Joint work with Yves Flix, Aniceto Murillo and Daniel Tanr If : is - - PowerPoint PPT Presentation

Joint work with Yves Félix, Aniceto Murillo and Daniel Tanré

If 𝒼: 𝒀 ⟶ 𝒁 is a continuous map between simply connected CW-complexes, the following properties are equivalent: 1. 𝜌𝑜 𝒼 ⨂ℚ ∶ 𝜌𝑜 𝒀 ⨂ℚ ⟶ 𝜌𝑜 𝒁 ⨂ℚ , 𝑜 ≥ 2.

≅

2. 𝐼𝑜 𝒼 ⨂ℚ ∶ 𝐼𝑜 𝒀 ; ℚ ⟶ 𝐼𝑜 𝒁 ; ℚ , 𝑜 ≥ 2.

≅

Such a map is called a rat atio iona nal ho homo motop topy equ equiv ival alenc ence.

𝒀 is ra ratio tiona nal if its homotopy groups are ℚ-vector spaces. A ra ratio tiona nali lisa satio tion of 𝒀 is a pair (𝒀ℚ , 𝑓), with 𝒀ℚ a rational space and and 𝑓 ∶ 𝒀 ⟶ 𝒀ℚ a rational homotopy equivalence. The study of the rational homotopy type of 𝒀 is the study of the homotopy type of its rationalisation 𝒀ℚ.

The study of the rational homotopy type of 𝒀 is the study of the homotopy type of its rationalisation 𝒀ℚ. Then, if 𝒀 is a finite simply connected CW-complex 𝜌𝑜 𝒀 = ⨁𝑠ℤ ⨁ ℤ𝑞1𝑠1⨁ ⋯ ⨁ ℤ𝑞𝑛𝑠𝑛 𝜌𝑜 𝒀ℚ ≅ 𝜌𝑜 𝒀 ⨂ℚ = ⨁𝑠ℚ

The rational homotopy type of 𝒀 is completely determined in algebraic terms.

DGL CDGA

A di diff ffer erent entia ial gr grad aded ed Li Lie a e alg lgeb ebra ra is a graded vector space 𝑀 = ⨁𝑞𝜗ℤ 𝑀𝑞 with: A bilinear operation . , . ∶ 𝑀 × 𝑀 ⟶ 𝑀 such that 𝑀𝑞 , 𝑀𝑟 ⊂ 𝑀𝑞+𝑟 satisfying: 𝑏, 𝑐 = − −1 𝑞𝑟 𝑐, 𝑏 , 𝑏 ∈ 𝑀𝑞 , 𝑐 ∈ 𝑀𝑟 𝑏, [𝑐, 𝑑] = 𝑏, 𝑐 , 𝑑 + −1 𝑞𝑟 𝑐, [𝑏, 𝑑] , 𝑏 ∈ 𝑀𝑞 , 𝑐 ∈ 𝑀𝑟 , 𝑑 ∈ 𝑀 a) a) b) b)

DGL

A linear map 𝜖 ∶ 𝑀 ⟶ 𝑀 such that 𝜖𝑀𝑞 ⊂ 𝑀𝑞−1 satisfying: 𝜖 ∘ 𝜖 = 0 𝜖 𝑏, 𝑐 = 𝜖𝑏, 𝑐 + −1 𝑞 𝑏, 𝜖𝑐 , 𝑏 ∈ 𝑀𝑞 , 𝑐 ∈ 𝑀 A di diff ffer erent entia ial gr grad aded ed Li Lie a e alg lgeb ebra ra is a graded vector space 𝑀 = ⨁𝑞𝜗ℤ 𝑀𝑞 with: a) a) b) b)

