The model Fulton–MacPherson operad Sketch of proof Factorization homology The Lambrechts–Stanley Model of Configuration Spaces Najib Idrissi Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Configuration spaces M : smooth closed n -manifold (+ future adjectives) Conf k ( M ) = { ( x 1 , . . . , x k ) ∈ M × k | x i � = x j ∀ i � = j } 4 2 1 3 Goal Obtain a CDGA model of Conf k ( M ) from a CDGA model of M Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Plan 1 The model 2 Action of the Fulton–MacPherson operad 3 Sketch of proof through Kontsevich formality 4 Computing factorization homology Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Models We are interested in rational/real models A ≃ Ω ∗ ( M ) “forms on M ” (de Rham, piecewise polynomial...) where A is an “explicit” CDGA M simply connected = ⇒ A contains all the rational/real homotopy type of M Conf k ( M ) smooth (but noncompact); we’re looking for a CDGA ≃ Ω ∗ ( Conf k ( M )) built from A Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Poincaré duality models Poincaré duality CDGA ( A , d , ε ) (example: A = H ∗ ( M )) • ( A , d ): finite type connected CDGA; • ε : A n → k such that ε ◦ d = 0; • A k ⊗ A n − k → k , a ⊗ b �→ ε ( ab ) non degenerate. Theorem (Lambrechts–Stanley 2008) ∼ ∼ Ω ∗ ( M ) · ∃ A Any simply connected manifold has such ∃ ε � k a model M Remark Reasonable assumption: ∃ non simply-connected L ≃ L ′ but Conf k ( L ) �≃ Conf k ( L ′ ) for k ≥ 2 [Longoni–Salvatore]. Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Diagonal class In cohomology, diagonal class [ M ] ∈ H n ( M ) �→ δ ∗ [ M ] ∈ H n ( M × M ) δ ( x ) = ( x , x ) ↔ ∆ M ∈ H 2 n − n ( M × M ) Representative in a Poincaré duality model ( A , d , ε ): � ( − 1) | a i | a i ⊗ a ∨ i ∈ ( A ⊗ A ) n ∆ A = { a i } : graded basis and ε ( a i a ∨ j ) = δ ij (independent of chosen basis) Properties • ( a ⊗ 1)∆ A = (1 ⊗ a )∆ A “concentrated around the diagonal” µ A • A ⊗ A − → A , ∆ A �→ e ( A ) = χ ( A ) · vol A Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology The Lambrechts–Stanley model Conf k ( R n ) is a formal space, with cohomology [Arnold–Cohen]: H ∗ ( Conf k ( R n )) = S ( ω ij ) 1 ≤ i � = j ≤ k / I , deg ω ij = n − 1 I = � ω ji = ± ω ij , ω 2 ij = 0 , ω ij ω jk + ω jk ω ki + ω ki ω ij = 0 � . G A ( k ) conjectured model of Conf k ( M ) = M × k \ � i � = j ∆ ij • “Generators”: A ⊗ k ⊗ S ( ω ij ) 1 ≤ i � = j ≤ k • Relations: • Arnold relations for the ω ij • p ∗ i ( a ) · ω ij = p ∗ j ( a ) · ω ij . ( p ∗ i ( a ) = 1 ⊗ · · · ⊗ 1 ⊗ a ⊗ 1 ⊗ · · · ⊗ 1) • d ω ij = ( p ∗ i · p ∗ j )(∆ A ). Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology First examples G A ( k ) = ( A ⊗ k ⊗ S ( ω ij ) 1 ≤ i < j ≤ k / J , d ω ij = ( p ∗ i · p ∗ j )(∆ A )) G A (0) = R : model of Conf 0 ( M ) = { ∅ } � G A (1) = A : model of Conf 1 ( M ) = M � � A ⊗ A ⊗ 1 ⊕ A ⊗ A ⊗ ω 12 � G A (2) = , d ω 12 = ∆ A ⊗ 1 1 ⊗ a ⊗ ω 12 ≡ a ⊗ 1 ⊗ ω 12 ∼ = ( A ⊗ A ⊗ 1 ⊕ A ⊗ A A ⊗ ω 12 , d ω 12 = ∆ A ⊗ 1) ∼ = ( A ⊗ A ⊗ 1 ⊕ A ⊗ ω 12 , d ω 12 = ∆ A ⊗ 1) ∼ → A ⊗ 2 / (∆ A ) − Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Brief history of G A 1969 [Arnold–Cohen] H ∗ ( Conf k ( R n )) ≈ “ G H ∗ ( R n ) ( k )” 1978 [Cohen–Taylor] E 2 = G H ∗ ( M ) ( k ) = ⇒ H ∗ ( Conf k ( M )) ~1994 For smooth projective complex manifolds (= ⇒ Kähler): • [Kříž] G H ∗ ( M ) ( k ) model of Conf k ( M ) • [Totaro] The Cohen–Taylor SS collapses 2004 [Lambrechts–Stanley] A ⊗ 2 / (∆ A ) model of Conf 