The survey towards universality

From now on, we set the stage in order to establish the universality and uniqueness of a universal Lie∞-algebroid of a singular foliation. The starting point of this survey is Lemma4.7.

Definition 4.4. Assume that a singular foliation F admits a geometric resolution(E, d, ρ). We say that a Lie ∞-algebroid(F, Q_F)with linear part (F, d⁰, ρ⁰,)defines a sub-foliation ofF if ρ⁰(Γ(F₋₁))⊂ F.

Remark 4.6. Note that (F, d⁰, ρ⁰)is not necessarily a geometric resolution of the sub-foliation.

Lemma 4.7. Let(E, QE)be a universal Lie∞-algebroid of a singular foliationF and let(F, QF) be another Lie∞-algebroid that induces a sub-foliation ofF. Then, there exists a chain map from the linear part of (F, QF)to the linear part of (E, QE).

Proof. We denote the linear parts of (E, QE) and (F, QF) by (E, d, ρ) and (F, d⁰, ρ⁰), respectively.

Since (E, QE) resolves the foliationF, the sequence ...→Γ(E₋₃) ^d

(3)

−−→Γ(E₋₂) ^d

(2)

−−→Γ(E₋₁)−→ F →^ρ 0 (4.25) is exact. In particular, we have thatρ(Γ(E−1)) =F. Moreover,ρ⁰(Γ(F−1))⊂ F by assumption.

We shall prove the statement by constructing the components of the chain map at each degree.

The image of a sectionσ ∈Γ(F₋₁) under the anchorρ⁰ lies inF =Im(ρ). Thus, there exists some section ˜σofE₋₁ which is a preimage ofρ⁰(σ), such thatρ⁰(σ) = ρ(˜σ). The assignment σ 7→ σ˜ depends smoothly onσ(i.e. it isC^∞(M)-linear), so we are actually describing a vector bundle map φ1:F₋₁→E₋₁ that makes the following diagram commute

F₋₁

T M.

E₋₁

ρ⁰ φ1

ρ

The linear part of (F, Q⁰) is not necessarily a resolution of the induced foliation, soIm(d⁰⁽²⁾) ⊂ Ker(ρ⁰). This implies thatφ1(Im(d⁰⁽²⁾)) ⊂ Ker(ρ) and sinceKer(ρ) = Im(d⁽²⁾), we can assign a section ofE₋₂to each section ofF₋₂. Indeed, forσ∈Γ(F₋₂),

ρ◦φ1◦d⁰⁽²⁾(σ) =ρ⁰◦d⁰⁽²⁾(σ) = 0.

So,φ1◦d⁰⁽²⁾(σ) lies in the kernel ofρand thus, there is a section ˜σ∈Γ(E₋₂) such that φ1◦d⁰⁽²⁾(σ) = ˜σ.

The assignmentσ7→σ˜ respects the multiplication of sections with smooth functions (by definition), so we have a bundle mapφ₂ : F₋₂ → E₋₂which, by construction, fits into the commutative dia- gram

F₋₂ F₋₁

E₋₂ E₋₁

φ₂ φ₁

d⁰⁽²⁾

d⁽²⁾

Next, note that the image ofφ2 ◦d⁰⁽³⁾ lies in the kernel ofd⁽²⁾. Using the equality Ker(d⁽²⁾) = Im(d⁽³⁾), an iteration of the previous argument gives an bundle mapφ3:F−3→E−3 and a commu- tative diagram

F₋₃ F₋₂ F₋₁

E₋₃ E₋₂ E₋₁

φ2 φ1

d⁰⁽²⁾

d⁽²⁾ d⁰⁽³⁾

φ3

d⁽³⁾

Recursively, we obtain a family of bundle maps (φi : F−i → E−i)i≥1 and these data sum up to a chain mapφ: (F, d⁰, ρ⁰)→(E, d, ρ) fitting in the following diagram

... F₋₃ F₋₂ F₋₁ T M

... E−3 E−2 E−1 T M

φ₂ φ₁

d⁰⁽²⁾

d⁽²⁾ d⁰⁽³⁾

φ₃ d⁽³⁾ d⁰⁽⁴⁾

d⁽⁴⁾

ρ⁰

ρ

idM

Remark 4.7. Consider the dual map ofφ,Φˆ⁽⁰⁾ : E^∗ → F^∗. We extend it, by 3.53, to a mor- phismΦ⁽⁰⁾ of graded algebras of degree 0, from the graded algebraE of functions onE →M to the graded algebraF of functions onF → M. This means thatΦ⁽⁰⁾ :E → F is a morphism of graded manifolds which, in addition, does not change the arity of functions onE, i.e. Φ⁽⁰⁾ has arity 0. Du- alizing the chain map condition,φ◦d⁰=d◦φ, we obtain that

Φ⁽⁰⁾◦Q⁽⁰⁾_E =Q⁽⁰⁾_F ◦Φ⁽⁰⁾ (4.26) and thus, Φ⁽⁰⁾ intertwines the linear parts of the homological vector fieldsQE andQF. However, this does not imply that it is a Lie∞-morphism.

Next, we see how to extend a chain map between the linear parts of two Lie∞-algebroids to a Lie

∞-morphism.

Proposition 4.2. Let (E, QE) and(F, QF)be Lie ∞-algebroids over a manifold M. Any chain map φ:F →E between the linear parts of the two algebroids is the linear part of a Lie∞-morphism from(F, QF) to(E, QE).

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

Proof. Denote byE the graded algebra of functions onE and byF the graded algebra of functions onF. Letφ: (F, d⁰, ρ,)→(E, d, ρ) be a chain map from the linear part of (F, QF) to the linear part of (E, QE)

· · · F₋₃ F₋₂ F₋₁ T M

· · · E₋₃ E₋₂ E₋₁ T M

d⁰

d

φ φ φ

d⁰ d⁰

d d

ρ⁰

id_{T M} ρ

We shall construct a morphism of graded algebras Φ : E → F of degree 0, whose linear part equals φ, that is compatible with the Lie∞-algebroids structures onE andF, i.e. such Φ must satisfy

QF ◦Φ = Φ◦QE, (4.27)

According to Lemma3.9, such a morphism Φ will admit a decomposition into the formal sum of arity components

Φ =X

k≥0

Φ^(k).

For everyk ≥ 0, Φ^(k)can be identified with anO-linear map, ˆΦ^(k) : Γ(E^∗) → Γ(S^k+1F^∗), which we consider as an element of Γ(S^k+1F^∗⊗E). According to Equations3.53,3.54,3.55, Φ^(k)depends only in Φ⁽⁰⁾, ...,Φ^(k−1)and on its restriction on Γ(E^∗). Furthermore, isolating the arity components in4.27, we obtain the following defining equations

X

0≤i,j≤n

i+j=n

Q⁽ⁱ⁾_F ◦Φ^(j)= X

0≤i,j≤n

i+j=n

Φ^(j)◦Q⁽ⁱ⁾_E, n≥0

which can be rewritten, in terms of the operator∂given in 4.23as

∂(Φ⁽⁰⁾) = 0, (4.28)

∂(Φ⁽ⁿ⁾) = X

1≤i,j+1≤n

i+j=n

Φ^(j)◦Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F ◦Φ^(j), n≥1. (4.29)

It is straightforward to define the component Φ⁽⁰⁾ as the dual map of the chain mapφ. According to Remark4.7, Φ⁽⁰⁾ satisfies4.28by definition. We follow the steps in the proof of Theorem4.1to define the components of Φ of higher arities. At each step, we check that the the right-hand side of 4.29is a cocycle in the bicomplex D⁽ⁿ⁾. As it turns out, these cocycles will always be coboundaries.