DGL

DGL

𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

DGL +

⟶

𝜇

⟶

< ∙ >𝑅

< 𝜇(𝒀) >𝑅 ≃ 𝒀ℚ

Rational homotopy equivalences Quasi-isomorphisms

(𝑀, ∙,∙ , 𝜖) is a DGL-mod

model el of 𝒀 if

𝜇(𝒀) 𝑀

⋯

⟶

≃

⟶

≃

⟶

≃

⟶

≃

DGL

𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

DGL +

⟶

𝜇

⟶

< ∙ >𝑅

< 𝜇(𝒀) >𝑅 ≃ 𝒀ℚ

Rational homotopy equivalences Quasi-isomorphisms

(𝑀, ∙,∙ , 𝜖) is a DGL-mod

model el of 𝒀 if

(𝑀, ∙,∙ , 𝜖)

⟶

≃

(𝕄(𝑋), 𝑒)

Minimal Quillen model

𝐼𝑜( ) 𝑀

Rational homotopy groups

≅ 𝜌𝑜+1(𝒀)⨂ℚ

Rational homology groups

𝑀

(𝕄 𝑋 , 𝜖) ⟶

≃

𝑡(𝑋 ⊕ ℚ) ≅ 𝐼∗(𝑌; ℚ)

DGL

DGL

Spheres Products Wedge products 𝕄 𝑤 , 0 , 𝑤 = 𝑜 − 1

is a model for the 𝑜-dimensional sphere 𝑇𝑜. If (𝑀, 𝜖) and (𝑀′, 𝜖′) are DGL-models for 𝒀 and 𝒁 respectively, then 𝑀 × 𝑀′, 𝜖 × 𝜖′ is a DGL-model of 𝒀 × 𝒁. If (𝕄 𝑊 , 𝑒) and (𝕄 𝑋 , 𝑒′) are minimal DGL models for 𝒀 and 𝒁 respectively, then (𝕄 𝑊 ⊕ 𝑋 , 𝐸) is a DGL-model of 𝒀 ∨ 𝒁.

A ba base sed to topo pologi logical cal sp space ace (𝒀, ∗)

( 𝕄 𝑊 , 𝜖 )

𝒀𝑜 𝑜-skeleton

( 𝕄(𝑊

<𝑜), 𝜖 )

= 𝑓𝛽

𝑜+1

𝑤𝛽 ∈ 𝑊

𝑜

𝑔

𝛽 ∶ 𝑇𝑜 → 𝒀𝑜 ;

[𝑔

𝛽] ∈ Π𝑜(𝒀𝑜)

𝜖𝑤𝛽 ∈ 𝐼𝑜−1(𝕄 𝑊

<𝑜 )

≅ Π𝑜(𝒀𝑜)⨂ℚ

CW-decomposition

A co commu mmuta tativ tive di diff ffer erent entia ial gr grad aded ed al alge gebra bra is a graded vector space 𝐵 = ⨁𝑞𝜗ℤ 𝐵𝑞 with: A bilinear operation ∙ ∶ 𝐵 × 𝐵 ⟶ 𝐵 such that 𝐵𝑞 ∙ 𝐵𝑟 ⊂ 𝐵𝑞+𝑟 satisfying: 𝑏 ∙ 𝑐 = −1 𝑞𝑟𝑐 ∙ 𝑏, 𝑏 ∈ 𝐵𝑞 , 𝑐 ∈ 𝐵𝑟 𝑏 ∙ (𝑐 ∙ 𝑑) = (𝑏 ∙ 𝑐) ∙ 𝑑 , 𝑏, 𝑐, 𝑑 ∈ 𝐵 a) a) b) b)

CDGA

CDGA

A co commu mmuta tativ tive di diff ffer erent entia ial gr grad aded ed al alge gebra bra is a graded vector space 𝐵 = ⨁𝑞𝜗ℤ 𝐵𝑞 with: A linear map 𝑒: 𝐵 ⟶ A such that 𝑒𝐵𝑞 ⊂ 𝐵𝑞+1 satisfying: 𝑒 ∘ 𝑒 = 0 𝑒(𝑏 ∙ 𝑐) = (𝑒𝑏) ∙ 𝑐 + −1 𝑞𝑏 ∙ (𝑒𝑐) , 𝑏 ∈ 𝑀𝑞 , 𝑐 ∈ 𝑀 a) a) b) b)

