Joint work with Yves Félix, Aniceto Murillo and Daniel Tanré

If 𝒻: 𝑿 ⟶ 𝒀 is a continuous map between simply

connected CW-complexes, the following properties are

equivalent:

1. 𝜋𝑛 𝒻 ⨂ℚ ∶ 𝜋𝑛 𝑿 ⨂ℚ⟶ 𝜋𝑛 𝒀 ⨂ℚ , 𝑛 ≥ 2.≅

2. 𝐻𝑛 𝒻 ⨂ℚ ∶ 𝐻𝑛 𝑿 ;ℚ ⟶ 𝐻𝑛 𝒀 ;ℚ , 𝑛 ≥ 2.≅

Such a map is called a rational homotopy equivalence.

𝑿 is rational if its homotopy groups are ℚ-vector spaces.

A rationalisation of 𝑿 is a pair (𝑿ℚ , 𝑒), with 𝑿ℚ a rational

space and and 𝑒 ∶ 𝑿 ⟶ 𝑿ℚ a rational homotopy equivalence.

The study of the rational homotopy type of 𝑿 is the

study of the homotopy type of its rationalisation 𝑿ℚ.

The study of the rational homotopy type of 𝑿 is the

study of the homotopy type of its rationalisation 𝑿ℚ.

Then, if 𝑿 is a finite simply connected CW-complex

𝜋𝑛 𝑿 = ⨁𝑟ℤ ⨁ ℤ𝑝1𝑟1⨁⋯⨁ ℤ𝑝𝑚𝑟𝑚

𝜋𝑛 𝑿ℚ ≅ 𝜋𝑛 𝑿 ⨂ℚ = ⨁𝑟ℚ

The rational homotopy

type of 𝑿 is completely

determined in algebraic

terms.

DGL CDGA

A differential graded Lie algebra is a

graded vector space 𝐿 = ⨁𝑝𝜖ℤ 𝐿𝑝 with:

A bilinear operation . , . ∶ 𝐿 × 𝐿 ⟶ 𝐿

such that 𝐿𝑝 , 𝐿𝑞 ⊂ 𝐿𝑝+𝑞 satisfying:

𝑎, 𝑏 = − −1 𝑝𝑞 𝑏, 𝑎 , 𝑎 ∈ 𝐿𝑝 , 𝑏 ∈ 𝐿𝑞

𝑎, [𝑏, 𝑐] = 𝑎, 𝑏 , 𝑐 + −1 𝑝𝑞 𝑏, [𝑎, 𝑐] ,

𝑎 ∈ 𝐿𝑝 , 𝑏 ∈ 𝐿𝑞 , 𝑐 ∈ 𝐿

a)

b)DGL

A linear map 𝜕 ∶ 𝐿 ⟶ 𝐿 such that

𝜕𝐿𝑝 ⊂ 𝐿𝑝−1 satisfying:

𝜕 ∘ 𝜕 = 0

𝜕 𝑎, 𝑏 = 𝜕𝑎, 𝑏 + −1 𝑝 𝑎, 𝜕𝑏 ,

𝑎 ∈ 𝐿𝑝 , 𝑏 ∈ 𝐿

A differential graded Lie algebra is a

graded vector space 𝐿 = ⨁𝑝𝜖ℤ 𝐿𝑝 with:

a)

b)DGL

DGL

𝐒𝐢𝐦𝐩𝐥𝐲 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 DGL +⟶

𝜆 ⟶

< ∙ >𝑄

< 𝜆(𝑿) >𝑄 ≃ 𝑿ℚ

Rational homotopy

equivalencesQuasi-isomorphisms

(𝐿, ∙,∙ , 𝜕) is a DGL-model of 𝑿 if

𝜆(𝑿) 𝐿⋯⟶≃ ⟶≃

⟶≃ ⟶≃

DGL

𝐒𝐢𝐦𝐩𝐥𝐲 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 DGL +⟶

𝜆 ⟶

< ∙ >𝑄

< 𝜆(𝑿) >𝑄 ≃ 𝑿ℚ

Rational homotopy

equivalencesQuasi-isomorphisms

(𝐿, ∙,∙ , 𝜕) is a DGL-model of 𝑿 if

(𝐿, ∙,∙ , 𝜕)⟶≃

(𝕃(𝑊), 𝑑)

