POLYHEDRAL PRODUCTS, TORIC MANIFOLDS, AND. Alex Suciu TWISTED COHOMOLOGY. Princeton Rider workshop Homotopy theory and toric spaces.

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1 POLYHEDRAL PRODUCTS, TORIC MANIFOLDS, AND TWISTED COHOMOLOGY Alex Suciu Northeastern University Princeton Rider workshop Homotopy theory and toric spaces February 23, 2012 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

2 TORIC MANIFOLDS AND SMALL COVERS TORIC MANIFOLDS AND SMALL COVERS Let P be an n-dimensional convex polytope; facets F 1,..., F m. Assume P is simple (each vertex is the intersection of n facets). Then P determines a dual simplicial complex, K = K BP, of dimension n 1: Vertex set [m] = t1,..., mu. Add a simplex σ = (i 1,..., i k ) whenever F i1,..., F ik intersect. FIGURE: A prism P and its dual simplicial complex K ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

3 TORIC MANIFOLDS AND SMALL COVERS Let χ be an n-by-m matrix with coefficients in G = Z or Z 2. χ is characteristic for P if, for each vertex v = F i1 X X F in, the n-by-n minor given by the columns i 1,..., i n of χ is unimodular. Let T = S 1 if G = Z, and T = S 0 = t 1u if G = Z 2. Given q P P, let F (q) = F j1 X X F jk be the maximal face so that q P F (q). The map χ yields a subtorus T F (q) T k inside T n. To the pair (P, χ), M. Davis and T. Januszkiewicz associate the toric manifold T n P/, where (t, p) (u, q) if p = q and t u 1 P T F (q). For G = Z, this is a complex toric manifold, denoted M P (χ): a closed, orientable manifold of dimension 2n. For G = Z 2, this is a real toric manifold (or, small cover), denoted N P (χ): a closed, not necessarily orientable manifold of dim n. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

4 TORIC MANIFOLDS AND SMALL COVERS EXAMPLE Let P = n be the n-simplex, and χ the n (n + 1) matrix Then M P (χ) = CP n and N P (χ) = RP n. ( ) P T P T P/ CP 1 RP 1 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

5 TORIC MANIFOLDS AND SMALL COVERS More generally, if X is a smooth, projective toric variety, then X (C) = M P (χ) and X (R) = N P (χ mod 2Z). But the converse does not hold: M = CP 2 7CP 2 is a toric manifold over the square, but it does not admit any (almost) complex structure. Thus, M X (C). If P is a 3-dim polytope with no triangular or quadrangular faces, then, by a theorem of Andreev, N P (χ) is a hyperbolic 3-manifold. (Characteristic χ exist for P = dodecahedron, by work of Garrison and Scott.) Then, by a theorem of Delaunay, N P (χ) X (R). Davis and Januszkiewicz found presentations for the cohomology rings H (M P (χ), Z) and H (N P (χ), Z 2 ), similar to the ones given by Danilov and Jurkiewicz for toric varieties. In particular, dim Q H 2i (M P (χ), Q) = dim Z2 H i (N P (χ), Z 2 ) = h i (P), where (h 0 (P),..., h n (P)) is the h-vector of P, depending only on the number of i-faces of P (0 i n). ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

6 TORIC MANIFOLDS AND SMALL COVERS In joint work with Alvise Trevisan, we compute H (N P (χ), Q), both additively and multiplicatively. The (rational) Betti numbers of N P (χ) no longer depend just on the h-vector of P, but also on the characteristic matrix χ. EXAMPLE There are precisely two small covers over a square P: The torus T 2 = N P (χ), with χ = ( ). The Klein bottle Kl = N P (χ 1 ), with χ 1 = ( ). Then b 1 (T 2 ) = 2, yet b 1 (Kl) = 1. Idea: use finite covers involving (up to homotopy) certain generalized moment-angle complexes: Z m n 2 Z n 2 Z K (D 1, S 0 ) N P (χ), N P (χ) Z K (RP 8, ). ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

7 GENERALIZED MOMENT-ANGLE COMPLEXES GENERALIZED MOMENT-ANGLE COMPLEXES Let (X, A) be a pair of topological spaces, and K a simplicial complex on vertex set [m]. The corresponding generalized moment-angle complex is Z K (X, A) = σpk(x, A) σ X m where (X, A) σ = tx P X m x i P A if i R σu. Construction interpolates between A m and X m. Homotopy invariance: (X, A) (X 1, A 1 ) ùñ Z K (X, A) Z K (X 1, A 1 ). Converts simplicial joins to direct products: Z K L (X, A) Z K (X, A) Z L (X, A). Takes a cellular pair (X, A) to a cellular subcomplex of X m. Particular case: Z K (X ) := Z K (X, ). ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

