Partial products of circles Alex Suciu Northeastern University Boston, Massachusetts Algebra and Geometry Seminar Vrije University Amsterdam October 13, 2009 Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 1 / 27
Outline Toric complexes 1 Partial products of spaces Toric complexes and right-angled Artin groups Graded Lie algebras associated to RAAGs Chen Lie algebras of RAAGs Artin kernels and Bestvina-Brady groups Resonance varieties 2 Resonance varieties Kähler and quasi-Kähler groups Kähler and quasi-Kähler RAAGs Kähler and quasi-Kähler BB groups Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 2 / 27
Toric complexes Partial products of spaces Partial product construction Input: K , a simplicial complex on [ n ] = { 1 , . . . , n } . ( X , A ) , a pair of topological spaces, A � = ∅ . Output: � ( X , A ) σ ⊂ X × n Z K ( X , A ) = σ ∈ K where ( X , A ) σ = { x ∈ X × n | x i ∈ A if i / ∈ σ } . Interpolates between Z ∅ ( X , A ) = Z K ( A , A ) = A × n and Z ∆ n − 1 ( X , A ) = Z K ( X , X ) = X × n Examples: Z n points ( X , ∗ ) = � n X (wedge) Z ∂ ∆ n − 1 ( X , ∗ ) = T n X (fat wedge) Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 3 / 27
Toric complexes Partial products of spaces Properties: L ⊂ K subcomplex ⇒ Z L ( X , A ) ⊂ Z K ( X , A ) subspace. ( X , A ) pair of (finite) CW-complexes ⇒ Z K ( X , A ) is a (finite) CW-complex. Z K ∗ L ( X , A ) ∼ = Z K ( X , A ) × Z L ( X , A ) . f : ( X , A ) → ( Y , B ) continuous map ⇒ f × n : X × n → Y × n restricts to a continuous map Z f : Z K ( X , A ) → Z K ( Y , B ) . Consequently, ( X , A ) ≃ ( Y , B ) ⇒ Z K ( X , A ) ≃ Z K ( Y , B ) . (Strickland) f : K → L simplicial � Z f : Z K ( X , A ) → Z L ( X , A ) continuous (if X connected topological monoid, A submonoid). (Denham–S. 2005) If ( M , ∂ M ) is a compact manifold of dim d , and K is a PL-triangulation of S m on n vertices, then Z K ( M , ∂ M ) is a compact manifold of dim ( d − 1 ) n + m + 1. (Bosio–Meersseman 2006) If K is a polytopal triangulation of S m , then Z K ( D 2 , S 1 ) if n + m + 1 is even, or Z K ( D 2 , S 1 ) × S 1 if n + m + 1 is odd, is a complex manifold. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 4 / 27
Toric complexes Toric complexes Toric complexes and right-angled Artin groups Definition Let L be simplicial complex on n vertices. The associated toric complex , T L , is the subcomplex of the n -torus obtained by deleting the cells corresponding to the missing simplices of L , i.e., T L = Z L ( S 1 , ∗ ) . k -cells in T L ← → ( k − 1 ) -simplices in L . C CW ( T L ) is a subcomplex of C CW ( T n ) ; thus, all ∂ k = 0, and ∗ ∗ H k ( T L , Z ) = C simplicial ( L , Z ) = Z # ( k − 1 ) -simplices of L . k − 1 H ∗ ( T L , k ) is the exterior Stanley-Reisner ring � V ∗ / J L , where ◮ V is the free k -module on the vertex set of L ; ◮ � k V ∗ is the exterior algebra on dual of V ; ◮ J L is the ideal generated by all monomials, v σ = v ∗ i 1 · · · v ∗ i k corresponding to simplices σ = { v i 1 , . . . , v i k } not belonging to L . Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 5 / 27
Toric complexes Toric complexes Right-angled Artin groups Definition Let Γ = ( V , E ) be a (finite, simple) graph. The corresponding right-angled Artin group is G Γ = � v ∈ V | vw = wv if { v , w } ∈ E � . Γ = K n ⇒ G Γ = Z n Γ = K n ⇒ G Γ = F n ; Γ = Γ ′ � Γ ′′ ⇒ G Γ = G Γ ′ ∗ G Γ ′′ ; Γ = Γ ′ ∗ Γ ′′ ⇒ G Γ = G Γ ′ × G Γ ′′ = Γ ′ ⇔ G Γ ∼ Γ ∼ = G Γ ′ (Kim–Makar-Limanov–Neggers–Roush 1980) π 1 ( T L ) = G Γ , where Γ = L ( 1 ) . K ( G Γ , 1 ) = T ∆ Γ , where ∆ Γ is the flag complex of Γ . (Davis–Charney 1995, Meier–VanWyk 1995) A := H ∗ ( G Γ , k ) = � k V ∗ / J Γ , where J Γ is quadratic monomial ideal ⇒ A is a Koszul algebra (Fröberg 1975). Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 6 / 27
Toric complexes Toric complexes Formality Definition (Sullivan) A space X is formal if its minimal model is quasi-isomorphic to ( H ∗ ( X , Q ) , 0 ) . Definition (Quillen) A group G is 1 -formal if its Malcev Lie algebra, m G = Prim ( � Q G ) , is a (complete, filtered) quadratic Lie algebra. Theorem (Sullivan) If X formal, then π 1 ( X ) is 1 -formal. Theorem (Notbohm–Ray 2005) T L is formal, and so G Γ is 1 -formal. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 7 / 27