Thus, for everyn ≥ 1, we obtain by Lemma4.7an element ˆΦ⁽ⁿ⁾ ∈ D⁽ⁿ⁾ that satisfies4.29. Then, we extend ˆΦ⁽ⁿ⁾ to anO-linear map Φ⁽ⁿ⁾:E → F by using3.53,3.54,3.55up to ordern. Finally, we take Φ to be the formal sum of Φ⁽ⁿ⁾.

We start by defining Φ⁽⁰⁾. Consider the dual map ofφ, ˆΦ⁽⁰⁾ : Γ(E^∗) → Γ(F^∗) and extend itO- linearly and by3.53to a morphism of graded algebras, Φ⁽⁰⁾:E → F, which has by definition degree and arity equal to 0. Dualizing the chain map condition forφgives that ˆΦ⁽⁰⁾ satisfies

Q⁽⁰⁾_F ◦Φˆ⁽⁰⁾= (φ◦d⁰)^∗= (d◦φ)^∗= ˆΦ⁽⁰⁾◦Q⁽⁰⁾_E and consequently, Φ⁽⁰⁾ satisfies4.28, i.e. ∂(Φ⁽⁰⁾) = 0.

Next, we want to define Φ⁽¹⁾. Forn= 1, Equation4.29reads

∂(Φ⁽¹⁾) = Φ⁽⁰⁾◦Q⁽¹⁾_E −Q⁽¹⁾_F ◦Φ⁽⁰⁾. (4.30) Denote byC1the map from E toF given by the right-hand side of4.30. Explicitly, we first define C1by its action on Γ(E^∗) and then extend it as a Φ⁽⁰⁾-derivation to E. Obviously, it has degree 1 and arity 1.

Let us see whyC1 isO-linear. Recall that a Φ⁽⁰⁾-derivation is O-linear if and only if it vanishes on smooth functions. Furthermore, note that, by definition,ρ^0∗ = ˆΦ⁽⁰⁾◦ρ^∗. Therefore, for everyf ∈ C^∞(M)

C₁(f) = Φ⁽⁰⁾◦Q⁽¹⁾_E (f)−Q⁽¹⁾_F ◦Φ⁽⁰⁾(f)

= ˆΦ⁽⁰⁾ ρ^∗(d_dRf)

−Q⁽¹⁾_F (f)

=ρ^0∗ d_dRf)−ρ^0∗ d_dRf)

= 0.

We shall show thatC₁is a cocycle in the bicomplex D⁽¹⁾ that is also a coboundary. Indeed

∂(C1) =Q⁽⁰⁾_F ◦ Φ⁽⁰⁾◦Q⁽¹⁾_E −Q⁽¹⁾_F ◦Φ⁽⁰⁾

−(−1) Φ⁽⁰⁾◦Q⁽¹⁾_E −Q⁽¹⁾_F ◦Φ⁽⁰⁾

◦Q⁽⁰⁾_E

=Q⁽⁰⁾_F ◦Φ⁽⁰⁾◦Q⁽¹⁾_E −Q⁽⁰⁾_F ◦Q⁽¹⁾_F ◦Φ⁽⁰⁾+ Φ⁽⁰⁾◦Q⁽¹⁾_E ◦Q⁽⁰⁾_E −Q⁽⁰⁾_F ◦Φ⁽⁰⁾◦Q⁽⁰⁾_E

= Φ⁽⁰⁾◦Q⁽⁰⁾_E ◦⁽¹⁾_E −Q⁽⁰⁾_F ◦Q⁽¹⁾_F ◦Φ⁽⁰⁾+ Φ⁽⁰⁾◦Q⁽¹⁾_E ◦Q⁽⁰⁾_E −Q⁽¹⁾_F ◦Q⁽⁰⁾_F ◦Φ⁽⁰⁾ by 4.28

= Φ⁽⁰⁾◦ [Q⁽⁰⁾, Q⁽¹⁾]

− [Q⁽⁰⁾_F , Q⁽¹⁾_F ]◦Φ⁽⁰⁾

= 0,

since the commutators [Q⁽⁰⁾_E , Q⁽¹⁾_E ] and [Q⁽⁰⁾_F , Q⁽¹⁾_F ] vanish, according to4.9.

The graded commutators [Q⁽¹⁾_E , Q⁽¹⁾_E ] and [Q⁽¹⁾_F , Q⁽¹⁾_F ] are both vertical (see the last part of the proof of Proposition3.6). So, for everyf ∈C^∞(M),

Φ⁽⁰⁾◦ [Q⁽¹⁾_E , Q⁽¹⁾_E ]

− [Q⁽¹⁾_F , Q⁽¹⁾_F ]

◦Φ⁽⁰⁾

f = 0.

Thus, for everyf ∈C^∞(M),

C1◦Q⁽¹⁾_E (f)−Q⁽¹⁾_F ◦C1(f) = Φ⁽⁰⁾◦Q⁽¹⁾_E ◦Q⁽¹⁾_E (f)−Q⁽¹⁾_F ◦Φ⁽⁰⁾◦Q⁽¹⁾_E (f)

−Q⁽¹⁾_F ◦Φ⁽⁰⁾◦Q⁽¹⁾_E (f) +Q⁽¹⁾_F ◦Q⁽¹⁾_F ◦Φ⁽⁰⁾(f)

=

Φ⁽⁰⁾◦ [Q⁽¹⁾_E , Q⁽¹⁾_E ]

+ [Q⁽¹⁾_F , Q⁽¹⁾_F ]

◦Φ⁽⁰⁾

f−[Q⁽¹⁾_F , Q⁽¹⁾_F ](f)

= 0.

SinceC1isO-linear, i.e. C1(f) = 0, we have that

C₁◦Q⁽¹⁾_E (f) = 0 (4.31)

which is dual to

id⊗ρ rt(C1)

(f) = 0. (4.32)

This implies thatC₁ is a rooted cocycle inD⁽¹⁾ whose root belongs in the kernel ofid⊗ρ. Accord- ing to Lemma4.7,C₁ is a coboundary in the bicomplexD⁽¹⁾: there exists an element ˆΦ⁽¹⁾ ∈ D⁽¹⁾ such that ˆΦ⁽¹⁾=C1. We extend ˆΦ⁽¹⁾ by3.54to a map Φ⁽¹⁾:E → F which satisfies4.29forn= 1.

Letn≥1 and assume that ˆΦ⁽ⁱ⁾has already been defined for 0≤i ≤n. For every 0≤ i≤ n, Φ⁽ⁱ⁾ is the (unique)O-linear map fromE to F whose restriction to Γ(E^∗) is ˆΦ⁽ⁱ⁾ and satisfies Equations 3.53, 3.54,3.55up to ordern. We want to define Φ⁽ⁿ⁺¹⁾, so that it obeys4.29fork=n+ 1:

∂(Φ⁽ⁿ⁺¹⁾) = X

1≤i,j+1≤n

i+j=n+1

Φ^(j)Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ^(j).

Denote the right-hand side byCn+1. More precisely,Cn+1 is anO-linear map fromE toF whose restriction on Γ(E^∗) is given by the right-hand side and then it is extended toE by the derivation property ofQ⁽ⁱ⁾_E, Q⁽ⁱ⁾_F and by equations3.54satisfied by Φ⁽ⁱ⁾. Cn+1 has degree 1 and arity n+ 1 and

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

1 2 3

1 2 3

0

C

₁

Φ

⁽¹⁾

Figure 4.6: C1is a∂-cocycle whose root lies in the kernel of the anchor mapρ, and a such it is a coboundary: C1=

∂(Φ⁽¹⁾).

thus, for degree reasons, it has depthn+ 1: ifω ∈ Γ(E_−j^∗ ), j ≤ n, thenCn+1(ω) must have arity n+2 and degreej+ 1≤n+ 1, but there exist no such functions. In particular, this means thatCn+1

is root-free.