8 GENERALIZED MOMENT-ANGLE COMPLEXES Functoriality properties Let f : (X, A) Ñ (Y, B) be a (cellular) map. Then f n : X n Ñ Y n restricts to a (cellular) map Z K (f ) : Z K (X, A) Ñ Z K (Y, B). Let f : (X, ) ãñ (Y, ) be a cellular inclusion. Then, Z K (f ) : C q (Z K (X )) ãñ C q (Z K (Y )) admits a 0. Let φ : K ãñ L be the inclusion of a full subcomplex. Then there are induced maps Z φ : Z L (X, A) Z K (X, A) and Z φ : Z K (X, A) ãñ Z L (X, A), such that Z φ Z φ = id. Fundamental group and asphericity (M. Davis) π 1 (Z K (X, )) is the graph product of G v = π 1 (X, ) along the graph Γ = K (1) = (V, E), where Prod Γ (G v ) = G v /t[g v, g w ] = 1 if tv, wu P E, g v P G v, g w P G w u. vpv Suppose X is aspherical. Then Z K (X ) is aspherical iff K is a flag complex. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

9 GENERALIZED MOMENT-ANGLE COMPLEXES Generalized Davis Januszkiewicz spaces G abelian topological group G GDJ space Z K (BG). We have a bundle G m Ñ Z K (EG, G) Ñ Z K (BG). If G is a finitely generated (discrete) abelian group, then π 1 (Z K (BG)) ab = G m, and thus Z K (EG, G) is the universal abelian cover of Z K (BG). G = S 1 : Usual Davis Januszkiewicz space, Z K (CP 8 ). π 1 = t1u. H (Z K (CP 8 ), Z) = S/I K, where S = Z[x 1,..., x m ], deg x i = 2. G = Z 2 : Real Davis Januszkiewicz space, Z K (RP 8 ). π 1 = W K, the right-angled Coxeter group associated to K (1). H (Z K (RP 8 ), Z 2 ) = R/I K, where R = Z 2 [x 1,..., x m ], deg x i = 1. G = Z: Toric complex, Z K (S 1 ). π 1 = G K, the right-angled Artin group associated to Γ = K (1). H (Z K (S 1 ), Z) = E/J K, where E = [e 1,..., e m ], deg e i = 1. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

10 GENERALIZED MOMENT-ANGLE COMPLEXES Standard moment-angle complexes Complex moment-angle complex, Z K (D 2, S 1 ) Z K (ES 1, S 1 ). π 1 = π 2 = t1u. H (Z K (D 2, S 1 ), Z) = Tor S (S/I K, Z). Real moment-angle complex, Z K (D 1, S 0 ) Z K (EZ 2, Z 2 ). π 1 = W 1 K, the derived subgroup of W K. H (Z K (D 1, S 0 ), Z 2 ) = Tor R (R/I K, Z 2 ) only additively! EXAMPLE Let K be a circuit on 4 vertices. Then: Z K (D 2, S 1 ) = S 3 S 3 Z K (D 1, S 0 ) = S 1 S 1 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

11 GENERALIZED MOMENT-ANGLE COMPLEXES THEOREM (BAHRI, BENDERSKY, COHEN, GITLER 2010) Let K a simplicial complex on m vertices. There is a natural homotopy equivalence ( Σ(Z K (X, A)) Σ ª ) pz KI (X, A), I [m] where K I is the induced subcomplex of K on the subset I [m]. COROLLARY If X is contractible and A is a discrete subspace consisting of p points, then à (p 1) H k (Z K à I (X, A); R) rh k 1 (K I ; R). I [m] 1 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

12 FINITE ABELIAN COVERS FINITE ABELIAN COVERS Let X be a connected, finite-type CW-complex, π = π 1 (X, x 0 ). Let p : Y Ñ X a (connected) regular cover, with group of deck transformations Γ. We then have a short exact sequence 1 π 1 (Y, y 0 ) p 7 π 1 (X, x 0 ) ν Γ 1. Conversely, every epimorphism ν : π Γ defines a regular cover X ν Ñ X (unique up to equivalence), with π 1 (X ν ) = ker(ν). If Γ is abelian, then ν = χ ab factors through the abelianization, while X ν = X χ is covered by the universal abelian cover of X: X ab X ν ÐÑ π 1 (X ) ab π 1 (X ) ab p X ν χ Γ ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

