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. 𝜌 𝑜 𝒼 ⨂ℚ ∶ 𝜌 𝑜 𝒀 ⨂ℚ ⟶ 𝜌 𝑜 𝒁 ⨂ℚ , 𝑜 ≥ 2. ≅ 2. 𝐼 𝑜 𝒼 ⨂ℚ ∶ 𝐼 𝑜 𝒀 ; ℚ ⟶ 𝐼 𝑜 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 (𝒀 ℚ , 𝑓) , with 𝒀 ℚ a rational space and and 𝑓 ∶ 𝒀 ⟶ 𝒀 ℚ a rational homotopy equivalence. The study of the rational homotopy type of 𝒀 is the study of the homotopy type of its rationalisation 𝒀 ℚ .

Then, if 𝒀 is a finite simply connected CW-complex 𝜌 𝑜 𝒀 = ⨁ 𝑠 ℤ ⨁ ℤ 𝑞 1𝑠1 ⨁ ⋯ ⨁ ℤ 𝑞 𝑛𝑠𝑛 𝜌 𝑜 𝒀 ℚ ≅ 𝜌 𝑜 𝒀 ⨂ℚ = ⨁ 𝑠 ℚ The study of the rational homotopy type of 𝒀 is the study of the homotopy type of its rationalisation 𝒀 ℚ .

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 𝑀 𝑞 with: A bilinear operation . , . ∶ 𝑀 × 𝑀 ⟶ 𝑀 such that 𝑀 𝑞 , 𝑀 𝑟 ⊂ 𝑀 𝑞+𝑟 satisfying: = − −1 𝑞𝑟 𝑐, 𝑏 a) a) 𝑏, 𝑐 , 𝑏 ∈ 𝑀 𝑞 , 𝑐 ∈ 𝑀 𝑟 𝑏, 𝑐 , 𝑑 + −1 𝑞𝑟 𝑐, [𝑏, 𝑑] , b) b) 𝑏, [𝑐, 𝑑] = DGL 𝑏 ∈ 𝑀 𝑞 , 𝑐 ∈ 𝑀 𝑟 , 𝑑 ∈ 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 𝑀 𝑞 with: A linear map 𝜖 ∶ 𝑀 ⟶ 𝑀 such that 𝜖𝑀 𝑞 ⊂ 𝑀 𝑞−1 satisfying: a) a) 𝜖 ∘ 𝜖 = 0 + −1 𝑞 𝑏, 𝜖𝑐 𝜖 𝑏, 𝑐 = 𝜖𝑏, 𝑐 , b) b) DGL 𝑏 ∈ 𝑀 𝑞 , 𝑐 ∈ 𝑀

𝜇 ⟶ 𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 ⟶ DGL + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑅 Rational homotopy Quasi-isomorphisms equivalences < 𝜇(𝒀) > 𝑅 ≃ 𝒀 ℚ (𝑀, ∙,∙ , 𝜖) is a DGL-mod model el of 𝒀 if ≃ ≃ ≃ ≃ DGL 𝜇(𝒀) ⟶ ⟶ 𝑀 ⟶ ⟶ ⋯

𝜇 ⟶ 𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 ⟶ DGL + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑅 Rational homotopy Quasi-isomorphisms equivalences < 𝜇(𝒀) > 𝑅 ≃ 𝒀 ℚ (𝑀, ∙,∙ , 𝜖) is a DGL-mod model el of 𝒀 if ≃ DGL (𝑀, ∙,∙ , 𝜖) (𝕄(𝑋), 𝑒) ⟶

Rational homotopy groups 𝐼 𝑜 ( ) ≅ 𝜌 𝑜+1 (𝒀)⨂ℚ 𝑀 Rational homology groups ≃ 𝑀 (𝕄 𝑋 , 𝜖) ⟶ Minimal Quillen model 𝑡(𝑋 ⊕ ℚ) ≅ 𝐼 ∗ (𝑌; ℚ) DGL

Spheres 𝕄 𝑤 , 0 , 𝑤 = 𝑜 − 1 is a model for the 𝑜 -dimensional sphere 𝑇 𝑜 . Products 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 . Wedge products If (𝕄 𝑊 , 𝑒) and (𝕄 𝑋 , 𝑒 ′ ) are minimal DGL DGL models for 𝒀 and 𝒁 respectively , then (𝕄 𝑊 ⊕ 𝑋 , 𝐸) is a DGL-model of 𝒀 ∨ 𝒁 .