Minimal Quillenmodel

𝐻𝑛( )𝐿

Rational homotopy groups

≅ 𝜋𝑛+1(𝑿)⨂ℚ

Rational homology groups

𝐿(𝕃 𝑊 , 𝜕) ⟶≃

𝑠(𝑊 ⊕ℚ) ≅ 𝐻∗(𝑋; ℚ)DGL

DGL

Spheres

Products

Wedge products

𝕃 𝑣 , 0 , 𝑣 = 𝑛 − 1is a model for the 𝑛-dimensional sphere 𝑆𝑛.

If (𝐿, 𝜕) and (𝐿′, 𝜕′) are DGL-models

for 𝑿 and 𝒀 respectively, then 𝐿 × 𝐿′, 𝜕 × 𝜕′

is a DGL-model of 𝑿 × 𝒀.

If (𝕃 𝑉 , 𝑑) and (𝕃 𝑊 , 𝑑′) are minimal DGL

models for 𝑿 and 𝒀 respectively, then

(𝕃 𝑉 ⊕𝑊 ,𝐷) is a DGL-model of 𝑿 ∨ 𝒀.

A based topological space (𝑿, ∗) ( 𝕃 𝑉 , 𝜕 )

𝑿𝑛 𝑛-skeleton ( 𝕃(𝑉<𝑛), 𝜕 )

= 𝑒𝛼𝑛+1 𝑣𝛼 ∈ 𝑉𝑛

𝑓𝛼 ∶ 𝑆𝑛 → 𝑿𝑛 ; [𝑓𝛼] ∈ Π𝑛(𝑿𝑛)

𝜕𝑣𝛼 ∈ 𝐻𝑛−1(𝕃 𝑉<𝑛 )

≅

Π𝑛(𝑿𝑛)⨂ℚ

CW-decomposition

A commutative differential graded algebra

is a graded vector space 𝐴 = ⨁𝑝𝜖ℤ 𝐴𝑝 with:

A bilinear operation ∙ ∶ 𝐴 × 𝐴 ⟶ 𝐴

such that 𝐴𝑝 ∙ 𝐴𝑞 ⊂ 𝐴𝑝+𝑞 satisfying:

𝑎 ∙ 𝑏 = −1 𝑝𝑞𝑏 ∙ 𝑎, 𝑎 ∈ 𝐴𝑝 , 𝑏 ∈ 𝐴𝑞

𝑎 ∙ (𝑏 ∙ 𝑐) = (𝑎 ∙ 𝑏) ∙ 𝑐 , 𝑎, 𝑏, 𝑐 ∈ 𝐴a)

b)CDGA

CDGA

A commutative differential graded algebra

is a graded vector space 𝐴 = ⨁𝑝𝜖ℤ 𝐴𝑝 with:

A linear map 𝑑: 𝐴 ⟶ A such that

𝑑𝐴𝑝 ⊂ 𝐴𝑝+1 satisfying:

𝑑 ∘ 𝑑 = 0

𝑑(𝑎 ∙ 𝑏) = (𝑑𝑎) ∙ 𝑏 + −1 𝑝𝑎 ∙ (𝑑𝑏) ,

𝑎 ∈ 𝐿𝑝 , 𝑏 ∈ 𝐿

a)

b)

CDGA

𝐍𝐢𝐥𝐩𝐨𝐭𝐞𝐧𝐭, 𝐟𝐢𝐧𝐢𝐭𝐞 𝐭𝐲𝐩𝒆𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 CDGA +⟶

