Description
String topology: operations on the free loop space LM = Map(S¹, M) of an oriented manifold M. Chas and Sullivan (1999) defined a loop product and coproduct on H*(LM), giving LM the structure of a 2D topological field theory (TQFT) with closed string compactification. Genus-zero part: ∞-Lie bialgebra; higher genus: quantum Lie bialgebra. Sullivan's contribution: field-theoretic interpretation, connections to rational homotopy.
Source: Wikipedia; Chas & Sullivan; Sullivan
Dependency Flowchart
graph TD
DefLoop["Def: Free loop space LM\nMap(S¹,M)"]
DefEq["Def: Equivariant homology\nH^S¹_*(LM)"]
DefProduct["Def: Loop product ⋆\nConcat at intersection"]
DefCoproduct["Def: Loop coproduct Δ\nSelf-intersection"]
ThmTQFT["Thm: 2D TQFT structure\nOn equivariant chains"]
ThmLie["Thm: Genus 0 ⇒ ∞-Lie bialgebra\n[Bracket, cobracket]"]
ThmQuantum["Thm: Higher genus ⇒ quantum Lie bialgebra"]
DefCompact["Def: Closed string compactification\nMod constant loops"]
DefLoop --> DefEq
DefLoop --> DefProduct
DefLoop --> DefCoproduct
DefProduct --> ThmTQFT
DefCoproduct --> ThmTQFT
DefEq --> ThmTQFT
ThmTQFT --> ThmLie
ThmTQFT --> ThmQuantum
DefCompact --> ThmTQFT
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
class DefLoop,DefEq,DefProduct,DefCoproduct,DefCompact definition
class ThmTQFT,ThmLie,ThmQuantum theorem
Color Scheme
Blue Definitions
Teal Theorems
Process Statistics
- Nodes: 9
- Edges: 11
- Definitions: 5
- Theorems: 3