CW-decomposition A ba base sed to topo pologi logical cal sp space ace (𝒀, ∗) ( 𝕄 𝑊 , 𝜖 ) 𝑤 𝛽 ∈ 𝑊 𝑜+1 = 𝑓 𝛽 𝑜 𝜖𝑤 𝛽 ∈ 𝐼 𝑜−1 (𝕄 𝑊 <𝑜 ) 𝛽 ∶ 𝑇 𝑜 → 𝒀 𝑜 ; 𝑔 [𝑔 𝛽 ] ∈ Π 𝑜 (𝒀 𝑜 ) ≅ Π 𝑜 (𝒀 𝑜 )⨂ℚ ( 𝕄(𝑊 <𝑜 ), 𝜖 ) 𝒀 𝑜 𝑜 - skeleton

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 𝐵 𝑞 with: A bilinear operation ∙ ∶ 𝐵 × 𝐵 ⟶ 𝐵 such that 𝐵 𝑞 ∙ 𝐵 𝑟 ⊂ 𝐵 𝑞+𝑟 satisfying: a) a) 𝑏 ∙ (𝑐 ∙ 𝑑) = (𝑏 ∙ 𝑐) ∙ 𝑑 , 𝑏, 𝑐, 𝑑 ∈ 𝐵 𝑏 ∙ 𝑐 = −1 𝑞𝑟 𝑐 ∙ 𝑏, b) b) 𝑏 ∈ 𝐵 𝑞 , 𝑐 ∈ 𝐵 𝑟 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 𝐵 𝑞 with: A linear map 𝑒: 𝐵 ⟶ A such that 𝑒𝐵 𝑞 ⊂ 𝐵 𝑞+1 satisfying: a) a) 𝑒 ∘ 𝑒 = 0 𝑒(𝑏 ∙ 𝑐) = (𝑒𝑏) ∙ 𝑐 + −1 𝑞 𝑏 ∙ (𝑒𝑐) , b) b) CDGA 𝑏 ∈ 𝑀 𝑞 , 𝑐 ∈ 𝑀

𝐵 𝑄𝑀 ⟶ 𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 ⟶ CDGA + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑇 Rational homotopy Quasi-isomorphisms equivalences < 𝐵 𝑄𝑀 (𝒀) > 𝑇 ≃ 𝒀 ℚ (𝐵, 𝑒) is a CDGA-model model of 𝒀 if ≃ ≃ ≃ ≃ CDGA ⟶ ⟶ ⟶ ⟶ 𝐵 𝐵 𝑄𝑀 (𝒀) ⋯

𝐵 𝑄𝑀 ⟶ 𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 ⟶ CDGA + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑇 Rational homotopy Quasi-isomorphisms equivalences < 𝐵 𝑄𝑀 (𝒀) > 𝑇 ≃ 𝒀 ℚ (𝐵, 𝑒) is a CDGA-model model of 𝒀 if ≃ CDGA (⋀𝑊, 𝑒) (𝐵, 𝑒) ⟶

Rational cohomology groups 𝐼 ∗ 𝐵, 𝑒 ≅ 𝐼 ∗ (𝒀; ℚ ) Rational homotopy groups ≃ (⋀𝑊, 𝑒) (𝐵, 𝑒) ⟶ Hom(𝑊 𝑙 , ℚ) ≅ 𝜌 𝑙 (𝒀)⨂ ℚ CDGA

Eilenberg-MacLane spaces ⋀𝑤, 0 , 𝑤 = 𝑜 is a model of the Eilenberg-MacLane space 𝐿(ℤ, 𝑜) . Wedge products 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 Products models for 𝒀 and 𝒁 respectively , then CDGA (⋀ 𝑊 ⊕ 𝑋 , 𝐸) is a CDGA-model of 𝒀 × 𝒁 .