𝐴𝑃𝐿 ⟶

< ∙ >𝑆Rational homotopy

equivalencesQuasi-isomorphisms

< 𝐴𝑃𝐿(𝑿) >𝑆 ≃ 𝑿ℚ

(𝐴, 𝑑) is a CDGA-model of 𝑿 if

𝐴𝑃𝐿(𝑿) 𝐴⋯⟶≃ ⟶≃

⟶≃ ⟶≃

CDGA

⟶𝐴𝑃𝐿

𝐍𝐢𝐥𝐩𝐨𝐭𝐞𝐧𝐭, 𝐟𝐢𝐧𝐢𝐭𝐞 𝐭𝐲𝐩𝒆𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 CDGA +

⟶

< ∙ >𝑆Rational homotopy

equivalencesQuasi-isomorphisms

< 𝐴𝑃𝐿(𝑿) >𝑆 ≃ 𝑿ℚ

(𝐴, 𝑑) is a CDGA-model of 𝑿 if

(𝐴, 𝑑)⟶≃

(⋀𝑉, 𝑑)

CDGA

Rational homotopy groups

Rational cohomology groups

𝐻∗ 𝐴, 𝑑 ≅ 𝐻∗(𝑿; ℚ )

(𝐴, 𝑑)⟶≃

(⋀𝑉, 𝑑)

Hom(𝑉𝑘 , ℚ) ≅ 𝜋𝑘(𝑿)⨂ ℚ

Eilenberg-MacLane spaces

Wedge products

Products

⋀𝑣, 0 , 𝑣 = 𝑛is a model of the Eilenberg-MacLane

space 𝐾(ℤ, 𝑛).

If (𝐴, 𝑑) and (𝐴′, 𝑑′) are CDGA-models

for 𝑿 and 𝒀 respectively, then 𝐴 × 𝐴′, 𝑑 × 𝑑′

is a CDGA-model of 𝑿 ∨ 𝒀.

If (⋀𝑉, 𝑑) and (⋀𝑊, 𝑑′) are minimal

models for 𝑿 and 𝒀 respectively, then

(⋀ 𝑉 ⊕𝑊 ,𝐷) is a CDGA-model of 𝑿 × 𝒀.CDGA

CDGA

Postnikov decomposition

𝐒𝐢𝐦𝐩𝐥𝐲 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 DGL +⟶

𝜆 ⟶

< ∙ >𝑄

𝐍𝐢𝐥𝐩𝐨𝐭𝐞𝐧𝐭, 𝐟𝐢𝐧𝐢𝐭𝐞 𝐭𝐲𝐩𝒆𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 CDGA +⟶

𝐴𝑃𝐿 ⟶

< ∙ >𝑆

𝐒𝐢𝐦𝐩𝐥𝐲 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 DGL +

⟶𝜆

⟶

< ∙ >𝑄

𝐍𝐢𝐥𝐩𝐨𝐭𝐞𝐧𝐭, 𝐟𝐢𝐧𝐢𝐭𝐞 𝐭𝐲𝐩𝒆𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬 CDGA +⟶

𝐴𝑃𝐿 ⟶

< ∙ >𝑆

⟶⟶ sLA⟶⟶sCHA⟶⟶sGrp

< 𝐴 >𝑆= Hom𝑪𝑫𝑮𝑨(𝐴, Ω )

< 𝐴 >𝑆 is defined for 𝐴 a ℤ-graded CDGA but

< 𝐿 >𝑄 has only sense for positively graded DGL’s

𝐒𝐢𝐦𝐩𝐥𝐲 𝐜𝐨𝐧𝐧𝐞𝐜𝐭𝐞𝐝, 𝐟𝐢𝐧𝐢𝐭𝐞 𝐭𝐲𝐩𝒆𝐂𝐖− 𝐜𝐨𝐦𝐩𝐥𝐞𝐱𝐞𝐬

⟶

⟶ℒ 𝒞∗

The goal is to understand the rational behaviour of the spaces:

map 𝑿, 𝒀 , map∗ 𝑿, 𝒀 , map𝑓 𝑿, 𝒀 , map𝑓∗ (𝑿, 𝒀)

If 𝑿 and 𝒀 are finite type

CW-complexes

map 𝑿, 𝒀 has the homotopy

type of a CW-complex.