CDGA

𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

CDGA +

⟶

𝐵𝑄𝑀

⟶

< ∙ >𝑇

Rational homotopy equivalences Quasi-isomorphisms

< 𝐵𝑄𝑀(𝒀) >𝑇 ≃ 𝒀ℚ (𝐵, 𝑒)

is a CDGA-model model of 𝒀 if

𝐵𝑄𝑀(𝒀) 𝐵

⋯

⟶

≃

⟶

≃

⟶

≃

⟶

≃

CDGA

⟶

𝐵𝑄𝑀

𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

CDGA +

⟶

< ∙ >𝑇

Rational homotopy equivalences Quasi-isomorphisms

< 𝐵𝑄𝑀(𝒀) >𝑇 ≃ 𝒀ℚ (𝐵, 𝑒)

is a CDGA-model model of 𝒀 if

(𝐵, 𝑒)

⟶

≃

(⋀𝑊, 𝑒)

CDGA

Rational homotopy groups Rational cohomology groups

𝐼∗ 𝐵, 𝑒 ≅ 𝐼∗(𝒀; ℚ ) (𝐵, 𝑒)

⟶

≃

(⋀𝑊, 𝑒) Hom(𝑊𝑙, ℚ) ≅ 𝜌𝑙(𝒀)⨂ ℚ

Eilenberg-MacLane spaces Wedge products Products ⋀𝑤, 0 , 𝑤 = 𝑜

is a model of the Eilenberg-MacLane space 𝐿(ℤ, 𝑜). If (𝐵, 𝑒) and (𝐵′, 𝑒′) are CDGA-models for 𝒀 and 𝒁 respectively, then 𝐵 × 𝐵′, 𝑒 × 𝑒′ is a CDGA-model of 𝒀 ∨ 𝒁. If (⋀𝑊, 𝑒) and (⋀𝑋, 𝑒′) are minimal models for 𝒀 and 𝒁 respectively, then (⋀ 𝑊 ⊕ 𝑋 , 𝐸) is a CDGA-model of 𝒀 × 𝒁.

CDGA

CDGA

Postnikov decomposition

𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

DGL +

⟶

𝜇

⟶

< ∙ >𝑅

𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

CDGA +

⟶

𝐵𝑄𝑀

⟶

< ∙ >𝑇

𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

DGL +

⟶

𝜇

⟶

< ∙ >𝑅

𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

CDGA +

⟶

𝐵𝑄𝑀

⟶

< ∙ >𝑇

⟶ ⟶

sLA

⟶ ⟶

sCHA

⟶ ⟶

sGrp < 𝐵 >𝑇= Hom𝑫𝑬𝑯𝑩(𝐵, Ω ) < 𝐵 >𝑇 is defined for 𝐵 a ℤ-graded CDGA but < 𝑀 >𝑅 has only sense for positively graded DGL’s

𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭

⟶ ⟶

ℒ 𝒟∗

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

map 𝒀, 𝒁 , map∗ 𝒀, 𝒁 , map𝑔 𝒀, 𝒁 , map𝑔

∗ (𝒀, 𝒁)

If 𝒀 and 𝒁 are finite type CW-complexes map 𝒀, 𝒁 has the homotopy type of a CW-complex. If 𝒀 and 𝒁 are nilpotent map𝑔 𝒀, 𝒁 is nilpotent. If 𝒀 is a finite CW-complex map𝑔 𝒀, 𝒁 is of finite type. If 𝒀 and 𝒁 are 1-connected map𝑔 𝒀, 𝒁 is 1-connected.