Toric complexes Graded Lie algebras Associated graded Lie algebra Let G be a finitely-generated group. Define: LCS series : G = G 1 ⊲ G 2 ⊲ · · · ⊲ G k ⊲ · · · , where G k + 1 = [ G k , G ] LCS quotients : gr k G = G k / G k + 1 (f.g. abelian groups) LCS ranks : φ k ( G ) = rank ( gr k G ) Associated graded Lie algebra : gr ( G ) = � k ≥ 1 gr k ( G ) , with Lie bracket [ , ]: gr k × gr ℓ → gr k + ℓ induced by group commutator. Example (Witt, Magnus) Let G = F n (free group of rank n ). Then gr G = Lie n (free Lie algebra of rank n ), with LCS ranks given by ∞ � ( 1 − t k ) φ k = 1 − nt . k = 1 � Explicitly: φ k ( F n ) = 1 d | k µ ( d ) n k / d , where µ is Möbius function. k Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 8 / 27
Toric complexes Graded Lie algebras Holonomy Lie algebra Definition (Chen 1977, Markl–Papadima 1992) Let G be a finitely generated group, with H 1 = H 1 ( G , Z ) torsion-free. The holonomy Lie algebra of G is the quadratic, graded Lie algebra h G = Lie ( H 1 ) / ideal ( im ( ∇ )) , where ∇ : H 2 ( G , Z ) → H 1 ∧ H 1 = Lie 2 ( H 1 ) is the comultiplication map. Let G = π 1 ( X ) and A = H ∗ ( X , Q ) . = � (Löfwall 1986) U ( h G ⊗ Q ) ∼ k ≥ 1 Ext k A ( Q , Q ) k . There is a canonical epimorphism h G ։ gr ( G ) . ≃ (Sullivan) If G is 1-formal, then h G ⊗ Q − → gr ( G ) ⊗ Q . Example G = F n , then clearly h G = Lie n , and so h G = gr ( G ) . Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 9 / 27
Toric complexes Graded Lie algebras Let Γ = ( V , E ) graph, and P Γ ( t ) = � k ≥ 0 f k (Γ) t k its clique polynomial. Theorem (Duchamp–Krob 1992, Papadima–S. 2006) For G = G Γ : gr ( G ) ∼ = h G . 1 Graded pieces are torsion-free, with ranks given by 2 ∞ � ( 1 − t k ) φ k = P Γ ( − t ) . k = 1 Idea of proof: A = � k V ∗ / J Γ ⇒ h G ⊗ k = L Γ := Lie ( V ) / ([ v , w ] = 0 if { v , w } ∈ E ) . 1 Shelton–Yuzvinsky: U ( L Γ ) = A ! (Koszul dual). 2 Koszul duality: Hilb ( A ! , t ) · Hilb ( A , − t ) = 1. 3 Hilb ( h G ⊗ k , t ) independent of k ⇒ h G torsion-free. 4 But h G ։ gr ( G ) is iso over Q (by 1-formality) ⇒ iso over Z . 5 LCS formula follows from (3) and PBW. 6 Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 10 / 27
Toric complexes Chen Lie algebras Chen Lie algebras Definition (Chen 1951) The Chen Lie algebra of a (finitely generated) group G is gr ( G / G ′′ ) , i.e., the assoc. graded Lie algebra of its maximal metabelian quotient. Write θ k ( G ) = rank gr k ( G / G ′′ ) for the Chen ranks. Facts: gr ( G ) ։ gr ( G / G ′′ ) , and so φ k ( G ) ≥ θ k ( G ) , with equality for k ≤ 3. The map h G ։ gr ( G ) induces epimorphism h G / h ′′ G ։ gr ( G / G ′′ ) . ≃ .–S. 2004) If G is 1-formal, then h G / h ′′ → gr ( G / G ′′ ) ⊗ Q . (P G ⊗ Q − Example (Chen) � n + k − 2 � θ k ( F n ) = ( k − 1 ) , for all k ≥ 2 . k Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 11 / 27
Toric complexes Chen Lie algebras The Chen Lie algebra of a RAAG Theorem (P .–S. 2006) For G = G Γ : gr ( G / G ′′ ) ∼ = h G / h ′′ G . 1 Graded pieces are torsion-free, with ranks given by 2 � � ∞ � t θ k t k = Q Γ , 1 − t k = 2 where Q Γ ( t ) = � j ≥ 2 c j (Γ) t j is the “cut polynomial" of Γ , with � ˜ c j (Γ) = b 0 (Γ W ) . W ⊂ V : | W | = j Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 12 / 27
Toric complexes Chen Lie algebras Idea of proof: Write A := H ∗ ( G , k ) = E / J Γ , where E = � k ( v ∗ 1 , . . . , v ∗ n ) . 1 Write h = h G ⊗ k . 2 By Fröberg and Löfwall (2002) 3 � h ′ / h ′′ � k ∼ = Tor E k − 1 ( A , k ) k , for k ≥ 2 By Aramova–Herzog–Hibi & Aramova–Avramov–Herzog (97-99): 4 � � i + 1 � � t dim k Tor E dim k Tor S k − 1 ( E / J Γ , k ) k = i ( S / I Γ , k ) i + 1 · , 1 − t k ≥ 2 i ≥ 1 where S = k [ x 1 , . . . , x n ] and I Γ = ideal � x i x j | { v i , v j } / ∈ E � . By Hochster (1977): 5 � dim k � dim k Tor S i ( S / I Γ , k ) i + 1 = H 0 (Γ W , k ) = c i + 1 (Γ) . W ⊂ V : | W | = i + 1 The answer is independent of k ⇒ h G / h ′′ G is torsion-free. 6 ≃ Using formality of G Γ , together with h G / h ′′ → gr ( G / G ′′ ) ⊗ Q G ⊗ Q − 7 ends the proof. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October 2009 13 / 27
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