If 𝑿 and 𝒀 are nilpotent map𝑓 𝑿, 𝒀 is nilpotent.

If 𝑿 is a finite CW-complex map𝑓 𝑿, 𝒀 is of finite type.

If 𝑿 and 𝒀 are 1-connected map𝑓 𝑿, 𝒀 is 1-connected.

Let 𝑿 be a finite CW-complex and 𝒀 be a nilpotent,

finite type CW-complex.

(𝐵⋕, 𝛿)

⋀ V⨂𝐵⋕ , 𝐷

𝐷 𝑣⨂𝛽 = σ𝑗(−1)𝑤 |𝛽𝑗

′|(𝑢⨂𝛽𝑗′) (𝑤⨂𝛽𝑗

′′) + 𝑣⨂𝛿𝛽

𝑿 (𝐵 , 𝑑) CDGA CDGC 𝐵⋕𝑛 = 𝐵−𝑛

(⋀𝑉, 𝑑)𝒀 Sullivan model

if 𝑣 ∈ V such that d𝑣 = 𝑢𝑤 and 𝛽 ∈ 𝐵⋕ with

∆𝛽 = σ𝒋𝛽𝒋′⨂𝛽𝒋

′′V⨂𝐵⋕

The following Sullivan algebra is a model for map 𝑿, 𝒀 .

Let ϕ: (⋀𝑉, 𝑑) ⟶ 𝐵 be a model of 𝑓: 𝑿 ⟶ 𝒀 .

Let 𝐾ϕ be the differential ideal generated by 𝐴1 ∪ 𝐴2 ∪ 𝐴3

𝐴1 = (𝑉 ⊗ 𝐵⋕)<0 𝐴2 = 𝐷(𝑉 ⊗ 𝐵⋕)0

𝐴3 = 𝛼 − 𝜙 𝛼 𝛼 ∈ (𝑉 ⊗ 𝐵⋕)0 }

The projection

map𝑓 𝑿, 𝒀 ⟶ map 𝑿, 𝒀

⋀ V⨂𝐵⋕ , 𝐷 ⟶ ⋀ V⨂𝐵⋕ , 𝐷 /𝐾ϕ

is a model of the injection

⋀ V⨂𝐵⋕ , 𝐷 /𝐾ϕ ≅ ⋀𝑉 ⊗𝐵⋕1⨁(𝑉 ⊗ 𝐵⋕)≥2, 𝐷𝝓

𝑿 , 𝒀 with DGL-models 𝐿 , 𝐿′ , respectively.

𝑓: 𝑿 ⟶ 𝒀 .

map𝑓 𝑿, 𝒀 ,DGL-model of map𝑓 𝑿, 𝒀 ?

(𝐵⋕, 𝛿)𝑿 CDGC 𝐵⋕, 𝛿 = 𝒞∗(𝐿) Cartan-Eilenberg

(⋀𝑉, 𝑑)𝒀 Sullivan model ⋀𝑉, 𝑑 = 𝒞∗ 𝐿′ cochains on 𝐿′

map𝑓 𝑿, 𝒀 ,DGL-model of map𝑓 𝑿, 𝒀 ?

(𝐵⋕, 𝛿)𝑿 CDGC 𝐵⋕, 𝛿 = 𝒞∗(𝐿) Cartan-Eilenberg

(⋀𝑉, 𝑑)𝒀 Sullivan model ⋀𝑉, 𝑑 = 𝒞∗ 𝐿′ cochains on 𝐿′

Consider the ℤ-graded vector space 𝐻𝑜𝑚ℚ(𝒞∗ 𝐿 ′ 𝐿′).

Let f ∶ 𝒞∗ 𝐿 ⟶ 𝐿′ be a linear map of degree 𝑝.