2 ( M ) for a 2-connected manifold H ∗ ( M ) ( k ) ∼ ~2004 [Félix–Thomas, Berceanu–Markl–Papadima] G ∨ = page E 2 of Bendersky–Gitler SS for H ∗ ( M × k , � i � = j ∆ ij ) 2008 [Lambrechts–Stanley] H ∗ ( G A ( k )) ∼ = Σ k − gVect H ∗ ( Conf k ( M )) 2015 [Cordova Bulens] A ⊗ 2 / (∆ A ) model of Conf 2 ( M ) for dim M = 2 m Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology First part of the theorem Theorem (I.) Let M be a smooth, closed, simply connected manifold of dimension ≥ 4. Then G A ( k ) is a model over R of Conf k ( M ) for all k ≥ 0. dim M ≥ 3 = ⇒ Conf k ( M ) is simply connected when M is (cf. Fadell–Neuwirth fibrations). Corollary All the real homotopy type of Conf k ( M ) is contained in ( A , d , ε ). Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Fulton–MacPherson compactification FM n ( k ): Fulton–MacPherson compactification of Conf k ( R n ) 6 7 8 3 5 4 2 1 (+ normalization to deal with R n being noncompact) Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Fulton–MacPherson compactification (2) FM M ( k ): similar compactification of Conf k ( M ) 5 6 4 3 7 1 2 Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Operads Idea Study all of { Conf k ( M ) } k ≥ 0 = ⇒ more structure. FM n = { FM n ( k ) } k ≥ 0 is an operad: we can insert an infinitesimal configuration into another 2 3 1 2 1 2 ◦ 2 = 1 FM n ( k ) × FM n ( l ) ◦ i − → FM n ( k + l − 1) , 1 ≤ i ≤ k Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Structure de module M framed = ⇒ FM M = { FM M ( k ) } k ≥ 0 is a right FM n -module: we can insert an infinitesimal configuration into a configuration on M 4 5 3 3 1 2 3 ◦ 3 = 1 1 2 2 FM M ( k ) × FM n ( l ) ◦ i − → FM M ( k + l − 1) , 1 ≤ i ≤ k Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Cohomology of FM n and coaction on G A H ∗ ( FM n ) inherits a Hopf cooperad structure One can rewrite: G A ( k ) = ( A ⊗ k ⊗ H ∗ ( FM n ( k )) / relations , d ) Proposition ⇒ G A = { G A ( k ) } k ≥ 0 Hopf right H ∗ ( FM n )-comodule χ ( M ) = 0 = Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Motivation We are looking for something to put here: ∼ − ? ∼ → Ω ∗ ( FM M ( k )) G A ( k ) ← − Hunch: if true, then hopefully it fits in something like this! ∼ ∼ Ω ∗ ( FM M ) ? G A � � � ∼ ∼ H ∗ ( FM n ) Ω ∗ ( FM n ) ? Fortunately, the bottom row is already known: formality of FM n Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Kontsevich’s graph complexes [Kontsevich] Hopf cooperad Graphs n = { Graphs n ( k ) } k ≥ 0 1 ∈ Graphs n (3) 2 3 1 1 1 · = 2 3 2 3 2 3 1 1 1 1 = ± ± ± d 2 3 2 3 2 3 2 3 Theorem (Kontsevich 1999, Lambrechts–Volić 2014) Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces ∼ ∼ H ( ) Ω ( )
The model Fulton–MacPherson operad Sketch of proof Factorization homology Labeled graph complexes ∼ ∼ Recall: Ω ∗ PA ( M ) R A ε � k M � labeled graph complex Graphs R : x y ∈ Graphs R (1) (where x , y ∈ R ) 1 2 xy � x y � dx y ± x dy ± = d 1 2 1 2 1 1 2 x ∆ ′ y ∆ ′′ R � R + ± 1 2 (∆ R ) � x � x y � ≡ σ ( y ) · 1 2 1 M Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
The model Fulton–MacPherson operad Sketch of proof Factorization homology Complete version of the theorem Theorem (I., complete version) ∼ ∼ Ω ∗ PA ( FM M ) G A Graphs R � † � † � ‡ ∼ ∼ H ∗ ( FM n ) Ω ∗ PA ( FM n ) Graphs n † When χ ( M ) = 0 ‡ When M is framed Najib Idrissi The Lambrechts–Stanley Model of Configuration Spaces
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