Hidden algebraic structure on cohomology of simplicial complexes, - PowerPoint PPT Presentation

Hidden algebraic structure on cohomology of simplicial complexes, and TFT Pavel Mnev University of Zurich Trinity College Dublin, February 4, 2013

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Simplicial complex T

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Simplicial complex T   � Simplicial cochains C 0 ( T ) → · · · → C top ( T ) , C k ( T ) = Span { k − simplices } , � d k : C k ( T ) → C k +1 ( T ) , e σ �→ ± e σ ′ ���� σ ′ ∈ T : σ ∈ faces( σ ′ ) basis cochain

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Simplicial complex T   � Simplicial cochains C 0 ( T ) → · · · → C top ( T ) , C k ( T ) = Span { k − simplices } , � d k : C k ( T ) → C k +1 ( T ) , e σ �→ ± e σ ′ ���� σ ′ ∈ T : σ ∈ faces( σ ′ ) basis cochain   � Cohomology H • ( T ) , H k ( T ) = ker d k / im d k − 1 — a homotopy invariant of T

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Cohomology carries a commutative ring structure, coming from (non-commutative) Alexander’s product for cochains.

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Cohomology carries a commutative ring structure, coming from (non-commutative) Alexander’s product for cochains. Massey operations on cohomology are a complete invariant of rational homotopy type in simply connected case (Quillen-Sullivan), i.e. rationalized homotopy groups Q ⊗ π k ( T ) can be recovered from them.

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Cohomology carries a commutative ring structure, coming from (non-commutative) Alexander’s product for cochains. Massey operations on cohomology are a complete invariant of rational homotopy type in simply connected case (Quillen-Sullivan), i.e. rationalized homotopy groups Q ⊗ π k ( T ) can be recovered from them. Example of use: linking of Borromean rings is detected by a non-vanishing Massey operation on cohomology of the complement. m 3 ([ α ] , [ β ] , [ γ ]) = [ u ∧ γ + α ∧ v ] ∈ H 2 where [ α ] , [ β ] , [ γ ] ∈ H 1 , du = α ∧ β , dv = β ∧ γ .

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Background Another example: nilmanifold M = H 3 ( R ) / H 3 ( Z )         1 x z 1 a c      | x, y, z ∈ R  | a, b, c ∈ Z = 0 1 y  / 0 1 b      0 0 1 0 0 1 Denote α = dx, β = dy, u = dz − y dx ∈ Ω 1 ( M ) Important point: α ∧ β = du . The cohomology is spanned by classes [1] , [ α ] , [ β ] , [ α ∧ u ] , [ β ∧ u ] , [ α ∧ β ∧ u ] ���� � �� � � �� � � �� � degree 0 degree 1 degree 2 degree 3 and m 3 ([ α ] , [ β ] , [ β ]) = [ u ∧ β ] ∈ H 2 ( M ) is a non-trivial Massey operation.

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Result Fix g a unimodular Lie algebra (i.e. with tr[ x, • ] = 0 for any x ∈ g ). Main construction (P.M.) Simplicial complex T   � local formula Unimodular L ∞ algebra structure on g ⊗ C • ( T )   � homotopy transfer Unimodular L ∞ algebra structure on g ⊗ H • ( T )

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Result Fix g a unimodular Lie algebra (i.e. with tr[ x, • ] = 0 for any x ∈ g ). Main construction (P.M.) Simplicial complex T   � local formula Unimodular L ∞ algebra structure on g ⊗ C • ( T )   � homotopy transfer Unimodular L ∞ algebra structure on g ⊗ H • ( T ) Main theorem (P.M.) Unimodular L ∞ algebra structure on g ⊗ H • ( T ) (up to isomorphisms) is an invariant of T under simple homotopy equivalence. horn filling collapse to a horn

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Result Main construction (P.M.) Simplicial complex T   � local formula Unimodular L ∞ algebra structure on g ⊗ C • ( T )   � homotopy transfer Unimodular L ∞ algebra structure on g ⊗ H • ( T ) Thom’s problem: lifting ring structure on H • ( T ) to a commutative product on cochains. Removing g , we get a homotopy commutative algebra on C • ( T ) . This is an improvement of Sullivan’s result with cDGA structure on cochains = Ω poly ( T ) . Local formulae for Massey operations. Our invariant is strictly stronger than rational homotopy type.