𝔇f = f ∘ δ − −1 f 𝜕′ ∘ f ∶ 𝒞∗ 𝐿 ⟶ 𝐿′

(𝐵⋕, 𝛿)𝑿 CDGC 𝐵⋕, 𝛿 = 𝒞∗(𝐿) Cartan-Eilenberg

(⋀𝑉, 𝑑)𝒀 Sullivan model ⋀𝑉, 𝑑 = 𝒞∗ 𝐿′ cochains on 𝐿′

Consider the ℤ-graded vector space 𝐻𝑜𝑚ℚ(𝒞∗ 𝐿 ′ 𝐿′).

Let f , g ∶ 𝒞∗ 𝐿 ⟶ 𝐿′ be linear maps of degree

𝑝 and 𝑞 respectively.

𝒞∗ 𝐿

𝒞∗ 𝐿 ⨂𝒞∗ 𝐿

⟶

∆

⟶f ⊗ g

𝐿′⨂ 𝐿′

⟶[f, g ]

⟶

𝐿′

[−,−]

( 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′ , 𝔇, −,− ) is a DGL-model of map (𝑿, 𝒀)

𝒞∗ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′ , 𝔇, −,− = ⋀ V⨂𝐵⋕ , 𝐷

U. Buijs, Y. Félix and A. Murillo, Trans. Amer. Math.Soc. 2008

𝒞∗ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′ , 𝔇, −,− = ⋀ V⨂𝐵⋕ , 𝐷

What about the components ⋀𝑉 ⊗ 𝐵⋕1⨁(𝑉 ⊗ 𝐵⋕)≥2, 𝐷𝝓 ?

ϕ|(𝑠𝐿′)⋕⋕ : 𝒞∗ 𝐿 ⟶ 𝑠𝐿′⟶ 𝐿′෨𝜙

𝒻: 𝑋 ⟶ 𝑌

𝜙: 𝒞∗(𝐿′) ⟶ 𝒞∗ 𝐿⋕

𝜙:⋀(𝑠𝐿′)⋕⟶ 𝒞∗ 𝐿⋕

𝜙|(𝑠𝐿′)⋕: (𝑠𝐿′)⋕⟶ 𝒞∗ 𝐿⋕

Continuous function

𝜙: (⋀𝑉, 𝑑) ⟶ 𝐵 CDGA-morphism

Linear map of degree 0

Linear map of degree -1

𝒞∗ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′ , 𝔇, −,− = ⋀ V⨂𝐵⋕ , 𝐷

What about the components ⋀𝑉 ⊗ 𝐵⋕1⨁(𝑉 ⊗ 𝐵⋕)≥2, 𝐷𝝓 ?

ϕ|(𝑠𝐿′)⋕⋕ : 𝒞∗ 𝐿 ⟶ 𝑠𝐿′⟶ 𝐿′෨𝜙 Linear map of degree -1

෩𝜙 ∈ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′−1 𝔇෪𝜙 = 𝔇+ 𝑎𝑑෪𝜙

Maurer-Cartan equation

is a differential if and only if

𝔇෩𝜙 = −1

2[ ෩𝜙 , ෩𝜙 ]

𝒞∗ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′ , 𝔇, −,− = ⋀ V⨂𝐵⋕ , 𝐷

What about the components ⋀𝑉 ⊗ 𝐵⋕1⨁(𝑉 ⊗ 𝐵⋕)≥2, 𝐷𝝓 ?

ϕ|(𝑠𝐿′)⋕⋕ : 𝒞∗ 𝐿 ⟶ 𝑠𝐿′⟶ 𝐿′෨𝜙 Linear map of degree -1

෩𝜙 ∈ 𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′−1 𝔇෪𝜙 = 𝔇+ 𝑎𝑑෪𝜙

𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′𝑛

෩𝜙=

0 if 𝑛 < 0

𝐻𝑜𝑚ℚ 𝒞∗ 𝐿 ′ 𝐿′𝑛 if 𝑛 > 0

if 𝑛 = 0Ker 𝔇෪𝜙

Let 𝑨 be an associative algebra.

𝑨 × 𝑨⟶ 𝑨 bilinear and associative.