Let 𝒀 be a finite CW-complex and 𝒁 be a nilpotent, finite type CW-complex. (𝐶⋕, 𝜀) ⋀ V⨂𝐶⋕ , 𝐸 𝐸 𝑤⨂𝛾 = σ𝑘(−1) 𝑥 |𝛾𝑘

′|(𝑣⨂𝛾𝑘

′) (𝑥⨂𝛾𝑘 ′′) + 𝑤⨂𝜀𝛾

𝒀 (𝐶 , 𝑒) CDGA CDGC 𝐶⋕𝑜 = 𝐶−𝑜 (⋀𝑊, 𝑒) 𝒁 Sullivan model if 𝑤 ∈ V such that d𝑤 = 𝑣𝑥 and 𝛾 ∈ 𝐶⋕ with ∆𝛾 = σ𝒌 𝛾𝒌

′⨂ 𝛾𝒌 ′′

V⨂𝐶⋕ The following Sullivan algebra is a model for map 𝒀, 𝒁 .

Let ϕ: (⋀𝑊, 𝑒) ⟶ 𝐶 be a model of 𝑔: 𝒀 ⟶ 𝒁 . Let 𝐿ϕ be the differential ideal generated by 𝐵1 ∪ 𝐵2 ∪ 𝐵3 𝐵1 = (𝑊 ⊗ 𝐶⋕)<0 𝐵2 = 𝐸(𝑊 ⊗ 𝐶⋕)0 𝐵3 = 𝛽 − 𝜚 𝛽 𝛽 ∈ (𝑊 ⊗ 𝐶⋕)0 } The projection

map𝑔 𝒀, 𝒁 ⟶ map 𝒀, 𝒁

⋀ V⨂𝐶⋕ , 𝐸 ⟶ ⋀ V⨂𝐶⋕ , 𝐸 /𝐿ϕ is a model of the injection ⋀ V⨂𝐶⋕ , 𝐸 /𝐿ϕ ≅ ⋀𝑊 ⊗ 𝐶⋕1⨁(𝑊 ⊗ 𝐶⋕)≥2, 𝐸𝝔

𝒀 , 𝒁 with DGL-models 𝑀 , 𝑀′ , respectively. 𝑔: 𝒀 ⟶ 𝒁 .

map𝑔 𝒀, 𝒁 ,

DGL-model of map𝑔 𝒀, 𝒁 ? (𝐶⋕, 𝜀) 𝒀 CDGC 𝐶⋕, 𝜀 = 𝒟∗(𝑀) Cartan-Eilenberg (⋀𝑊, 𝑒) 𝒁 Sullivan model ⋀𝑊, 𝑒 = 𝒟∗ 𝑀′ cochains on 𝑀′

map𝑔 𝒀, 𝒁 ,

DGL-model of map𝑔 𝒀, 𝒁 ? (𝐶⋕, 𝜀) 𝒀 CDGC 𝐶⋕, 𝜀 = 𝒟∗(𝑀) Cartan-Eilenberg (⋀𝑊, 𝑒) 𝒁 Sullivan model ⋀𝑊, 𝑒 = 𝒟∗ 𝑀′ cochains on 𝑀′ Consider the ℤ-graded vector space 𝐼𝑝𝑛ℚ(𝒟∗ 𝑀 ′ 𝑀′). Let f ∶ 𝒟∗ 𝑀 ⟶ 𝑀′ be a linear map of degree 𝑞. 𝔈f = f ∘ δ − −1

f 𝜖′ ∘ f ∶ 𝒟∗ 𝑀 ⟶ 𝑀′

(𝐶⋕, 𝜀) 𝒀 CDGC 𝐶⋕, 𝜀 = 𝒟∗(𝑀) Cartan-Eilenberg (⋀𝑊, 𝑒) 𝒁 Sullivan model ⋀𝑊, 𝑒 = 𝒟∗ 𝑀′ cochains on 𝑀′ Consider the ℤ-graded vector space 𝐼𝑝𝑛ℚ(𝒟∗ 𝑀 ′ 𝑀′). Let f , g ∶ 𝒟∗ 𝑀 ⟶ 𝑀′ be linear maps of degree 𝑞 and 𝑟 respectively. 𝒟∗ 𝑀 𝒟∗ 𝑀 ⨂𝒟∗ 𝑀