We will show thatCn+1 is a cocycle in the bicomplexD⁽ⁿ⁺¹⁾and then apply Lemma4.7. We have that

∂(Cn+1) = X

1≤i,j+1≤n+1

i+j=n+1

Q⁽⁰⁾_F Φ^(j)Q⁽ⁱ⁾_E −Q⁽⁰⁾_F Q⁽ⁱ⁾_F Φ^(j)+ Φ^(j)Q⁽ⁱ⁾_EQ⁽⁰⁾_E −Q⁽ⁱ⁾_F Φ^(j)Q⁽⁰⁾_E . (4.33)

For every 0≤j≤n, Φ^(j)satisfies the induction hypothesis4.29. Explicitly, for 0≤j≤n,

∂(Φ^(j)) = X

1≤k,l+1≤j

k+l=j

Φ^(l)Q^(k)_E −Q^(k)_F Φ^(l)⇐⇒

Q⁽⁰⁾_F Φ^(j)−Φ^(j)Q⁽⁰⁾_E = X

1≤k,l+1≤j

k+l=j

Φ^(l)Q^(k)_E −Q^(k)_F Φ^(l).

Then, the first term in the sum at the right hand side of4.33gives X

1≤i,j+1≤n+1

i+j=n+1

Q⁽⁰⁾_F Φ^(j)Q⁽ⁱ⁾_E =Q⁽⁰⁾_F Φ⁽⁰⁾Q⁽ⁿ⁺¹⁾_E + X

1≤i,j≤n

i+j=n+1

Q⁽⁰⁾_F Φ^(j)Q⁽ⁱ⁾_E

= Φ⁽⁰⁾Q⁽⁰⁾_E Q⁽ⁿ⁺¹⁾_E + X

1≤i,j≤n

i+j=n+1

Φ^(j)Q⁽⁰⁾_E Q⁽ⁱ⁾_E

+ X

1≤i,j≤n

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Φ^(l)Q^(k)_E −Q^(k)_F Φ^(l) Q⁽ⁱ⁾_E

(4.34)

where for the last equation we used that Φ⁽⁰⁾ satisfiesQ⁽⁰⁾_F Φ⁽⁰⁾= Φ⁽⁰⁾Q⁽⁰⁾_E . The last term in4.33is:

X

1≤i,j+1≤n+1

i+j=n+1

Q⁽ⁱ⁾_F Φ^(j)Q⁽⁰⁾_E =Q⁽ⁿ⁺¹⁾_F Φ⁽⁰⁾Q⁽⁰⁾_E + X

1≤i,j≤n

i+j=n+1

Q⁽ⁱ⁾_F Φ^(j)Q⁽⁰⁾_E

=Q⁽ⁿ⁺¹⁾_F Q⁽⁰⁾_F Φ⁽⁰⁾+ X

1≤i,j≤n

i+j=n+1

Q⁽ⁱ⁾_F Q⁽⁰⁾_F Φ^(j)

− X

1≤i,j≤n

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Q⁽ⁱ⁾_F Φ^(l)Q^(k)_E −Q^(k)_F Φ^(l) .

(4.35)

The second term in4.33can be written as X

1≤i,j+1≤n+1

i+j=n+1

Q⁽⁰⁾_F Q⁽ⁱ⁾_F Φ^(j)=Q⁽⁰⁾_F Q⁽ⁿ⁺¹⁾_F Φ⁽⁰⁾+ X

1≤i,j≤n

i+j=n+1

Q⁽⁰⁾_F Q⁽ⁱ⁾_F Φ^(j) (4.36)

and the third one as X

1≤i,j+1≤n+1

i+j=n+1

Φ^(j)Q⁽ⁱ⁾_EQ⁽⁰⁾_E = Φ⁽⁰⁾Q⁽ⁿ⁺¹⁾_E Q⁽⁰⁾_E + X

1≤i,j≤n

i+j=n+1

Φ^(j)Q⁽ⁱ⁾_E Q⁽⁰⁾_E . (4.37)

Taking into consideration the signs in4.33, the first term in4.34combined with the first term in 4.37give Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽ⁿ⁺¹⁾_E ]. Moreover, the second term in4.34combined with the second term in 4.37give X

1≤i,j≤n

i+j=n+1

Φ^(j)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E]. A similar computation with the respective terms of 4.35and4.36 implies that we can write equation4.33as:

∂(C_n+1) = Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽ⁿ⁺¹⁾_E ] + X

1≤i,j≤n

i+j=n+1

Φ^(j)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E] + X

1≤i,j≤n+1

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Φ^(l)Q^(k)_E Q⁽ⁱ⁾_E

−[Q⁽⁰⁾_F , Q⁽ⁿ⁺¹⁾_F ]Φ⁽⁰⁾− X

1≤i,j≤n

i+j=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ]Φ^(j)− X

1≤i,j≤n

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Q⁽ⁱ⁾_F Q^(k)_F Φ^(l)

− X

1≤i,j≤n

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Q^(k)_F Φ^(l)Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ^(l)Q^(k)_E .

(4.38)

A closer look at the last term in4.38reveals that is equal to zero. Indeed, since 0 ≤ j ≤ n, the range of values for the indexl goes from 0 ton−1 and in particular, we have thati+k=n+ 1−l, for every value ofl. Thus,k takes values from 1 tonand we can rewrite the last term in 4.38as:

X

0≤l≤n−1

X

1≤i,k≤n

i+k=n+1−l

Q^(k)_F Φ^(l)Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ^(l)Q^(k)_E . (4.39)

For every 0≤l≤n−1, the summandQ^(k)_F Φ^(l)Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ^(l)Q^(k)_E is antisymmetric with respect toi andk, which means that the term in4.39vanishes.

Therefore,4.38reads

∂(Cn+1) = Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽ⁿ⁺¹⁾_E ] + X

1≤i,j≤n

i+j=n+1

Φ^(j)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E] + X

1≤i,j≤n

i+j=n+1

X

1≤k,l+1≤j

k+l=j

Φ^(l)Q^(k)_E Q⁽ⁱ⁾_E

−[Q⁽⁰⁾_F , Q⁽ⁿ⁺¹⁾_F ]Φ⁽⁰⁾− X

1≤i,j≤n

i+j=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ]Φ^(j)− X

1≤i,j≤n

i+j=n+

X

1≤k,l+1≤j

k+l=j

Q⁽ⁱ⁾_F Q^(k)_F Φ^(l).

(4.40)

Finally we split the study of4.40in two cases.

• Forn= 1, we have that

∂(C2) = Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽²⁾_E ] + Φ⁽¹⁾[Q⁽⁰⁾_E , Q⁽¹⁾_E ] + Φ⁽⁰⁾Q⁽¹⁾_E Q⁽¹⁾_E

−[Q⁽⁰⁾_F , Q⁽²⁾_F ]Φ⁽⁰⁾−[Q⁽⁰⁾_F , Q⁽¹⁾_F ]Φ⁽⁰⁾−Q⁽¹⁾_F Q⁽¹⁾_F Φ⁽⁰⁾

= Φ⁽⁰⁾

[Q⁽⁰⁾_E , Q⁽²⁾_E ] + 1

2[Q⁽¹⁾_E , Q⁽¹⁾_E ]

+ Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽¹⁾_E ]

−

[Q⁽⁰⁾_F , Q⁽²⁾_F ] +1

2[Q⁽¹⁾_F , Q⁽¹⁾_F ]

Φ⁽⁰⁾−[Q⁽⁰⁾_F , Q⁽¹⁾_F ]Φ⁽⁰⁾.

(4.41)

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

SinceQE andQF are homological vector fields, according to4.9, [Q⁽⁰⁾_E , Q⁽¹⁾_E ] = 0 and [Q⁽⁰⁾_F , Q⁽¹⁾_F ] = 0. Moreover, the parentheses in4.41are zero beacuseQE andQF satisfy4.10fork= 2.

