ∼= H∗(ΛM)
by the inclusion.
Proposition 6.3. There is an isomorphism of vector spaces
H∗(Hom(F
pB(A), A)) ∼= H∗−2(Hom(F
pB(A), A∨)).
Proof. We define P : H∗−2(Hom(F
pB(A), A))→ H∗(Hom(F
pB(A), A∨)) by
P (ϕ)(ω1, · · · , ωp)(ω) =
ω ∧ ϕ(ω1, · · · , ωp).
This map preserves the filtrations. On E1-level, the map
Hom(⊗qH(A), H(A))→ Hom(⊗qH(A), H(A)∨)
is isomorphic. Therefore we obtain the proposition.
Now we construct a bracket on H0(Hom(B(A), A
∨)). First, we define the
Connes’s operator B : H∗(Hom(F
pB(A), A∨)) → H∗+1(Hom(F
p−1B(A), A∨))
B(ϕ)(ω1, · · · , ωp−1)(ω)
0≤k≤p−1
(−1)(εk+1)(εp−1−εk)ϕ(ωk+1, · · · , ωp−1, ω, ω1, · · ·ωk)(1).
Composing these maps and the cup product, we can define a bracket on
H0(Hom(F
pB(A), A∨)) by
[ϕ1, ϕ2] = −P (P
−1Bϕ1 ∪ P
−1Bϕ2) ∈ H0(Hom(F
p−1B(A), A∨)).
Take 2g closed 1-forms on M , α1, · · · , αg, β1, · · ·βg, such that
αi ∧ βj =
δij . Let {E
p.q} denote the spectral sequence of Hom(B(A), A
∨) associated with
F pB(A). Notice that the cyclic group Z/pZ acts on E
∼= Hom(⊗pH1(A),R)
ιϕ(ω1, · · · , ωp) = ϕ(ω2, · · · , ωp, ω1)
where ι is a generator of Z/pZ. The bracket [ , ] : E
p,−p⊗E
q,−q → E
p+q−2,−p−q+2
[ϕ1, ϕ2](ω1, · · · , ωp+q−2)
i,m,n
ιmϕ1(αi, ω1, · · · , ωp−1)̺
nϕ2(βi, ωp, · · · , ωp+q−2)
−ιmϕ1(βi, ω1, · · · , ωp−1)̺
nϕ2(αi, ωp, · · · , ωp+q−2)
where ι and ̺ are generators of Z/pZ and Z/qZ, respectively.
Proposition 6.4. The following diagram commutes for p, q ≥ 1:
J̃p/ ˜Jp+1 ⊗ J̃q/ ˜Jq+1 −−−−→ E
∞ ⊗ E
[ , ]
[ , ]
J̃p+q−2/J̃p+q−1 −−−−→ E
p+q−2,−p−q+2
Proof. Take σ = (σ1 − x) · · · (σp − x) ∈ FpC0(LM), τ = (τ1 − y) · · · (τq − y) ∈
FqC0(LM). Take 2g curves in M , ai, bi, as in Figure 1. Assume that σi and τj ,
ak, or bk, intersect transversally for any i, j, k. Also assume that τj and ak, or
bk, intersect transversally for any j, k.
Assume that all the intersection points are distinct. Then for any i, j, k, we
can take each tubular neighborhoods of ai and bi so that it does not include some
neighborhoods of intersection points of σj and τk. We fix such neighborhoods
of intersection points and denote them by Up for each p. We can also take
a tubular neighborhood of the diagonal map from M to M×M outside those
neighborhoods of intersection points of σi and τj for any i, j i.e.
S1 \ ∪pσ
i (Up)
S1 \ ∪pτ
j (Up)
= φ, ∀i, j.
Here N∆ denotes the tubular neighborhood of the diagonal map. Thom class Φ
of this tubular neighborhood satisfies
Φ = −ε(p;σi, τj),
where ε(p;σi, τj) is the intersecion number of σi and τj at p.
Fig. 1
・ ・ ・
Define e♯ : C0(LM) → C1(LM) by e♯γ(ξ)(t) = γ(ξ + t). Let ωk, 1 ≤
k ≤ n, be differential forms on M which has its support inside the tubular
neighborhoods of ai and bi. Then
[σi,τj]
ω1 · · ·ωn
p∈σi♯τj,k
ε(p;σi, τj)
(σi)p
ω1 · · ·ωk
(τj)p
ωk+1 · · ·ωn
p∈σi♯τj ,k
×τj |
(σi)p
ω1 · · ·ωk
(τj)p
ωk+1 · · ·ωn
e♯σi×e♯τj
π∗Φ ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωn.
Here p1, p2 : LM×LM → LM are the projections. The last equality is obtained
by the following lemma.
Lemma 6.5. If p ∈ σi♯τj and p
′ ∈ Up ∩ σi([0, 1]), then
(σi)p
ω1 · · ·ωn =
(σi)p′
ω1 · · ·ωn.
Proof. F Let γ be the curve from p to p′ along σi inside Up. If γ and σ are in
the same direction, then
(σi)p′
ω1 · · ·ωn =
γ·(σi)p′
ω1 · · ·ωn =
(σ)p·γ
ω1 · · ·ωn
ω1 · · ·ωn.