⟶

∆

⟶

f ⊗ g

𝑀′⨂ 𝑀′

⟶

[f, g ]

⟶

𝑀′

[−, −]

( 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ , 𝔈, −, − ) is a DGL-model of map (𝒀, 𝒁) 𝒟∗ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ , 𝔈, −, − = ⋀ V⨂𝐶⋕ , 𝐸

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

𝒟∗ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ , 𝔈, −, − = ⋀ V⨂𝐶⋕ , 𝐸 What about the components

⋀𝑊 ⊗ 𝐶⋕1⨁(𝑊 ⊗ 𝐶⋕)≥2, 𝐸𝝔 ?

ϕ|(𝑡𝑀′)⋕

⋕

: 𝒟∗ 𝑀 ⟶ 𝑡𝑀′⟶ 𝑀′

෨ 𝜚 𝒼: 𝑌 ⟶ 𝑍

𝜚: 𝒟∗(𝑀′) ⟶ 𝒟∗ 𝑀 ⋕ 𝜚: ⋀(𝑡𝑀′)⋕⟶ 𝒟∗ 𝑀 ⋕ 𝜚|(𝑡𝑀′)⋕: (𝑡𝑀′)⋕⟶ 𝒟∗ 𝑀 ⋕ Continuous function 𝜚: (⋀𝑊, 𝑒) ⟶ 𝐶 CDGA-morphism Linear map of degree 0 Linear map of degree -1

𝒟∗ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ , 𝔈, −, − = ⋀ V⨂𝐶⋕ , 𝐸 What about the components

⋀𝑊 ⊗ 𝐶⋕1⨁(𝑊 ⊗ 𝐶⋕)≥2, 𝐸𝝔 ?

ϕ|(𝑡𝑀′)⋕

⋕

: 𝒟∗ 𝑀 ⟶ 𝑡𝑀′⟶ 𝑀′

෨ 𝜚

Linear map of degree -1

෩ 𝜚 ∈ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ −1 𝔈෪

𝜚 = 𝔈 + 𝑏𝑒෪ 𝜚

Ma Maur urer er-Ca Cart rtan an equ equat atio ion

is a differential if and only if 𝔈 ෩ 𝜚 = −

1 2 [ ෩

𝜚 , ෩ 𝜚 ]

𝒟∗ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ , 𝔈, −, − = ⋀ V⨂𝐶⋕ , 𝐸 What about the components

⋀𝑊 ⊗ 𝐶⋕1⨁(𝑊 ⊗ 𝐶⋕)≥2, 𝐸𝝔 ?

ϕ|(𝑡𝑀′)⋕

⋕

: 𝒟∗ 𝑀 ⟶ 𝑡𝑀′⟶ 𝑀′

෨ 𝜚

Linear map of degree -1

෩ 𝜚 ∈ 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ −1 𝔈෪

𝜚 = 𝔈 + 𝑏𝑒෪ 𝜚

𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′

𝑜 ෩ 𝜚 =

if 𝑜 < 0 𝐼𝑝𝑛ℚ 𝒟∗ 𝑀 ′ 𝑀′ 𝑜 if 𝑜 > 0 if 𝑜 = 0 Ker 𝔈෪

𝜚

Let 𝑩 be an associative algebra. 𝑩 × 𝑩 ⟶ 𝑩 bilinear and associative. (𝒃, 𝒄) ⟼ 𝒃 ∙ 𝒄

𝑩 𝑢 = { ෍

𝑗=0 ∞

𝒃𝑗𝑢𝑗 , 𝒃𝑗 ∈ 𝑩 }

A def defor

• rma

matio tion of 𝑩 in 𝑩 𝑢 is a new bilinear and associative product ∗: 𝑩 𝑢 × 𝑩 𝑢 ⟶ 𝑩 𝑢