(𝒂, 𝒃) ⟼ 𝒂 ∙ 𝒃

𝑨 𝑡 = {

𝑖=0

∞

𝒂𝑖𝑡𝑖 , 𝒂𝑖 ∈ 𝑨 }

A deformation of 𝑨 in 𝑨 𝑡 is a new bilinear and

associative product ∗: 𝑨 𝑡 × 𝑨 𝑡 ⟶ 𝑨 𝑡

(σ𝑖=0∞ 𝒂𝑖𝑡

𝑖) ∗ (σ𝑖=0∞ 𝒃𝑖𝑡

𝑖) = 𝒂0 ∙ 𝒃0 + 𝒄1𝑡1 + 𝒄2𝑡

2 + 𝒄3𝑡3 +⋯ 𝒄𝑖 ∈ 𝑨

We can deform any algebraic structure on a vector sace 𝑨.

Instead of 𝑨 𝑡 , we can consider a local ring 𝑅 with a

unique maximal ideal 𝔐 with 𝑅/𝔐 ≅ 𝕂.

A deformation of 𝑨 in 𝑨⨂𝑅 is an operation:

∗: 𝑨⨂𝑅 × 𝑨⨂𝑅 ⟶ (𝑨⨂𝑅)

satisfying the same properties of the original such that

we recover the original operation taking quotient by 𝔐

𝑨× 𝑨⟶ 𝑨.

We denote by Def 𝑨, 𝑅 the set of equivalence

clases of deformations of 𝑨 in 𝑨⨂𝑅

“In characteristic 0 every deformation problem

is governed by a differential graded Lie algebra”

There exists a differential graded Lie algebra

𝐿 = 𝐿(𝑨 , 𝑅) such that we have a bijection

Def 𝑨, 𝑅 ≅ 𝑀𝐶(𝐿)/𝒢

Let 𝐿 be a DGL. 𝑎 ∈ 𝐿−1 is a Maurer-Cartan

element if satisfies the equation

𝜕𝑎 = −1

2[ 𝑎 , 𝑎 ]

Two Maurer-Cartan elements 𝑎, 𝑏 ∈ 𝑀𝐶(𝐿) are

gauge-equivalent 𝑎 ∼𝒢 𝑏 if there is an element

𝑥 ∈ 𝐿0 such that

𝑏 = 𝑒ad𝑥 𝑎 −𝑒ad𝑥 𝑎 − id

ad𝑥(𝜕𝑥)

For each 𝑛 ≥ 0 , consider the standard 𝑛-simplex ∆𝑛

∆𝑝𝑛= 𝑖0, … , 𝑖𝑝 0 ≤ 𝑖0 ≤ ⋯ ≤ 𝑖𝑝 ≤ 𝑛 }, if 𝑝 ≤ 𝑛 ,

∆𝑝𝑛= ∅, if 𝑝 > 𝑛.

Let 𝕃 𝑠−1∆𝑛 , 𝑑 be the complete free DGL on the desuspended

rational simplicial chain complex

𝑑𝑎𝑖0,…,𝑖𝑝 =

𝑗=0

𝑝

(−1)𝑗𝑎𝑖0,…,𝑖𝑗,…,𝑖𝑝

where 𝑎𝑖0,…,𝑖𝑝 denotes the generator of degree 𝑝 − 1 represented

by the 𝑝-simplex 𝑖0, … , 𝑖𝑝 ∈ ∆𝑝𝑛.

Sullivan’s question

Problem:

Define a new differential in 𝕃 𝑠−1∆𝑛 such that:

For each 𝑖 = 0,… , 𝑛 the generator 𝑎𝑖 ∈ ∆0𝑛 is

a Maurer-Cartan element.

1.

𝜕𝑎𝑖 = −1

2[𝑎𝑖 , 𝑎𝑖]

2. The linear part 𝜕1 of 𝜕 is precisely 𝑑

Sullivan’s question

Case 𝑛 = 0 𝑎0

𝕃 𝑠−1Δ0 , 𝑑 = 𝕃 𝑎0 , 𝑑

where 𝑑𝑎0 = 0.