Consequently,C₂is a ∂-cocycle of depth 2, and therefore, according to Lemma 4.7, there ex- ists an element ˆΦ⁽²⁾∈D⁽²⁾ such that

C2=∂( ˆΦ⁽²⁾).

This is equation4.29fork = 2. We extend ˆΦ⁽²⁾ to anO-linear map Φ⁽²⁾ : E → F by using Equation3.55for k=2.

• Assume now thatn ≥2. Forl = 0, the third sum in4.40gives X

1≤i,k≤n

i+k=n+1

Φ⁽⁰⁾Q^(k)_E Q⁽ⁱ⁾_E which in turn, combined with the first term in4.40, gives:

Φ⁽⁰⁾[Q⁽⁰⁾_E , Q⁽ⁿ⁺¹⁾_E ] + X

1≤i,k≤n

i+k=n+1

Φ⁽⁰⁾Q^(k)_E Q⁽ⁱ⁾_E = Φ⁽⁰⁾

[Q⁽⁰⁾_E , Q⁽ⁿ⁺¹⁾_E ] + X

1≤i,k≤n

i+k=n+1

1

2[Q⁽ⁱ⁾_E, Q^(j)_E ]

= 0

because QE satisfies 4.10fork=n+ 1. A repetition of this argument for the last sum in4.40 yields

−[Q⁽⁰⁾_F , Q⁽ⁿ⁺¹⁾_F ]Φ⁽⁰⁾− X

1≤i,j≤n

i+j=n+1

Q⁽ⁱ⁾_F Q^(j)_F Φ⁽⁰⁾=−

[Q⁽⁰⁾_F , Q⁽ⁿ⁺¹⁾_F ] + 1 2

X

1≤i,j≤n

i+j=n+1

[Q⁽ⁱ⁾_F , Q^(j)_F ]

Φ⁽⁰⁾= 0.

Whilej ranges from 0 ton, the values of the index l in4.40range from 1 ton−1. Moreover, we have thati+k=n+ 1−land that the range of values for the index krange from 1 ton.

Finally, Equation 4.40becomes:

∂(Cn+1) = X

1≤i,j≤n

i+j=n+1

Φ^(j)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E ] + X

1≤l≤l−1

X

1≤i,k≤n

i+k=n+1−l

Φ^(l)Q^(k)_E Q⁽ⁱ⁾_E

− X

1≤i,j≤n

i+j=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ]Φ^(j)− X

1≤l≤l−1

X

1≤i,k≤n

i+k=n+1−l

Q⁽ⁱ⁾_F Q^(k)_F Φ^(l)

= Φ⁽ⁿ⁾[Q⁽⁰⁾_E , Q⁽¹⁾_E ]

| {z }

=0

+ X

2≤i,j+1≤n

i+j=n+1

Φ^(j)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E ] + X

1≤l≤l−1

X

1≤i,k≤n

i+k=n+1−l

Φ^(l)Q^(k)_E Q⁽ⁱ⁾_E

−[Q⁽⁰⁾_F , Q⁽¹⁾_F ]

| {z }

=0

Φ⁽ⁿ⁾− X

2≤i,j+≤n

i+j=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ]Φ^(j)− X

1≤l≤l−1

X

1≤i,k≤n

i+k=n+1−l

Q⁽ⁱ⁾_F Q^(k)_F Φ^(l)

= X

1≤l≤n−1

X

2≤i≤n

i+l=n+1

Φ^(l)[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E] + X

1≤i,k≤n

i+k=n+1−l

Φ^(l)1

2[Q^(k)_E , Q⁽ⁱ⁾_E]

since Q^(k)_E Q⁽ⁱ⁾_E = 2[Q^(k)_E , Q⁽ⁱ⁾_E]

− X

1≤l≤n−1

X

2≤i≤n

i+l=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ]Φ^(l)+ X

1≤i,k≤n

i+k=n+1−l

1

2[Q⁽ⁱ⁾_F , Q^(k)_F ]Φ^(l)

= X

1≤l≤n−1

X

2≤i≤n

i+l=n+1

Φ^(l)

[Q⁽⁰⁾_E , Q⁽ⁱ⁾_E] + 1 2

X

1≤j,k≤i

j+k=i

[Q^(j)_E , Q^(k)_E ]

− X

1≤l≤n−1

X

2≤i≤n

i+l=n+1

[Q⁽⁰⁾_F , Q⁽ⁱ⁾_F ] +1 2

X

1≤j,k≤i

j+k=i

[Q^(j)_F , Q^(k)_F ]

Φ^(l)

= X

1≤l≤n−1

X

2≤i≤n

i+l=n+1

Φ^(l)[QE, QE]⁽ⁱ⁾−[QF, QF]⁽ⁱ⁾Φ^(l)

which vanishes for everyiandl, because [QE, QE] = 0 and [QF, QF] = 0. Thus, we have proved thatCn+1 is a cocycle in the bicomplexD⁽ⁿ⁺¹⁾and has depthn+ 1≥3. According to Lemma 4.7, there exists an element inD⁽ⁿ⁺¹⁾ such that

Cn+1=∂( ˆΦⁿ⁺¹).

We extend ˆΦ⁽ⁿ⁺¹⁾ to a map Φ⁽ⁿ⁺¹⁾ : E → F by using Equations3.55. This concludes the induction argument.

Taking the formal sum of these Φ⁽ⁱ⁾, set

Φ =X

k≥0

Φ^(k),

and thus we obtain anO-linear map Φ : E → F that is by construction a Lie ∞-morphism. More- over, since this procedure started by considering the dual of a chain mapφ : (F, d⁰, ρ⁰) → (E, d, ρ), i.e. ˆΦ⁽⁰⁾ =φ^∗, we have that the linear part of Φ coincides with the initial chain mapφ.

n+ 1 n+ 2

n+ 2 n+ 3

0

C

_n+1

Φ

⁽ⁿ⁺¹⁾

Figure 4.7: IfCn+1is a root-free cocycle, then it is a coboundary:Cn+1=∂(Φ⁽ⁿ⁺¹⁾)

The construction of Φ involves a choice, at each step, of an element ˆΦ⁽ⁿ⁾∈D⁽ⁿ⁾such that∂( ˆΦ⁽ⁿ⁾) = Cn. A different sequence of choices of arity components would result to a another Lie -morphism with the same linear part. This situation forces us to investigate under which conditions two Lie

∞-morphisms are homotopic.

It is actually the case that any two Lie∞-morphisms with the same linear part are homotopic.

This requires some build up. Recall the operator [Q,·] on the space of mapsa: E → F, defined in Section 3.4, which is given by

a7→[Q, a] =QF◦a−(−1)^|a|a◦QE.

Proposition 4.3. Let Φ,Ψ : E → F be two Lie∞-moprhims such that Φ⁽ⁱ⁾ = Ψ⁽ⁱ⁾, for every 0≤i≤n, for somen∈N∩ {0}. Then, there exists a Lie∞-morphismχ:E → F that is homotopic toΦand satisfiesχ⁽ⁱ⁾= Ψ⁽ⁱ⁾, for every0≤i≤n+ 1.

Proof. Consider the map Ψ−Φ : E → F that is given by (Ψ−Φ)(ω) = Ψ(ω)−Φ(ω),ω ∈ E. For everyk≥0, the arity components of Ψ−Φ are given by (Ψ−Φ)^(k)= Ψ^(k)−Φ^(k). Let us study the component (Ψ−Φ)⁽ⁿ⁺¹⁾. Since both Ψ and Φ are Lie∞-morphisms, they satisfy3.55fork=n+ 1.