(σ𝑗=0

∞ 𝒃𝑗𝑢𝑗) ∗ ( σ𝑗=0 ∞ 𝒄𝑗𝑢𝑗) = 𝒃0 ∙ 𝒄0 + 𝒅1𝑢1 + 𝒅2𝑢2 + 𝒅3𝑢3 + ⋯

𝒅𝑗 ∈ 𝑩

We can deform any algebraic structure on a vector sace 𝑩. Instead of 𝑩 𝑢 , we can consider a local ring 𝑆 with a unique maximal ideal 𝔑 with 𝑆/𝔑 ≅ 𝕃. A def defor

• rmat

mation ion of 𝑩 in 𝑩⨂𝑆 is an operation: ∗: 𝑩⨂𝑆 × 𝑩⨂𝑆 ⟶ (𝑩⨂𝑆) satisfying the same properties of the original such that we recover the original operation taking quotient by 𝔑 𝑩 × 𝑩 ⟶ 𝑩.

We denote by Def 𝑩, 𝑆 the set of equivalence clases of deformations of 𝑩 in 𝑩⨂𝑆 “In characteristic 0 every deformation problem is governed by a differential graded Lie algebra” There exists a differential graded Lie algebra 𝑀 = 𝑀(𝑩 , 𝑆) such that we have a bijection Def 𝑩, 𝑆 ≅ 𝑁𝐷(𝑀)/𝒣

Let 𝑀 be a DGL. 𝑏 ∈ 𝑀−1 is a Maurer-Cartan element if satisfies the equation

𝜖𝑏 = − 1 2 [ 𝑏 , 𝑏 ]

Two Maurer-Cartan elements 𝑏, 𝑐 ∈ 𝑁𝐷(𝑀) are gauge-equivalent 𝑏 ∼𝒣 𝑐 if there is an element 𝑦 ∈ 𝑀0 such that 𝑐 = 𝑓ad𝑦 𝑏 − 𝑓ad𝑦 𝑏 − id ad𝑦 (𝜖𝑦)

For each 𝑜 ≥ 0 , consider the standard 𝑜-simplex ∆𝑜 ∆𝑞

𝑜=

𝑗0, … , 𝑗𝑞 0 ≤ 𝑗0 ≤ ⋯ ≤ 𝑗𝑞 ≤ 𝑜 }, if 𝑞 ≤ 𝑜 , ∆𝑞

𝑜= ∅, if 𝑞 > 𝑜.

Let ෡ 𝕄 𝑡−1∆𝑜 , 𝑒 be the complete free DGL on the desuspended rational simplicial chain complex

𝑒𝑏𝑗0,…,𝑗𝑞 = ෍

𝑘=0 𝑞

(−1)𝑘𝑏𝑗0,…,෡

𝑗𝑘,…,𝑗𝑞

where 𝑏𝑗0,…,𝑗𝑞 denotes the generator of degree 𝑞 − 1 represented by the 𝑞-simplex 𝑗0, … , 𝑗𝑞 ∈ ∆𝑞

𝑜.

Sullivan’s question

Problem:

Define a new differential in ෡ 𝕄 𝑡−1∆𝑜 such that: For each 𝑗 = 0, … , 𝑜 the generator 𝑏𝑗 ∈ ∆0

𝑜 is

a Maurer-Cartan element.

1.

𝜖𝑏𝑗 = − 1 2 [𝑏𝑗, 𝑏𝑗]

• 2. The linear part 𝜖1 of 𝜖 is precisely 𝑒

Sullivan’s question

Case 𝑜 = 0 𝑏0 ෡ 𝕄 𝑡−1Δ0 , 𝑒 = ෡ 𝕄 𝑏0 , 𝑒 where 𝑒𝑏0 = 0. The first condition implies: ෡ 𝕄 𝑏0 , 𝜖𝑏0 = − 1 2 [𝑏0, 𝑏0]

