Partial products of circles

Transcription

1 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 / 27

2 Outline 1 Toric complexes 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 2 Resonance varieties 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 / 27

3 Toric complexes Partial products of spaces Partial product construction Input: Output: K, a simplicial complex on [n] = {1,..., n}. (X, A), a pair of topological spaces, A. Z K (X, A) = (X, A) σ X n 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 Z n 1(X, ) = T n X (wedge) (fat wedge) Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

4 Properties: Toric complexes Partial products of spaces 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 / 27

5 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. (T L ) is a subcomplex of C CW (T n ); thus, all k = 0, and C CW H k (T L, Z) = C simplicial k 1 (L, Z) = Z # (k 1)-simplices of L. 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 ; JL is the ideal generated by all monomials, v σ = vi 1 vi k corresponding to simplices σ = {v i1,..., v ik } not belonging to L. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

6 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 = F n ; Γ = K n G Γ = Z n Γ = Γ Γ G Γ = G Γ G Γ ; Γ = Γ Γ G Γ = G Γ G Γ Γ = Γ G Γ = GΓ (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 / 27

7 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( QG), 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 / 27

8 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 Explicitly: φ k (F n ) = 1 k (1 t k ) φ k = 1 nt. k=1 d k µ(d)nk/d, where µ is Möbius function. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

9 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 Extk 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 / 27

10 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 : 1 gr(g) = h G. 2 Graded pieces are torsion-free, with ranks given by Idea of proof: (1 t k ) φ k = P Γ ( t). k=1 1 A = k V /J Γ h G k = L Γ := Lie(V)/([v, w] = 0 if {v, w} E). 2 Shelton Yuzvinsky: U(L Γ ) = A! (Koszul dual). 3 Koszul duality: Hilb(A!, t) Hilb(A, t) = 1. 4 Hilb(h G k, t) independent of k h G torsion-free. 5 But h G gr(g) is iso over Q (by 1-formality) iso over Z. 6 LCS formula follows from (3) and PBW. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

11 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 ). (P. S. 2004) If G is 1-formal, then h G /h G Q gr(g/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 / 27

12 Toric complexes Chen Lie algebras The Chen Lie algebra of a RAAG Theorem (P. S. 2006) For G = G Γ : 1 gr(g/g ) = h G /h G. 2 Graded pieces are torsion-free, with ranks given by ( 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) = W V: W =j b 0 (Γ W ). Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

13 Idea of proof: Toric complexes Chen Lie algebras 1 Write A := H (G, k) = E/J Γ, where E = k (v 1,..., v n ). 2 Write h = h G k. 3 By Fröberg and Löfwall (2002) ( h /h ) k = Tor E k 1 (A, k) k, for k 2 4 By Aramova Herzog Hibi & Aramova Avramov Herzog (97-99): dim k Tor E k 1 (E/J Γ, k) k = ( ) t i+1 dim k Tor S 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. 5 By Hochster (1977): dim k Tor S i (S/I Γ, k) i+1 = dim k H0 (Γ W, k) = c i+1 (Γ). W V: W =i+1 6 The answer is independent of k h G /h G is torsion-free. 7 Using formality of G Γ, together with h G /h G Q gr(g/g ) Q ends the proof. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

14 Toric complexes Chen Lie algebras Example Let Γ be a pentagon, and Γ a square with an edge attached to a vertex. Then: P Γ = P Γ = 1 5t + 5t 2, and so φ k (G Γ ) = φ k (G Γ ), for all k 1. Q Γ = 5t 2 + 5t 3 but Q Γ = 5t 2 + 5t 3 + t 4, and so θ k (G Γ ) θ k (G Γ ), for k 4. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

15 Toric complexes Bestvina-Brady groups Artin kernels Definition Given a graph Γ, and an epimorphism χ: G Γ Z, the corresponding Artin kernel is the group N χ = ker(χ: G Γ Z) Note that N χ = π 1 (T χ L ), where T χ L T L is the regular Z-cover defined by χ. A classifying space for N χ is T χ Γ, where Γ = L (1). Noteworthy is the case when χ is the diagonal" homomorphism ν : G Γ Z, which assigns to each vertex the value 1. The corresponding Artin kernel, N Γ = N ν, is called the Bestvina Brady group associated to Γ. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