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

Thus, for everyf, g∈ E, we have that

(Ψ−Φ)⁽ⁿ⁺¹⁾(f g) = Ψ⁽ⁿ⁺¹⁾(f g)−Φ⁽ⁿ⁺¹⁾(f g)

=

n+1

X

i=0

Ψ⁽ⁱ⁾(f)Ψ^(n+1−i)(g)−Φ⁽ⁱ⁾(f)Φ^(n+1−i)(g)

= Ψ⁽⁰⁾(f)Ψ⁽ⁿ⁺¹⁾(g)−Φ⁽⁰⁾(f)Φ⁽ⁿ⁺¹⁾(g) + Ψ⁽ⁿ⁺¹⁾(f)Ψ⁽⁰⁾(g)−Φ⁽ⁿ⁺¹⁾(f)Φ⁽⁰⁾(g) +

n

X

i=1

Ψ⁽ⁱ⁾(f)Ψ^(n+1−i)(g)−Φ⁽ⁱ⁾(f)Φ^(n+1−i)(g).

(4.42) By assumption, Φ⁽ⁱ⁾= Ψ⁽ⁱ⁾, for every 1≤i≤n. Thus,

Ψ⁽ⁱ⁾)(f)Ψ^(n+1−i)(g)−Φ⁽ⁱ⁾(f)Φ^(n+1−i)(g) = Φ⁽ⁱ⁾(f)Φ^(n+1−i)(g)−Φ⁽ⁱ⁾(f)Φ^(n+1−i)(g) = 0 and the sum in expression4.42vanishes. Substituting Ψ⁽⁰⁾ with Φ⁽⁰⁾ gives

(Ψ−Φ)⁽ⁿ⁺¹⁾(f g) = Φ⁽⁰⁾(f)Ψ⁽ⁿ⁺¹⁾(g)−Φ⁽⁰⁾(f)Φ⁽ⁿ⁺¹⁾(g) + Ψ⁽ⁿ⁺¹⁾(f)Φ⁽⁰⁾(g)−Φ⁽ⁿ⁺¹⁾(f)Φ⁽⁰⁾(g)

= (Ψ−Φ)⁽ⁿ⁺¹⁾(f)Φ⁽⁰⁾(g) + Φ⁽⁰⁾(f)(Ψ−Φ)⁽ⁿ⁺¹⁾(g)

which means that (Ψ−Φ)⁽ⁿ⁺¹⁾is a Φ⁽⁰⁾-derivation. Moreover, Ψ−Φ is a Lie ∞-morphism, since both Ψ and Φ are such. First,

QF(Ψ−Φ) =QFΨ−QFΦ = ΨQE−ΦQE= (Ψ−Φ)QE, and a closer look at the components of arityn+ 1 in the last equation, reveals that

n+1

X

i=0

Q⁽ⁱ⁾_F (Ψ−Φ)^(n+1−i)=

n+1

X

i=0

(Ψ−Φ)^(n+1−i)Q⁽ⁱ⁾_E ⇐⇒

Q⁽⁰⁾_F (Ψ−Φ)⁽ⁿ⁺¹⁾+

n

X

i=0

Q⁽ⁱ⁾_F (Ψ−Φ)^(n+1−i)= (Ψ−Φ)⁽ⁿ⁺¹⁾Q⁽⁰⁾_E +

n

X

i=0

(Ψ−Φ)^(n+1−i)Q⁽ⁱ⁾_E ⇐⇒

Q⁽⁰⁾_F (Ψ−Φ)⁽ⁿ⁺¹⁾−(Ψ−Φ)⁽ⁿ⁺¹⁾Q⁽⁰⁾_E = 0, that is,∂ (Ψ−Φ)⁽ⁿ⁺¹⁾

= 0. Thus, (Ψ−Φ)⁽ⁿ⁺¹⁾is a cocycle in the bicomplex D⁽ⁿ⁺¹⁾ of degree 0 and depthn+ 1, so it is a root-free cocycle. According to Lemma2.6, there exists an element Hˆ⁽ⁿ⁺¹⁾∈D⁽ⁿ⁺¹⁾of degree -1 such that

(Ψ−Φ)⁽ⁿ⁺¹⁾=∂( ˆH⁽ⁿ⁺¹⁾) (4.43)

which we extend to a Φ⁽⁰⁾-derivationH⁽ⁿ⁺¹⁾:E → F of degree -1:

H⁽ⁿ⁺¹⁾(f g) =H⁽ⁿ⁺¹⁾(f)Φ⁽⁰⁾(g) + (−1)^|f|Φ⁽⁰⁾(f)H⁽ⁿ⁺¹⁾(g).

Moreover, fork≥n+ 2, consider theO-linear map ˆH^(k): Γ(E^∗)→Γ(Sⁿ⁺³F^∗) given by

Hˆ^(k)(f) =H⁽ⁿ⁺¹⁾(Φ^(k−n−1)(f)) (4.44)

We want to construct a Lie∞-morphismχthat is homotopic to Φ and satisfiesχ⁽ⁱ⁾ = Ψ⁽ⁱ⁾, for every 0≤i≤n+ 1. Roughly speaking, the idea is to deform the morphism Φ to a Lie∞-morphism with the desired property. A homotopy (Φt, Ht) will arise from this deformation.

First, we extend the Φ⁽⁰⁾-derivationH⁽ⁿ⁺¹⁾as follows. For everyt∈[0,1], consider the mapH_2n+1|t: E → F, given by

∀0≤k≤n, H_2n+1|t^(k) = 0,

∀n+ 1≤k≤2n+ 1, H_2n+1|t^(k) (f g) =

k

X

i=0

H_2n+1|t⁽ⁱ⁾ (f)Φ^(k−i)(g) + (−1)^|f|Φ^(k−i)(f)H_2n+1|t⁽ⁱ⁾ (g).

(4.45)

Practically, for everyt ∈ [0,1] and everyk ∈ {n+ 1, ...,2n+ 1}, ˆH_2n+1|t^(k) is equal to ˆH^(k), given by4.44, and is extended it to anO-linear mapH_2n+1|t^(k) : E → F by3.60, so thatH_2n+1|tis a Φ- derivation. Note thatH_2n+1|tdoes not actually depend on the variabletfor any arity components and thatH_2n+1|t⁽ⁿ⁺¹⁾ =H⁽ⁿ⁺¹⁾. Moreover, for everyt∈[0,1], we define a map Φ_2n+1|t:E → F by

∀0≤k≤2n+ 1, Φ^(k)_2n+1|t= Φ^(k)+t[Q, H_2n+1|t]^(k). (4.46) Observe that the following boundary conditions are then satisfied:

Φ⁽ⁿ⁺¹⁾_2n+1|t=0= Φ⁽ⁿ⁺¹⁾ and Φ⁽ⁿ⁺¹⁾_2n+1|t=1⁽ⁿ⁺¹⁾= Ψ⁽ⁿ⁺¹⁾.

The first condition is obvious, actually we have that Φ_2n+1|t=0 = Φ. For the second one, we take a look at [Q, H_2n+1|t]⁽ⁿ⁺¹⁾:

[Q, H_2n+1|t]⁽ⁿ⁺¹⁾=

n+1

X

i=0

Q^(k−i)_F H_2n+1|t⁽ⁱ⁾ +H_2n+1|t⁽ⁱ⁾ Q⁽ⁱ⁾_E =Q⁽⁰⁾_F H⁽ⁿ⁺¹⁾+H⁽ⁿ⁺¹⁾Q⁽⁰⁾_E =∂H⁽ⁿ⁺¹⁾ sinceH_2n+1|t⁽ⁿ⁺¹⁾ = H⁽ⁿ⁺¹⁾. In particular, [Q, H_2n+1|t]^(k) = 0, for everyk ≤ n. Finally, with a use of identity4.43, we obtain that

Φ⁽ⁿ⁺¹⁾_2n+1|t=1= Φ⁽ⁿ⁺¹⁾+ [Q, H_2n+1|t]⁽ⁿ⁺¹⁾= Φ⁽ⁿ⁺¹⁾+∂H⁽ⁿ⁺¹⁾= Ψ⁽ⁿ⁺¹⁾. (4.47) Note however that Φ_2n+1|t=1 may not coincide with Ψ for higher arities.