Sullivan’s question

Case 𝑜 = 1 ෡ 𝕄 𝑡−1Δ1 , 𝑒 = ෡ 𝕄 𝑏0, 𝑏1, 𝑏01 , 𝑒 where 𝑒𝑏0 = 𝑒𝑏1 = 0. 𝑒𝑏01 = 𝑏1 − 𝑏0. The first condition implies:

𝜖𝑏0 = − 1 2 𝑏0, 𝑏0 , 𝜖𝑏1 = − 1 2 [𝑏1, 𝑏1]

The second condition implies: 𝜖1𝑏01 = 𝑏1 − 𝑏0. 𝑏0 𝑏01 𝑏1

𝜖2 𝑏01 = 𝜖(𝑏1 − 𝑏0) = − 1 2 𝑏0, 𝑏0 + 1 2 𝑏1, 𝑏1 ≠ 0

𝜖2 𝑏01 = 𝜖(𝑏1 − 𝑏0 + 1 2 𝑏01, 𝑏0 + 1 2 [𝑏01, 𝑏1]) = 1 4 𝑏01, 𝑏0 , 𝑏0 − 1 4 𝑏01, 𝑏1 , 𝑏1 + 1 4 [𝑏01, 𝑏1, 𝑏0 ] ≠ 0

𝜖𝑏01 = 𝑏1 − 𝑏0 +𝜇1 𝑏01, 𝑏0 + 𝜈1[𝑏01, 𝑏1] + 1 2 + 1 2 +𝜇2 𝑏01, [𝑏01, 𝑏0] + 𝜈2[𝑏01, 𝑏01, 𝑏1 ]

⋯

Sullivan’s question

Case 𝑜 = 1 ෡ 𝕄 𝑡−1Δ1 , 𝑒 = ෡ 𝕄 𝑏0, 𝑏1, 𝑏01 , 𝑒 where 𝑒𝑏0 = 𝑒𝑏1 = 0. 𝑒𝑏01 = 𝑏1 − 𝑏0. The first condition implies:

𝜖𝑏0 = − 1 2 𝑏0, 𝑏0 , 𝜖𝑏1 = − 1 2 [𝑏1, 𝑏1]

The second condition implies: 𝜖1𝑏01 = 𝑏1 − 𝑏0. 𝑏0 𝑏01 𝑏1

𝜖𝑏01 = 𝑏01, 𝑏1 + ෍

𝑗=0 ∞ 𝐶𝑗

𝑗! ad𝑏01

𝑗

( 𝑏1 − 𝑏0) Bernoulli numbers are defined by the series 𝑦 𝑓𝑦 − 1 = ෍

𝑜=0 ∞ 𝐶𝑜

𝑜! 𝑦𝑜

𝑓𝑦 − 1 𝑦 = ෍

𝑜=0 ∞

1 𝑜 + 1 ! 𝑦𝑜

Since

𝐶0 = 1, 𝐶1 = − 1 2 , 𝐶2 = 1 6 , 𝐶3 = 0, 𝐶4 = − 1 30 , 𝐶5 = 0, 𝐶6 = 1 42 , ⋯

Sullivan’s question

Sullivan’s question

Case 𝑜 = 2

෡ 𝕄 𝑡−1Δ2 , 𝑒 = ෡ 𝕄 𝑏0, 𝑏1, 𝑏2, 𝑏01, 𝑏02, 𝑏12, 𝑏012 , 𝑒 𝑒𝑏0 = 𝑒𝑏1 = 𝑒𝑏2 = 0 𝑒𝑏01 = 𝑏1 − 𝑏0 𝑒𝑏02 = 𝑏2 − 𝑏0 𝑒𝑏12 = 𝑏2 − 𝑏1 𝑒𝑏012 = 𝑏12 − 𝑏02 + 𝑏01 𝑏0 𝑏1 𝑏2 𝑏01 𝑏02 𝑏12 𝑏012 MC MC MC LS LS LS 𝜖𝑏012 = 𝑏12 − 𝑏02 + 𝑏01 + Ψ 𝜖𝑏012 = 𝑏12 ∗ 𝑏01 ∗ −𝑏02 − [𝑏012, 𝑏0] BCH