16 Toric complexes Bestvina-Brady groups Stallings, Bieri, Bestvina and Brady: geometric and homological finiteness properties of N Γ topology of Γ, e.g.: N Γ is finitely generated Γ is connected N Γ is finitely presented Γ is simply-connected. More generally, it follows from Meier Meinert VanWyk (1998) and Bux Gonzalez (1999) that: Theorem Assume L is a flag complex. Let W = {v V χ(v) 0} be the support of χ. Then: 1 N χ is finitely generated L W is connected, and, v V \ W, there is a w W such that {v, w} L. 2 N χ is finitely presented L W is 1-connected and, σ L V\W, the space lk LW (σ) = {τ L W τ σ L} is (1 σ )-acyclic. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

17 Toric complexes Bestvina-Brady groups Theorem (P. S. 2009) Let Γ be a graph, and N χ and Artin kernel. 1 If H 1 (N χ, Q) is a trivial QZ-module, then N χ is finitely generated. 2 If both H 1 (N χ, Q) and H 2 (N χ, Q) have trivial Z-action, then N χ is 1-formal. Thus, if Γ is connected, and H 1 ( Γ, Q) = 0, then N Γ is 1-formal. Theorem (P. S. 2009) Suppose H 1 (N, Q) has trivial Z-action. Then, both gr(n) and gr(n/n ) are torsion-free, with graded ranks, φ k and θ k, given by (1 t k ) φ k = P Γ( t), 1 t k=1 ( t θ k t k = Q Γ 1 t k=2 Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27 ).

18 Resonance varieties Resonance varieties Resonance varieties Let X be a connected CW-complex with finite k-skeleton (k 1). Let k be a field; if char k = 2, assume H 1 (X, Z) has no 2-torsion. Let A = H (X, k). Then: a A 1 a 2 = 0. Thus, get cochain complex (A, a): A 0 a A 1 a A 2 Definition (Falk 1997, Matei S. 2000) The resonance varieties of X (over k) are the algebraic sets R i d (X, k) = {a A1 dim k H i (A, a) d}, defined for all integers 0 i k and d > 0. R i d are homogeneous subvarieties of A1 = H 1 (X, k) R i 1 Ri 2 Ri b i +1 =, where b i = b i (X, k). R 1 d (X, k) depends only on G = π 1(X), so denote it by R d (G, k). Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

19 Resonance varieties Resonance varieties Resonance of toric complexes Recall A = H (T L, k) is the exterior Stanley-Reisner ring of L. Using a formula of Aramova, Avramov, and Herzog (1999), we prove: Theorem (P. S. 2009) R i d (T L, k) = W V P σ L dim k Hi 1 σ e (lk LW (σ),k) d V\W k W, where L W is the subcomplex induced by L on W, and lk K (σ) is the link of a simplex σ in a subcomplex K L. In particular: R 1 1 (G Γ, k) = W V Γ W disconnected k W. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

20 Resonance varieties Resonance varieties Example Let Γ and Γ be the two graphs above. Both have P(t) = 1 + 6t + 9t 2 + 4t 3, and Q(t) = t 2 (6 + 8t + 3t 2 ). Thus, G Γ and G Γ have the same LCS and Chen ranks. Each resonance variety has 3 components, of codimension 2: R 1 (G Γ, k) = k 23 k 25 k 35, R 1 (G Γ, k) = k 15 k 25 k 26. Yet the two varieties are not isomorphic, since dim(k 23 k 25 k 35 ) = 3, but dim(k 15 k 25 k 26 ) = 2. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

21 Resonance varieties Kaehler manifolds Kähler manifolds Definition A compact, connected, complex manifold M is called a Kähler manifold if M admits a Hermitian metric h for which the imaginary part ω = I(h) is a closed 2-form. Examples: Riemann surfaces, CP n, and, more generally, smooth, complex projective varieties. Definition A group G is a Kähler group if G = π 1 (M), for some compact Kähler manifold M. G is projective if M is actually a smooth projective variety. G finite G is a projective group (Serre 1958). G 1, G 2 Kähler groups G 1 G 2 is a Kähler group G Kähler group, H < G finite-index subgroup H is a Kähler gp Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