Next, we show that, for everyt ∈ [0,1], Φ_2n+1|t is a morphism of algebras and thatH_2n+1|tis a Φ_2n+1|t-derivation. The second claim is almost obvious: for everyt∈[0,1], it follows from4.45that

H_2n+1|t^(k) (f g) =

k

X

i=0

H_2n+1|t⁽ⁱ⁾ (f)Φ^(k−i)_2n+1|t(g) + (−1)^|f|Φ^(k−i)_2n+1|t(f)H_2n+1|t⁽ⁱ⁾ (g) (4.48) where we have used the following to modify the sum in4.45and end up in 4.48:

• For 0≤i≤n,H_2n+1|t⁽ⁱ⁾ is zero, so we can writeH_2n+1|t⁽ⁱ⁾ (f)Φ^(k−i)_2n+1|t(g) instead ofH_2n+1|t⁽ⁱ⁾ (f)Φ^(k−i)(g) and respectively Φ^(k−i)_2n+1|t(f)H_2n+1|t⁽ⁱ⁾ (g) instead of Φ^(k−i)(f)H_2n+1|t⁽ⁱ⁾ (g).

• Observe that the n-first arity components of Φ and Φ_2n+1|tcoincide. Thus, forn+ 1 ≤ i ≤ k, we have that 0 ≤ k−i ≤ nand we can write Φ^(k−i)_2n+1|t(f) (resp. Φ^(k−i)_2n+1|t(g) ) instead of Φ^(k−i)(f) (resp. Φ^(k−i)(g).

The proof of Φ_2n+1|tbeing a morphism of graded algebras is more complicated and consists of two parts.

• First, we prove that [Q, H_2n+1|t] is a Φ_2n+1|t-derivation of degree 0. This would come imme- diately by Lemma3.9, only if we knew a priori that Φ_2n+1|tis a Lie∞-morphism. By the derivation property 4.48forH_2n+1|t, we have that

[Q, H_2n+1|t](f g) =QFH_2n+1|t(f g) +H_2n+1|tQE(f g)

=QF

H_2n+1|t(f)Φ_2n+1|t(g) + (−1)^|f|Φ_2n+1|t(f)H_2n+1|t(g)

+H_2n+1|t

Q_E(f)g+ (−1)^|f|f Q_E(g)

=QF H_2n+1|t(f)

Φ_2n+1|t(g) + (−1)^|f|−1H_2n+1|t(f)QF Φ_2n+1|t(g) + (−1)^|f|

QF Φ_2n+1|t(f)

H_2n+1|t(g) + (−1)^|f|Φ_2n+1|t(f)QF H_2n+1|t(g)

+H_2n+1|t QE(f)

Φ_2n+1|t(g) + (−1)^|f|+1Φ_2n+1|t QE(f)

H_2n+1|t(g) (−1)^|f|

H_2n+1|t(f)Φ_2n+1|t QE(g)

+ (−1)^|f|Φ_2n+1|t(f)H_2n+1|t QE(g)

.

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

Then, a factorisation of the above expression yields

[Q, H_2n+1|t](f g) = [Q, H_2n+1|t](f)Φ_2n+1|t(g) + Φ_2n+1|t(f)[Q, H_2n+1|t](g) + (−1)^|f|

H_2n+1|t(f)[Q,Φ_2n+1|t](g) + [Q,Φ_2n+1|t](f)H_2n+1(g)

. (4.49) However, the last term in4.49vanishes because [Q,Φ_2n+1|t] is equal to zero. Indeed, it suffices a look at its arity components. For everyk∈N,

[Q,Φ_2n+1|t]^(k)=

k

X

i=0

Q^(k−i)_F Φ⁽ⁱ⁾_2n+1|t−Φ⁽ⁱ⁾_2n+1|tQ^(k−i)_E

=

k

X

i=0

Q^(k−i)_F Φ⁽ⁱ⁾+t[Q, H_2n+1|t]⁽ⁱ⁾

− Φ⁽ⁱ⁾+t[Q, H_2n+1|t]⁽ⁱ⁾ Q^(k−i)_E

=

k

X

i=0

Q^(k−i)_F Φ⁽ⁱ⁾−Φ⁽ⁱ⁾Q^(k−i)_E +t

k

X

i=n0

Q^(k−i)_F [Q, H_2n+1|t]⁽ⁱ⁾−[Q, H_2n+1|t]⁽ⁱ⁾Q^(k−i)_E

= [Q,Φ]^(k)+t[Q,[Q, H_2n+1|t]]^(k)

(4.50) which vanishes because Φ is a Lie∞-morphism and [Q,·] squares to zero. This concludes the proof of [Q, H_2n+1|t] being a Φ_2n+1|t-derivation.

• We apply the following trick to show that Φ2n+1 is a morphism of graded algebras. For every t∈[0,1] and everyf, g∈ E, we have that

d dt

Φ^(k)_2n+1|t(f g)−

k

X

i=0

Φ⁽ⁱ⁾_2n+1|t(f)Φ^(k−i)_2n+1|t(g)

= [Q, H_2n+1|t]^(k)(f g)−

k

X

i=0

[Q, H_2n+1|t⁽ⁱ⁾ ](f)Φ^(k−i)_2n+1|t(g)−Φ⁽ⁱ⁾_2n+1|t(f)[Q, H_2n+1|t]^(k−i)(g)

(4.51)

which vanishes since [Q, H_2n+1|t] satisfies the Φ_2n+1|t-derivation property. Thus, for everyt∈ [0,1],

Φ^(k)_2n+1|t(f g)−

k

X

i=0

Φ⁽ⁱ⁾_2n+1|t(f)Φ^(k−i)_2n+1|t(g) (4.52) is constant and since Φ_2n+1|t=0 = Φ is a morphism of graded algebras, it follows that4.52is equal to zero for everyk∈N, i.e. Φ_2n+1|tis a morphism of graded algebras.

We must indicate that4.50now implies that Φ_2n+1|t is in fact a Lie∞-morphism. Moreover, it is evident thatt7→Φ_2n+1|t is a differentiable path of Lie∞-morphisms.

For arity 2n+ 2, we extendH_2n+1|tto a mapH_2n+2|t:E → F given by

∀0≤k≤2n+ 1, H_2n+2|t^(k) =H_2n+1|t^(k) H_2n+2|t⁽²ⁿ⁺²⁾(f g) =

2n+2

X

i=0

H_2n+1|t⁽ⁱ⁾ (f)Φ^(2n+2−i)_2n+1|t (g) + (−1)^|f|Φ^(2n+2−i)_2n+1|t (f)H_2n+1|t⁽ⁱ⁾ (g)

(4.53)

In analogy withH_2n+1|t, ˆH_2n+2|t⁽²ⁿ⁺²⁾ equalsH⁽²ⁿ⁺²⁾ˆ , which is given by4.44for k=2n+1, and then is extended to anO-linear mapH_2n+2|t⁽²ⁿ⁺²⁾ : E → F according to3.60, so thatH_2n+2|tis a Φ_2n+1|t- derivation. Thus we are able to define Φ⁽²ⁿ⁺²⁾_t as the solution to the following boundary value prob- lem

(Φ⁽²ⁿ⁺²⁾_t=0 = Φ⁽²ⁿ⁺²⁾,

dΦ⁽²ⁿ⁺²⁾_t

dt = [Q, H_2n+2|t]²ⁿ⁺².