The first condition implies:

𝕃 𝑎0 , 𝜕𝑎0 = −1

2[𝑎0, 𝑎0]

Sullivan’s question

Case 𝑛 = 1

𝕃 𝑠−1Δ1 , 𝑑 = 𝕃 𝑎0, 𝑎1, 𝑎01 , 𝑑

where 𝑑𝑎0 = 𝑑𝑎1 = 0.

𝑑𝑎01 = 𝑎1 − 𝑎0.

The first condition implies:

𝜕𝑎0 = −1

2𝑎0, 𝑎0 , 𝜕𝑎1 = −

1

2[𝑎1, 𝑎1]

The second condition implies:

𝜕1𝑎01 = 𝑎1 − 𝑎0.

𝑎0 𝑎01 𝑎1

𝜕2 𝑎01 = 𝜕(𝑎1 − 𝑎0)

= −1

2𝑎0, 𝑎0 +

1

2𝑎1, 𝑎1 ≠ 0

𝜕2 𝑎01 = 𝜕(𝑎1 − 𝑎0 +1

2𝑎01, 𝑎0 +

1

2[𝑎01, 𝑎1])

=1

4𝑎01, 𝑎0 , 𝑎0 −

1

4𝑎01, 𝑎1 , 𝑎1

+1

4[𝑎01, 𝑎1, 𝑎0 ] ≠ 0

𝜕𝑎01 = 𝑎1 − 𝑎0 +𝜆1 𝑎01, 𝑎0 + 𝜇1[𝑎01, 𝑎1]+1

2+1

2

+𝜆2 𝑎01, [𝑎01, 𝑎0] + 𝜇2[𝑎01, 𝑎01, 𝑎1 ]

⋯

Sullivan’s question

Case 𝑛 = 1

𝕃 𝑠−1Δ1 , 𝑑 = 𝕃 𝑎0, 𝑎1, 𝑎01 , 𝑑

where 𝑑𝑎0 = 𝑑𝑎1 = 0.

𝑑𝑎01 = 𝑎1 − 𝑎0.

The first condition implies:

𝜕𝑎0 = −1

2𝑎0, 𝑎0 , 𝜕𝑎1 = −

1

2[𝑎1, 𝑎1]

The second condition implies:

𝜕1𝑎01 = 𝑎1 − 𝑎0.

𝑎0 𝑎01 𝑎1

𝜕𝑎01 = 𝑎01, 𝑎1 +

𝑖=0

∞𝐵𝑖𝑖!ad𝑎01

𝑖 ( 𝑎1 − 𝑎0)

Bernoulli numbers are defined by theseries 𝑥

𝑒𝑥 − 1=

𝑛=0

∞𝐵𝑛𝑛!𝑥𝑛

𝑒𝑥 − 1

𝑥=

𝑛=0

∞1

𝑛 + 1 !𝑥𝑛Since

𝐵0 = 1, 𝐵1 = −1

2, 𝐵2 =

1

6, 𝐵3 = 0, 𝐵4 = −

1

30, 𝐵5 = 0, 𝐵6 =

1

42, ⋯

Sullivan’s question

Sullivan’s question

Case 𝑛 = 2

𝕃 𝑠−1Δ2 , 𝑑 = 𝕃 𝑎0, 𝑎1, 𝑎2, 𝑎01, 𝑎02, 𝑎12, 𝑎012 , 𝑑

𝑑𝑎0 = 𝑑𝑎1 = 𝑑𝑎2 = 0

𝑑𝑎01 = 𝑎1 − 𝑎0

𝑑𝑎02 = 𝑎2 − 𝑎0

𝑑𝑎12 = 𝑎2 − 𝑎1

𝑑𝑎012 = 𝑎12 − 𝑎02 + 𝑎01

𝑎0

𝑎1

𝑎2

𝑎01

𝑎02

𝑎12𝑎012

MC

MC MC

LS

LS

LS

𝜕𝑎012 = 𝑎12 − 𝑎02 + 𝑎01 +Ψ

𝜕𝑎012 = 𝑎12 ∗ 𝑎01 ∗ −𝑎02 − [𝑎012, 𝑎0]