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Unimodular L ∞ algebras Definition A unimodular L ∞ algebra is the following collection of data: (a) a Z -graded vector space V • , (b) “classical operations” l n : ∧ n V → V , n ≥ 1 , (c) “quantum operations” q n : ∧ n V → R , n ≥ 1 ,

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Unimodular L ∞ algebras Definition A unimodular L ∞ algebra is the following collection of data: (a) a Z -graded vector space V • , (b) “classical operations” l n : ∧ n V → V , n ≥ 1 , (c) “quantum operations” q n : ∧ n V → R , n ≥ 1 , subject to two sequences of quadratic relations: � 1 r ! s ! l r +1 ( • , · · · , • , l s ( • , · · · , • )) = 0 , n ≥ 1 1 r + s = n (anti-symmetrization over inputs implied), 1 n ! Str l n +1 ( • , · · · , • , − )+ 2 + � 1 r ! s ! q r +1 ( • , · · · , • , l s ( • , · · · , • )) = 0 r + s = n

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Unimodular L ∞ algebras Definition A unimodular L ∞ algebra is the following collection of data: (a) a Z -graded vector space V • , (b) “classical operations” l n : ∧ n V → V , n ≥ 1 , (c) “quantum operations” q n : ∧ n V → R , n ≥ 1 , subject to two sequences of quadratic relations: � 1 r ! s ! l r +1 ( • , · · · , • , l s ( • , · · · , • )) = 0 , n ≥ 1 1 r + s = n (anti-symmetrization over inputs implied), 1 n ! Str l n +1 ( • , · · · , • , − )+ 2 + � 1 r ! s ! q r +1 ( • , · · · , • , l s ( • , · · · , • )) = 0 r + s = n Note: First classical operation satisfies ( l 1 ) 2 = 0 , so ( V • , l 1 ) is a complex. A unimodular L ∞ algebra is in particular an L ∞ algebra (as introduced by Lada-Stasheff), by ignoring q n . Unimodular Lie algebra is the same as unimodular L ∞ algebra with l � =2 = q • = 0 .

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Unimodular L ∞ algebras An alternative definition A unimodular L ∞ algebra is a graded vector space V endowed with a vector field Q on V [1] of degree 1, a function ρ on V [1] of degree 0, satisfying the following identities: [ Q, Q ] = 0 , div Q = Q ( ρ )

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Homotopy transfer Homotopy transfer theorem (P.M.) If ( V, { l n } , { q n } ) is a unimodular L ∞ algebra and V ′ ֒ → V is a deformation retract of ( V, l 1 ) , then V ′ carries a unimodular L ∞ structure given by 1 n = � : ∧ n V ′ → V ′ l ′ 1 Γ 0 | Aut(Γ 0 ) | n = � + � : ∧ n V ′ → R q ′ 1 1 Γ 1 | Aut(Γ 1 ) | Γ 0 | Aut(Γ 0 ) | where Γ 0 runs over rooted trees with n leaves and Γ 1 runs over 1-loop graphs with n leaves.

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Homotopy transfer Homotopy transfer theorem (P.M.) If ( V, { l n } , { q n } ) is a unimodular L ∞ algebra and V ′ ֒ → V is a deformation retract of ( V, l 1 ) , then V ′ carries a unimodular L ∞ structure given by 1 n = � : ∧ n V ′ → V ′ l ′ 1 Γ 0 | Aut(Γ 0 ) | n = � + � : ∧ n V ′ → R q ′ 1 1 Γ 1 | Aut(Γ 1 ) | Γ 0 | Aut(Γ 0 ) | where Γ 0 runs over rooted trees with n leaves and Γ 1 runs over 1-loop graphs with n leaves. Decorations: i : V ′ ֒ p : V ։ V ′ leaf → V root − s : V • → V •− 1 edge ( m + 1) -valent vertex l m cycle super-trace over V m -valent ◦ -vertex q m where s is a chain homotopy, l 1 s + s l 1 = id − i p .

Unimodular L ∞ algebra associated to a simplicial complex TFT perspective Conclusion Homotopy transfer Homotopy transfer theorem (P.M.) If ( V, { l n } , { q n } ) is a unimodular L ∞ algebra and V ′ ֒ → V is a deformation retract of ( V, l 1 ) , then V ′ carries a unimodular L ∞ structure given by 1 n = � : ∧ n V ′ → V ′ l ′ 1 Γ 0 | Aut(Γ 0 ) | n = � + � : ∧ n V ′ → R q ′ 1 1 Γ 1 | Aut(Γ 1 ) | Γ 0 | Aut(Γ 0 ) | where Γ 0 runs over rooted trees with n leaves and Γ 1 runs over 1-loop graphs with n leaves. Decorations: i : V ′ ֒ p : V ։ V ′ leaf → V root − s : V • → V •− 1 edge ( m + 1) -valent vertex l m cycle super-trace over V m -valent ◦ -vertex q m where s is a chain homotopy, l 1 s + s l 1 = id − i p . Algebra ( V ′ , { l ′ n } , { q ′ n } ) changes by isomorphisms under changes of 2 induction data ( i, p, s ) .