This ordinary differential equation admits a unique solution and thus such a map Φ⁽²ⁿ⁺²⁾_t exists in- deed. In fact, it is straightforward that the solution to this ODE is actually Φ⁽²ⁿ⁺²⁾+t[Q, H_2n+2|t].

Finally, we extend Φ_2n+1|tto a map Φ_2n+2|t:E → F by

∀0≤k≤2n+ 1, Φ^(k)_2n+2|t= Φ^(k)_2n+1|t Φ⁽²ⁿ⁺²⁾_2n+2|t = Φ⁽²ⁿ⁺²⁾_t .

(4.54)

We check with similar computations that Φ_2n+2|t is a morphism of graded algebras and thatH_2n+2|t is a Φ_2n+2|t-derivation.

Recursively, we can define Φ_i|tandH_i|tfori ≥2n+ 3. For everyt ∈ [0,1], we consider the unique morphism of graded algebras Φtwhose restriction to components up to arityi(fori ≥ 2n+ 1) coincides with Φ_i|t. Moreover, we denote byHtthe unique Φt-derivation that coincides withH_i|tup to arityi, for everyi≥2n+ 1. By definition, the two maps are related by

dΦ⁽ⁱ⁾_t

dt = [Q, Ht]⁽ⁱ⁾

for everyt∈[0,1] and everyi≥0. In particular, this indicates that Φtis differentiable on the inter- val [0,1]. We have that Φt=0 = Φ, by construction, and we defineχto be the morphism Φt=1. The commutator [Q,Φt] is differentiable for everyt∈[0,1] and its derivative with respect to the param- etertis equal to zero, since ^dΦ_dt^t is a [Q,·]-exact term. This means that [Q,Φt] is constant and then Φt=0 = Φ being a Lie∞-morphism, i.e. [Q,Φ] = 0, implies that [Q,Φt] is equal to zero for every t∈[0,1]. In particular, this implies that (Φt)_t∈[0,1] is a differentiable path of Lie ∞-morphisms and thatχ = Φt=1is also a Lie∞-morphism. By construction, (Φt, Ht) is a homotopy from Φ toχ and χcoincides with Ψ up to arity n+ 1.

Before proving Proposition4.5, we need to introduce the concept of homotopy deformations up to arityn.

Definition 4.5. Let n ∈ N∪ {0,+∞} letΦandΨbe two Lie∞-morphisms from(F, Q_F)to

(E, QE). A homotopy deformation ofΦtoΨup to arityn is a sequence of Lie∞-morphisms(Φk)_−1≤k≤n such that

(i) Φ₋₁= Φ,

(ii) For every 0≤k≤n,Φk andΨcoincide up to arityk, i.e. Φ⁽ⁱ⁾_k = Ψ⁽ⁱ⁾ for every0≤i≤k.

(iii) For every−1 ≤ k ≤ n−1,Φk andΦk+1 are homotopic Lie ∞-morphisms with homotopy (Φτ, Hτ)_τ∈[k,k+1]such that Hτ is aΦτ-derivation whose components of arities lesser than or equal to k are equal to zero.

At infinity, a homotopy deformation should be a homotopy between two Lie∞-morphisms.

Lemma 4.8. Let ΦandΨbe Lie∞-morphisms such that there exists a homotopy deformation of ΦtoΨof arity +∞. Letf : [0,1]→ [−1,∞]be a strictly increasing surjective C¹ function. Then, (Φ_f(t), f⁰(t)H_f(t))is a homotopy between ΦandΦ_f(1):= Ψ.

Proof. Let (Φk)_k≥−1be a homotopy deformation of arity +∞of Φ to Ψ.

For everyn≥0, items (ii),(iii) in Definition4.5imply that Φ⁽ⁿ⁾τ = Ψ⁽ⁿ⁾, forτ ≥n. Given an arbi- traryτ ≥n, there exists some integerk0≥nsuch thatτ∈[k0, k0+ 1]. Since Φ⁽ⁿ⁾_k

0 = Φ⁽ⁿ⁾_k

0+1= Ψ⁽ⁿ⁾, the homotopy (Φτ, Hτ)_τ∈[k0,k₀+1]that deforms Φk₀ to Φk₀+1must be constant at aritiesτ lesser than or equal tok0, i.e. Φ⁽ⁿ⁾τ = Ψ⁽ⁿ⁾.

Moreover, for a fixedn, there exists a neighborhoodUn ⊂ [0,1] of 1 such that Φ⁽ⁿ⁾_f(t) = Ψ⁽ⁿ⁾, for everyt∈Un and we can takeUn to be the semi-closed interval (tn,1] withf(tn) =n.

4.3. THE UNIVERSALITY OF A LIE∞-ALGEBROID RESOLVING A FOLIATION

We shall show thatt7→Φ⁽ⁿ⁾_f(t)is a piecewise-C¹path. In particular, we only have to prove that Φ⁽ⁿ⁾_f(t) is piecewise-C¹in [0, tn] sincet7→Φ⁽ⁿ⁾_f(t)is constant and henceC¹on [tn,1]. According to Definition 3.18, for every 0 ≤ i ≤ n, there exists a subdivision{i = tⁱ₀ < tⁱ₁ < ... < t_ki = i+ 1}with re- spect to which (Φ⁽ⁿ⁾τ )_τ∈[i,i+1]is piecewise-C¹. Now, we consider the partition of [0,1] into the closed intervals

[f⁻¹(tⁱ_j), f⁻¹(tⁱ_j+1)], wherei = −1, ..., nandj = 0,1, ..., ki. Actually,tⁿ_k

n = tn. This is indeed a partition becausef is surjective. Thus, fort ∈ [f⁻¹(tⁱ_j), f⁻¹(tⁱ_j+1)],f(t) takes values in [tⁱ_j, tⁱ_j+1] on which Φ_τ isC¹. Consequently,t7→Φ⁽ⁿ⁾_f(t) is piecewise-C¹ with respect to the partition

{f⁻¹(tⁱ_j) :j = 0,1, ..., ki, i=−1,0, ..., n} ∪ {tn}

Similarly, one can construct a partition of [0,1] with respect to whicht7→f⁰(t)H_f(t)⁽ⁿ⁾ is piecewise-C¹ Φ_f(t)-derivation. For anyt∈[0,1] at which Φ_f(t)is differentiable, sayf(t)∈[i, i+ 1], we have that

d

dt(Φ_f(t)) =dΦτ

dτ |_τ=f(t)·df

dt(t) =f⁰(t)[Q, H_f(t)] = [Q, f⁰(t)H_f(t)]

because (Φτ, Hτ)_τ∈[i,i+1]is a homotopy. Eventually, it is now clear that (Φ_f(t), f⁰(t)H_f(t)) is a ho- motopy from Φ to Ψ.

The following result derives immediately from Lemma4.8.

Corollary 4.1. Let ΦandΨbe two Lie ∞-morphisms such that there exists a homotopy deforma- tion of arity +∞of ΦtoΨ. Then,ΦandΨare homotopic Lie ∞-morphisms.

Remark 4.8. A homotopy deformation of ΦintoΨof arity n can be thought of as an approxima- tion ofΨby Φup to ordern. We believe that it is an interesting generalisation of the concept of homotopy, as developed in the previous Chapter.

We are now in position to prove the following important result.

Proposition 4.4. Two Lie ∞-morphisms whose linear parts coincide are homotopic.

Proof. Let Φ and Ψ be two such morphisms. By assumption, we have that Φ⁽⁰⁾ = Ψ⁽⁰⁾. According to Lemma4.8, there exists a Lie∞-morphism Φ1 that is homotopic to Φ with homotopy (Φ0,t, H0,t).