We can also verify the case where γ is in the direction opposite to σ in the same
We have the equality
e♯σ×e♯τ
π∗Φ ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
e♯σ×e♯τ
− p∗1(α1 ∧ β1)− p
2(α1 ∧ β1) + p
1αj ∧ p
2βj − p
1βj ∧ p
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
In fact, if η ∈ Λ(M ×M) then
(−1)|η|+1
e♯σ×e♯τ
π∗dη ∧ p∗1
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
e♯σ×e♯τ
π∗η ∧ d
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωp+q−2
+(e♯σ)
ω1 · · ·ωk
(e♯τ)
ωk+1 · · ·ωj ∧ ωj+1 · · ·ωp+q−2
The last equality is obtained by the following lemma.
Lemma 6.6. If σ ∈ FpC0(LM), then
(e♯σ)
ω1 · · ·ωp−2 = 0.
Proof. It suffices to show the case σ = (τ1 − x) · · · (τp − x) where x ∈ M and τi
∈ ΩxM . We define τ̄i ∈ ΩxM by
τ̄i(t) =
τi(pt), if (i − 1)/p ≤ t ≤ i/p
0, otherwise.
Let σ̄ denote (τ̄1 − x) · · · (τ̄p − x). It can be shown that e♯σ̄ restricted on [(i −
1)/p, i/p] is contained in Fp−1C1(LM) for any i. Therefore
(e♯σ)
ω1 · · ·ωp−2 = (e♯σ̄)
ω1 · · ·ωp−2 = 0.
Jones, Geztler, and Petrack describes the map e♯ in terms of iterated inte-
grals by the following theorem.
Theorem 6.7 (Geztler-Jones-Petrack [8]). If σ ∈ C0(LM) and ω, ωi ∈ Λ
1 ≤ i ≤ p, then
π∗ω ∧
ω1 · · ·ωp =
ωk · · ·ωpωω1 · · ·ωk−1.
This theorem asserts the equality
e♯σ×e♯τ
− p∗1(α1 ∧ β1)− p
2(α1 ∧ β1) + p
1αj ∧ p
2βj − p
1βj ∧ p
ω1 · · ·ωk ∧ p
ωk+1 · · ·ωn
j,k,l
ωk+1 · · ·ωp−1αjω1 · · ·ωk
ωl+1 · · ·ωp+q−2βjωp · · ·ωl
ωk+1 · · ·ωp−1βjω1 · · ·ωk
ωl+1 · · ·ωp+q−2αjωp · · ·ωl
Finally we obtain the equality
[σ,τ ]
ω1 · · ·ωp+q−2
j,k,l
ωk+1 · · ·ωp−1αjω1 · · ·ωk
ωl+1 · · ·ωp+q−2βjωp · · ·ωl
ωk+1 · · ·ωp−1βjω1 · · ·ωk
ωl+1 · · ·ωp+q−2αjωp · · ·ωl
Since we can take ωi ∈ H
1(M), 1 ≤ i ≤ p + q − 2, so that their support are
inside the tubular neighborhoods of aj and bj, we obtain the proposition.
Proof of theorem 6.1. We obtain the following isomorphism of Lie algebras by
proposition 6.4.
Rπ̃/J̃p ∼= H0(Hom(B(A), A
To obtain the isomorphism of Lie algebras
H0(Hom(B(A), A
∨) ∼= H0(Hom(B(H
∗(M)), H∗(M)∨),
we introduce the following lemma, which asserts the formality of the compact
Kähler manifolds.
Lemma 6.8 (ddcLemma, Deligne-Griffiths-Morgan-Sullivan [7]). Let X be a
compact Kähler manifold and dc = J−1dJ where J gives the complex structure
in the cotangent bundle. If α is a differential form on X such that dα = 0 and
dcα = 0, and such that α = dγ, then α = ddcβ for some β.
Cor. There are quasi-isomorphisms of differential graded algebras
(ΛX, d)← (kerdc, d)→ (H∗dc(X), 0).
Notice that a closed oriented surface endowed with a complex structure become
a Kähler manifolds for the dimensional reason. Therefore the following lemma
completes the proof of the theorem.
Lemma 6.9. If f : A1 → A2 is a quasi-isomorphism of differential graded
algebras, then the map induced by f
H0(Hom(B(A1), A
1 )→ H0(Hom(B(A2), A
is an isomorphism.
Proof. It suffices to verify that the map induced by f
f : H0(Hom(F
pB(A1), A
1 )→ H0(Hom(F
pB(A2), A
is an isomorphism for any p. On E1-level, the map induced by f
Hom(⊗sH(A1), H(A1)
∨)→ Hom(⊗sH(A2), H(A2)
is an isomorphism because f is quasi-isomorphism. Therefore we obtain the
lemma.
Therefore we obtain the theorem.
References
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	Introduction
	Chen's iterated integrals
	Preliminaries
	Proof of Theorem 1.1
	The conjugacy classes of fundamental groups
	The Goldman bracket