22 Resonance varieties Kaehler manifolds Problem (Serre 1958) Which finitely presented groups are Kähler (or projective) groups? The Kähler condition puts strong restrictions on M: 1 H (M, Z) admits a Hodge structure 2 Hence, the odd Betti numbers of M are even 3 M is formal, i.e., (Ω(M), d) (H (M, R), 0) (Deligne Griffiths Morgan Sullivan 1975) The Kähler condition also puts strong restrictions on G = π 1 (M): 1 b 1 (G) is even 2 G is 1-formal, i.e., its Malcev Lie algebra m(g) is quadratic 3 G cannot split non-trivially as a free product (Gromov 1989) Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

23 Resonance varieties Kaehler manifolds Quasi-Kähler manifolds Definition A manifold X is called quasi-kähler if X = X \ D, where X is a compact Kähler manifold and D is a divisor with normal crossings. Similar definition for X quasi-projective. The notions of quasi-kähler group and quasi-projective group are defined as above. X quasi-projective H (X, Z) has a mixed Hodge structure (Deligne ) X = CP n \ {hyperplane arrangement} X is formal (Brieskorn 1973) X quasi-projective, W 1 (H 1 (X, C)) = 0 π 1 (X) is 1-formal (Morgan 1978) X = CP n \ {hypersurface} π 1 (X) is 1-formal (Kohno 1983) Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

24 Resonance varieties Kaehler manifolds Resonance varieties of quasi-kähler manifolds Theorem (D. P. S. 2009) Let X be a quasi-kähler manifold, and G = π 1 (X). Let {L α } α be the non-zero irred components of R 1 (G). If G is 1-formal, then 1 Each L α is a p-isotropic linear subspace of H 1 (G, C), with dim L α 2p + 2, for some p = p(α) {0, 1}. 2 If α β, then L α L β = {0}. 3 R d (G) = {0} α L α, where the union is over all α for which dim L α > d + p(α). Furthermore, 4 If X is compact Kähler, then G is 1-formal, and each L α is 1-isotropic. 5 If X is a smooth, quasi-projective variety, and W 1 (H 1 (X, C)) = 0, then G is 1-formal, and each L α is 0-isotropic. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

25 Resonance varieties Kaehler and quasi-kaehler RAAGs Theorem (Dimca Papadima S. 2009) The following are equivalent: 1 G Γ is a quasi-kähler group 2 Γ = K n1,...,n r := K n1 K nr 3 G Γ = F n1 F nr 1 G Γ is a Kähler group 2 Γ = K 2r 3 G Γ = Z 2r Example Let Γ be a linear path on 4 vertices. The maximal disconnected subgraphs are Γ {134} and Γ {124}. Thus: R 1 (G Γ, C) = C {134} C {124}. But C {134} C {124} = C {14}, which is a non-zero subspace. Thus, G Γ is not a quasi-kähler group. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

26 Resonance varieties Kaehler and quasi-kaehler BB groups Theorem (D. P. S. 2008) For a Bestvina Brady group N Γ = ker(ν : G Γ Z), the following are equivalent: 1 N Γ is a quasi-kähler group 2 Γ is either a tree, or Γ = K n1,...,n r, with some n i = 1, or all n i 2 and r 3. 1 N Γ is a Kähler group 2 Γ = K 2r+1 3 N Γ = Z 2r Example Γ = K 2,2,2 G Γ = F 2 F 2 F 2 N Γ = the Stallings group = π 1 (CP 2 \ {6 lines}) N Γ is finitely presented, but H 3 (N Γ, Z) has infinite rank, so N Γ not FP 3. Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27

27 Resonance varieties Kaehler and quasi-kaehler BB groups References G. Denham, A. Suciu, Moment-angle complexes, monomial ideals, and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, A. Dimca, S. Papadima, A. Suciu, Quasi-Kähler Bestvina-Brady groups, J. Algebraic Geom. 17 (2008), no. 1, , Topology and geometry of cohomology jump loci, Duke Math. Journal 148 (2009), no. 3, S. Papadima, A. Suciu, Algebraic invariants for right-angled Artin groups, Math. Annalen 334 (2006), no. 3, , Algebraic invariants for Bestvina-Brady groups, J. London Math. Soc. 76 (2007), no. 2, , Toric complexes and Artin kernels, Adv. Math. 220 (2009), no. 2, Alex Suciu (Northeastern University) Partial products of circles VU Amsterdam, October / 27