Postnikov decomposition CDGA

𝜇 ⟶ 𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 ⟶ DGL + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑅 𝐵 𝑄𝑀 ⟶ 𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 ⟶ CDGA + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑇

𝜇 ⟶ ⟶ 𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞 ⟶ ⟶ ⟶ ⟶ sGrp ⟶ ⟶ sCHA sLA DGL + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑅 ⟶ ⟶ 𝐓𝐣𝐧𝐪𝐦𝐳 𝐝𝐩𝐨𝐨𝐟𝐝𝐮𝐟𝐞, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 𝒟 ∗ ℒ 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 𝐵 𝑄𝑀 ⟶ 𝐎𝐣𝐦𝐪𝐩𝐮𝐟𝐨𝐮, 𝐠𝐣𝐨𝐣𝐮𝐟 𝐮𝐳𝐪𝒇 < 𝐵 > 𝑇 = Hom 𝑫𝑬𝑯𝑩 (𝐵, Ω ) ⟶ CDGA + 𝐃𝐗 − 𝐝𝐩𝐧𝐪𝐦𝐟𝐲𝐟𝐭 < ∙ > 𝑇 < 𝐵 > 𝑇 is defined for 𝐵 a ℤ -graded CDGA but < 𝑀 > 𝑅 has only sense for positively graded DGL’s

The goal is to understand the rational behaviour of the spaces: ∗ (𝒀, 𝒁) map 𝒀, 𝒁 , map ∗ 𝒀, 𝒁 , map 𝑔 𝒀, 𝒁 , map 𝑔 If 𝒀 and 𝒁 are finite type map 𝒀, 𝒁 has the homotopy CW-complexes type of a CW-complex. If 𝒀 and 𝒁 are nilpotent map 𝑔 𝒀, 𝒁 is nilpotent. If 𝒀 is a finite CW-complex map 𝑔 𝒀, 𝒁 is of finite type. If 𝒀 and 𝒁 are 1-connected map 𝑔 𝒀, 𝒁 is 1-connected.

Let 𝒀 be a finite CW-complex and 𝒁 be a nilpotent, finite type CW-complex. (𝐶 , 𝑒) CDGA 𝐶 ⋕𝑜 = 𝐶 −𝑜 (𝐶 ⋕ , 𝜀) CDGC 𝒀 Sullivan model (⋀𝑊, 𝑒) 𝒁 The following Sullivan algebra is a model for map 𝒀, 𝒁 . if 𝑤 ∈ V such that d𝑤 = 𝑣𝑥 and 𝛾 ∈ 𝐶 ⋕ with ⋀ V⨂𝐶 ⋕ , 𝐸 V⨂𝐶 ⋕ ′ ⨂ 𝛾 𝒌 ′′ ∆𝛾 = σ 𝒌 𝛾 𝒌 ′ | (𝑣⨂𝛾 𝑘 𝐸 𝑤⨂𝛾 = σ 𝑘 (−1) 𝑥 |𝛾 𝑘 ′ ) (𝑥⨂𝛾 𝑘 ′′ ) + 𝑤⨂𝜀𝛾

Let ϕ: (⋀𝑊, 𝑒) ⟶ 𝐶 be a model of 𝑔: 𝒀 ⟶ 𝒁 . Let 𝐿 ϕ be the differential ideal generated by 𝐵 1 ∪ 𝐵 2 ∪ 𝐵 3 𝐵 2 = 𝐸(𝑊 ⊗ 𝐶 ⋕ ) 0 𝐵 1 = (𝑊 ⊗ 𝐶 ⋕ ) <0 𝛽 ∈ (𝑊 ⊗ 𝐶 ⋕ ) 0 } 𝐵 3 = 𝛽 − 𝜚 𝛽 ⋀ V⨂𝐶 ⋕ , 𝐸 ⟶ ⋀ V⨂𝐶 ⋕ , 𝐸 / 𝐿 ϕ The projection map 𝑔 𝒀, 𝒁 ⟶ map 𝒀, 𝒁 is a model of the injection ⋀ V⨂𝐶 ⋕ , 𝐸 / 𝐿 ϕ ≅ ⋀𝑊 ⊗ 𝐶 ⋕1 ⨁(𝑊 ⊗ 𝐶 ⋕ ) ≥2 , 𝐸 𝝔

𝒀 , 𝒁 with DGL-models 𝑀 , 𝑀 ′ , respectively. 𝑔: 𝒀 ⟶ 𝒁 . DGL-model of map 𝑔 𝒀, 𝒁 ? map 𝑔 𝒀, 𝒁 , 𝐶 ⋕ , 𝜀 = 𝒟 ∗ (𝑀) Cartan-Eilenberg (𝐶 ⋕ , 𝜀) CDGC 𝒀 ⋀𝑊, 𝑒 = 𝒟 ∗ 𝑀 ′ Sullivan model cochains on 𝑀 ′ (⋀𝑊, 𝑒) 𝒁