In particular, Φ₁satisfies Φ⁽⁰⁾₁ = Ψ⁽⁰⁾ and Φ⁽¹⁾₁ = Ψ⁽¹⁾. By a further application of Lemma4.8, we obtain a Lie∞-morphism Φ₂that is homotopic to Φ₁with homotopy (Φ_1,t, H_1,t) and satisfies Φ⁽ⁱ⁾₂ = Ψ⁽ⁱ⁾, fori= 0,1,2.

Recursively, for everyk ≥ 0, we have a family of Lie∞-morphisms (Φi)−1≤i≤k, where we have set Φ₋₁ = Φ₀ = Φ. For every 0 ≤ i ≤ k, (Φ_τ, H_τ)_τ∈[i,i+1] = (Φ_i|i+it, H_i|i+it)_t∈[0,1] and by construc- tion (recall the inductive steps of the proof of Lemma4.8) the components ofH_i|i+it of arity less than or equal toiare zero. In particular, this is a homotopy deformation of Φ into Ψ of arity +∞.

According to Proposition4.5, Φ and Ψ are homotopic Lie∞-morphisms.

However, we have a somehow more general condition under which two Lie∞-morphisms are homo- topic.

Proposition 4.5. Let φ, ψ : (F, d⁰, ρ⁰)→(E, d, ρ)be two homotopic chain maps between the linear parts of a Lie∞-algebroid(F, QF)that covers F⁰ ⊂ F and a universal Lie∞-algebroid(E, QE)of F. Then any two Lie∞-morphismsΦ,Ψfrom(F, QF)to(E, QE)whose respective linear parts are φandψ are homotopic.

Proof. For eveyt∈[0,1], we consider the chain mapφt= (1−t)φ+tψ. Moreover, for everyt∈[0,1], we consider a Lie∞-morphism Φt : E → F whose linear part is the dual map ofφt: according to Proposition4.2such a morphism exists for everyt. In particular, we set Φt=0 = Φ but there is no

need to assume that Φt=1 is Ψ. However, note that in any case Φ⁽⁰⁾_t=1 = Ψ⁽⁰⁾ since the chain mapφ1

is equal toψ. Thus, we have a patht 7→ Φtof Lie∞-morphisms which we can assume that isC¹ for everyt ∈ [0,1], i.e. t 7→ Φ⁽ⁱ⁾_t isC¹for every i ≥0. This is achievable because of the technique that we deployed in the proof of Lemma4.3.

Let us make this statement more precise. For eacht ∈ [0,1], Φ⁽⁰⁾_t being the dual map ofφ_tmeans that it isC¹:

Φ⁽⁰⁾_t = (1−t)Φ⁽⁰⁾+tΨ⁽⁰⁾.

Furthermore, for everyt∈[0,1], the righ-hand side of Equation4.30which is Φ⁽⁰⁾_t Q⁽¹⁾_E −Q⁽¹⁾_F Φ⁽⁰⁾_t

isC¹as well. So, we can define a path t 7→ Φ⁽¹⁾_t which isC¹ and satisfies the relation∂(Φ⁽¹⁾_t ) = Φ⁽⁰⁾_t Q⁽¹⁾_E −Q⁽¹⁾_F Φ⁽⁰⁾_t . Similarly, for everyt ∈ [0,1], the right-hand side of Equation4.30forn = 2, which reads

Φ⁽¹⁾_t Q⁽¹⁾_E −Q⁽¹⁾_F Φ⁽¹⁾_t + Φ⁽²⁾_t Q⁽²⁾_E −Q⁽²⁾_F Φ⁽²⁾_t ,

isC¹since it only involves Φ⁽⁰⁾_t and Φ⁽¹⁾_t . Thus, we can consider aC¹ patht 7→ Φ⁽²⁾_t that solves Equation4.30forn= 2, for every t∈[0,1]. Recursively, we can show that the quantityCn+1which is defined by the right-hand side of4.30:

C_n+1= X

1≤i,j+1≤n+1

i+j=n+1

Φ⁽ⁱ⁾_t Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ⁽ⁱ⁾_t ,

isC¹for every t∈[0,1] and hence, we can choose aC¹ patht 7→Φ⁽ⁿ⁺¹⁾_t that solves Equation 4.30, i.e.

∂(Φ⁽ⁿ⁺¹⁾_t ) = X

1≤i,j+1≤n+1

i+j=n+1

Φ⁽ⁱ⁾_t Q⁽ⁱ⁾_E −Q⁽ⁱ⁾_F Φ⁽ⁱ⁾_t .

Finally, this method provides a path of Lie∞-morphismst 7→ Φ_t whose components of homoge- neous arities areC¹. In particular, we have that Φ⁽⁰⁾_t=0 = Φ⁽⁰⁾ and Φ⁽⁰⁾_t=1 = Ψ⁽⁰⁾, but Φ_t=1 may not coincide with Ψ at higher arities. To obtain Φ as Φ_t=0, we only have to impose the initial condition Φ⁽ⁱ⁾_t=0= Φ⁽ⁱ⁾, for everyi≥1.

Next, we show that ^dΦ_dt^t is [Q,·]-exact for everyt ∈ [0,1], i.e. we construct aC¹patht 7→ Htof Φt-derivations of degree -1 such that

dΦ_t

dt = [Q, H_t]. (4.55)

Leth: F →E be the homotopy between the chain mapsφ, ψ and for everyt ∈ [0,1] consider the mapH_t⁽⁰⁾ : E → F which is the dual map ofhextended as a Φ_t-derivation (as a Φ⁽⁰⁾_t -derivation actually). The homotopy equationψ−φ=d◦h+h◦d⁰ translates into

Ψ⁽⁰⁾−Φ⁽⁰⁾=Q⁽⁰⁾_F H_t⁽⁰⁾+H_t⁽⁰⁾Q⁽⁰⁾_E = [Q⁽⁰⁾, H_t⁽⁰⁾]. (4.56) The left-hand size is equal to dΦ⁽⁰⁾_t

dt , so 4.56is dΦ⁽⁰⁾_t

dt = [Q⁽⁰⁾, H_t⁽⁰⁾]. For the rest of the proof, we write [Q⁽⁰⁾, H_t⁽⁰⁾] forQ⁽⁰⁾_F ◦H_t⁽⁰⁾+H_t⁽⁰⁾◦Q⁽⁰⁾_E . Then, whenever it is defined we see that

∂ dΦ⁽¹⁾_t dt ) = d

dt∂(Φ⁽¹⁾_t ) =dΦ⁽⁰⁾_t

dt Q⁽¹⁾_E −Q⁽¹⁾_F dΦ⁽⁰⁾_t

dt = [Q⁽⁰⁾, H_t⁽⁰⁾]Q⁽¹⁾_E −Q⁽¹⁾_F [Q⁽⁰⁾, H_t⁽⁰⁾]

=Q⁽⁰⁾_F H_t⁽⁰⁾Q⁽¹⁾_E +H_t⁽⁰⁾Q⁽⁰⁾_E Q⁽¹⁾_E −Q⁽¹⁾_F Q⁽⁰⁾_F H_t⁽⁰⁾−Q⁽¹⁾_F H_t⁽⁰⁾Q⁽⁰⁾_E .

(4.57)

Moreover, the action of the operator∂ on [Q⁽¹⁾, H_t⁽⁰⁾] =Q⁽¹⁾_F H_t⁽⁰⁾+H_t⁽⁰⁾Q⁽¹⁾_E gives:

∂([Q⁽¹⁾, H_t⁽⁰⁾]) =Q⁽⁰⁾_F Q⁽¹⁾_F H_t⁽⁰⁾+Q⁽⁰⁾_F H_t⁽⁰⁾Q⁽¹⁾_E −Q⁽¹⁾_F H_t⁽⁰⁾Q⁽⁰⁾_E −H_t⁽⁰⁾Q⁽¹⁾_E Q⁽⁰⁾_E . (4.58)

No documento A thesis presented for the degree of Master of Science (páginas 78-96)