Sullivan’s question

The Baker-Campbell-Hausdorff formula Let 𝑦, 𝑧 be two non-commuting variables. ෠ 𝑈(𝑦, 𝑧) The Ba Bake ker-Camp Campbell bell-Haus Hausdorff dorff fo form rmula ula 𝑦 ∗ 𝑧 is the solution to 𝑨 = log(exp 𝑦 exp 𝑧 ) Explicitely 𝑦 ∗ 𝑧 = σ𝑜=0

∞

𝑨𝑜(𝑦, 𝑧)

𝑨𝑜 𝑦, 𝑧 = ෍

𝑙=1 𝑜 (−1)𝑙−1

𝑙 ෍ 𝑦𝑞1𝑧𝑟1 ⋯ 𝑦𝑞𝑙𝑧𝑟𝑙 𝑞1! 𝑟1! ⋯ 𝑞𝑙! 𝑟𝑙! 𝑞1 + 𝑟1 > 0; ⋯ ; 𝑞𝑙+𝑟𝑙 > 0. 𝑞1 + 𝑟1 + ⋯ + 𝑞𝑙 + 𝑟𝑙 = 𝑜

Sullivan’s question

The Baker-Campbell-Hausdorff formula

The Baker-Campbell-Hausdorff formula satisfies the following properties: ∗ is an associative product: 𝑦 ∗ 𝑧 ∗ 𝑨 = 𝑦 ∗ (𝑧 ∗ 𝑨) The inverse of 𝑦 is −𝑦, i.e. 𝑦 ∗ −𝑦 = 0. 𝑦 ∗ 𝑧 can be written as a linear combination of nested commutators 𝑦 ∗ 𝑧 = 𝑦 + 𝑧 + 1 2 𝑦, 𝑧 + 1 12 𝑦, 𝑦, 𝑧 − 1 12 𝑧, 𝑦, 𝑧 + ⋯

𝑦 ∗ 𝑧 ∈ ෡ 𝕄(𝑦, 𝑧) ⊂ ෠ 𝑈(𝑦, 𝑧)

Theorem

𝔐 = { 𝔐𝑜 } 𝑜≥0

is a cosimplicial DGL

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

Definition

< 𝑀 >𝑜= 𝐼𝑝𝑛𝑑𝐸𝐻𝑀( 𝔐𝑜 , 𝑀) cDGL ⟶

< ∙ >

Theorem 𝜌0 < 𝑀 >≅ MC(𝑀)/𝒣 < 𝑀 > ≃ ∐𝑨∈MC(𝑀)/𝒣 < 𝑀𝑨 > 𝜌𝑜 < 𝑀 >≅ 𝐼𝑜−1(𝑀) 𝑜 > 1 𝜌1 < 𝑀 >≅ 𝐼0(𝑀)

For any cDGL 𝑀 we have

1. 2.

𝜌𝑜 < 𝑀 >≅ 𝐼𝑜−1(𝑀) 𝑜 > 1 𝜌1 < 𝑀 >≅ 𝐼0(𝑀)

𝜌1 < 𝑀 >× 𝜌𝑜 < 𝑀 > ⟶ 𝜌𝑜 < 𝑀 >

𝜐

≅ ≅

𝐼0 𝑀 × 𝐼𝑜−1(𝑀) 𝐼𝑜−1(𝑀)

?

(𝑦 , 𝑨 )

𝑓ad𝑦(𝑨)

Reduced Complete HomcDGL(𝔐∙, 𝑀) Reduced Nilpotent, finite-type Nilpotent, finite-type Complete 𝑂∗෢ 𝒱∙𝒣∙ ഥ 𝑋(𝑀) 𝒟(𝑀) 𝑇 MC(𝑀⨂Ω∙) Neisendorfer,

• Pac. J. M., 1978.

Getzler’s

• bservation

Buijs, Félix, Murillo, Tanré, Preprint 2017 Work in progress