Description
Homotopy equivalence, homotopy groups πₙ(X), CW complexes, and rationalization Xℚ. Rational homotopy theory studies spaces when "all primes are ignored"—i.e., up to rational homotopy equivalence. Whitehead's and Hurewicz theorems connect homotopy and homology. Foundation for Sullivan's minimal model approach.
Source: Wikipedia; Hatcher, Algebraic Topology
Dependency Flowchart
graph TD
DefHomotopy["Def: Homotopy\nf ≃ g : X×I → Y"]
DefHEquiv["Def: Homotopy equivalence\nX ≃ Y"]
DefPi["Def: πₙ(X) = [Sⁿ, X]\nHomotopy groups"]
DefCW["Def: CW complex\nCells, skeletons"]
DefRational["Def: Rationalization Xℚ\nπₙ(Xℚ) = πₙ(X)⊗ℚ"]
ThmWhitehead["Thm: Whitehead\nf* iso on πₙ ⇒ f homotopy equiv"]
ThmHurewicz["Thm: Hurewicz\nπₙ ≅ Hₙ for n-connected"]
LemPiAbelian["Lem: πₙ abelian for n≥2"]
DefHomotopy --> DefHEquiv
DefHomotopy --> DefPi
DefPi --> LemPiAbelian
DefCW --> ThmWhitehead
DefPi --> ThmWhitehead
DefPi --> ThmHurewicz
DefRational --> DefPi
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef lemma fill:#74c0fc,color:#fff,stroke:#4dabf7
class DefHomotopy,DefHEquiv,DefPi,DefCW,DefRational definition
class ThmWhitehead,ThmHurewicz theorem
class LemPiAbelian lemma
Color Scheme
Blue Definitions
Teal Theorems
Purple Lemmas
Process Statistics
- Nodes: 8
- Edges: 8
- Definitions: 5
- Lemmas: 1
- Theorems: 2