BCH

Sullivan’s question

The Baker-Campbell-Hausdorff formula

Let 𝑥, 𝑦 be two non-commuting variables. 𝑇(𝑥, 𝑦)

The Baker-Campbell-Hausdorff formula 𝑥 ∗ 𝑦 is

the solution to 𝑧 = log(exp 𝑥 exp 𝑦 )

Explicitely 𝑥 ∗ 𝑦 = σ𝑛=0∞ 𝑧𝑛(𝑥, 𝑦)

𝑧𝑛 𝑥, 𝑦 =

𝑘=1

𝑛(−1)𝑘−1

𝑘

𝑥𝑝1𝑦𝑞1⋯𝑥𝑝𝑘𝑦𝑞𝑘

𝑝1! 𝑞1!⋯𝑝𝑘! 𝑞𝑘!

𝑝1 + 𝑞1 > 0; ⋯ ; 𝑝𝑘+𝑞𝑘 > 0. 𝑝1 + 𝑞1 +⋯+ 𝑝𝑘 + 𝑞𝑘 = 𝑛

Sullivan’s question

The Baker-Campbell-Hausdorff formula

The Baker-Campbell-Hausdorff formula satisfies the following properties:

∗ is an associative product: 𝑥 ∗ 𝑦 ∗ 𝑧 = 𝑥 ∗ (𝑦 ∗ 𝑧)

The inverse of 𝑥 is −𝑥, i.e. 𝑥 ∗ −𝑥 = 0.

𝑥 ∗ 𝑦 can be written as a linear combination of

nested commutators

𝑥 ∗ 𝑦 = 𝑥 + 𝑦 +1

2𝑥, 𝑦 +

1

12𝑥, 𝑥, 𝑦 −

1

12𝑦, 𝑥, 𝑦 + ⋯

𝑥 ∗ 𝑦 ∈ 𝕃(𝑥, 𝑦) ⊂ 𝑇(𝑥, 𝑦)

Theorem

𝔏 = { 𝔏𝑛 } 𝑛≥0 is a cosimplicial DGL

U. Buijs, Y. Félix, A. Murillo, D. Tanré. Israel J. of Math. 2019

Definition

< 𝐿 >𝑛= 𝐻𝑜𝑚𝑐𝐷𝐺𝐿( 𝔏𝑛 , 𝐿)

cDGL ⟶< ∙ >

Theorem

𝜋0 < 𝐿 >≅ MC(𝐿)/𝒢

< 𝐿 > ≃ ∐𝑧∈MC(𝐿)/𝒢 < 𝐿𝑧 >

𝜋𝑛 < 𝐿 >≅ 𝐻𝑛−1(𝐿) 𝑛 > 1

𝜋1 < 𝐿 >≅ 𝐻0(𝐿)

For any cDGL 𝐿 we have

1.

2.

𝜋𝑛 < 𝐿 >≅ 𝐻𝑛−1(𝐿) 𝑛 > 1

𝜋1 < 𝐿 >≅ 𝐻0(𝐿)𝜋1 < 𝐿 >× 𝜋𝑛 < 𝐿 > ⟶ 𝜋𝑛 < 𝐿 >

𝜏

≅ ≅

𝐻0 𝐿 × 𝐻𝑛−1(𝐿) 𝐻𝑛−1(𝐿)?

(𝑥 , 𝑧 ) 𝑒ad𝑥(𝑧)

Reduced

Complete

HomcDGL(𝔏∙, 𝐿)

Reduced

Nilpotent, finite-type

Nilpotent, finite-type

Complete𝑁∗𝒰∙𝒢∙ ഥ𝑊(𝐿)

𝒞(𝐿) 𝑆

MC(𝐿⨂Ω∙)Neisendorfer,

Pac. J. M., 1978.Getzler’s

observation

Buijs, Félix, Murillo, Tanré, Preprint 2017

Work in progress