Description
Sullivan's rational homotopy theory: algebraic models for spaces when primes are ignored. A Sullivan minimal model is a CDGA (∧V, d) with d: V → ∧²V, quasi-isomorphic to the algebra of piecewise polynomial forms. Equivalence: simply-connected rational homotopy types ↔ minimal CDGAs. H(X;ℚ) ≅ H(∧V); πₙ(X)⊗ℚ ≅ Hom(Vⁿ,ℚ). Parallel to Quillen's DGL approach.
Source: Wikipedia; Sullivan; Félix–Halperin–Thomas
Dependency Flowchart
graph TD
DR["De Rham cohomology"]
Homotopy["Homotopy theory basics"]
DefCDGA["Def: CDGA\nCommutative dg algebra"]
DefSullAlg["Def: Sullivan algebra\n(A,d) → (A⊗∧V,d')"]
DefMinimal["Def: Minimal\n d: V → ∧²V"]
DefModel["Def: Sullivan minimal model\n(∧V,d) for space X"]
ThmEquiv["Thm: Equivalence\nRational ht types ↔ minimal CDGAs"]
ThmH["Thm: H(X;ℚ) ≅ H(∧V)"]
ThmPi["Thm: πₙ(X)⊗ℚ ≅ Hom(Vⁿ,ℚ)"]
DR --> DefCDGA
Homotopy --> DefModel
DefCDGA --> DefSullAlg
DefSullAlg --> DefMinimal
DefMinimal --> DefModel
DefModel --> ThmEquiv
DefModel --> ThmH
DefModel --> ThmPi
classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class DR,Homotopy axiom
class DefCDGA,DefSullAlg,DefMinimal,DefModel definition
class ThmEquiv,ThmH,ThmPi theorem
Color Scheme
Red Prerequisites
Blue Definitions
Teal Theorems
Process Statistics
- Nodes: 10
- Edges: 11
- Axioms: 2
- Definitions: 4
- Theorems: 3