13 FINITE ABELIAN COVERS Let C q (X ν ; k) be the group of cellular q-chains on X ν, with coefficients in a field k. We then have natural isomorphisms C q (X ν ; k) C q (X; kγ) C q (r X ) bkπ kγ. Now suppose Γ is finite abelian, k = k, and char k = 0. Then, all k-irreps of Γ are 1-dimensional, and so C q (X ν ; k) à ρphom(γ,k ) C q (X; k ρ ν ), where k ρ ν denotes the field k, viewed as a kπ-module via the character ρ ν : π Ñ k. Thus, H q (X ν ; k) À ρphom(γ,k ) H q(x; k ρ ν ). ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

14 FINITE ABELIAN COVERS Now let P be an n-dimensional, simple polytope with m facets, and let K = K BP be the simplicial complex dual to BP. Let χ : Z m 2 Ñ Zn 2 be a characteristic matrix for P. Then ker(χ) Z m n 2 acts freely on Z K (D 1, S 0 ), with quotient the real toric manifold N P (χ). N P (χ) comes equipped with an action of Z m 2 / ker(χ) Zn 2 ; the orbit space is P. Furthermore, Z K (D 1, S 0 ) is homotopy equivalent to the maximal abelian cover of Z K (RP 8 ), corresponding to the sequence 1 W 1 K W K ab Z m 2 1. Thus, N P (χ) is, up to homotopy, a regular Z n 2 -cover of Z K (RP 8 ), corresponding to the sequence 1 π 1 (N P (χ)) W K ν=χ ab Z n 2 1. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

15 THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES Let K be a simplicial complex on m vertices. Identify π 1 (Z K (BZ p )) ab = Z m p, with generators x 1,..., x m. Let λ : Z m p Ñ k be a character; supp(λ) := ti P [m] λ(x i ) 1u. Let K λ be the induced subcomplex on vertex set supp(λ). PROPOSITION (A.S. TREVISAN) H q (Z K (BZ p ); k λ ) r Hq 1 (K λ ; k). Idea: The inclusion ι : (S 1, ) ãñ (BZ p, ) induces a cellular inclusion Z K (ι) : T K = Z K (S 1 ) ãñ Z K (BZ p ). Moreover, φ : K λ ãñ K induces a cellular inclusion Z φ : T Kλ ãñ T K. Let λ : Z m Z m λ p ÝÑ k. We then get (chain) retractions C q (T K ; k λ ) C q (Z K (BZ p ); k λ) C q (T Kλ ; k λ ) r Cq 1 (K λ ; k) ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

16 THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES This shows that dim k H q (Z K (BZ p ); k λ ) dim k Hq 1 r (K λ ; k). For the reverse inequality, we use [BBCG], à which, in this case, says (p 1) H q (Z K (EZ p, Z p à I ); k) rh q 1 (K I ; k), I [m] and the fact that Z K (EZ p, Z p ) (Z K (BZ p )) ab, which gives H q (Z K (EZ p, Z p ); k) à 1 ρphom(z m p,k ) H q (Z K (BZ p ); k ρ ). THEOREM (A.S. TREVISAN) Let G be a prime-order cyclic group, and let Z K (BG) χ be the abelian cover defined by an epimorphism χ : G m Γ. Then H q (Z K (BG) χ ; k) à ρphom(γ;k ) rh q 1 (K ρ χ ; k), where K ρ χ is the induced subcomplex of K on vertex set supp(ρ χ). ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

17 THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS THE Q-HOMOLOGY OF REAL TORIC MANIFOLDS Let again P be a simple polytope, and set K = K BP. Let χ : Z m 2 Ñ Zn 2 be a characteristic matrix for P. For each subset S of [n] = t1,..., nu: Compute χ S = ips χ i, where χ i is the i-th row of χ. Find the induced subcomplex K χ,s of K on vertex set supp(χ S ) = tj P [m] the j-th entry of χ S is non-zerou. Compute the reduced simplicial Betti numbers b q (K χ,s ) = dim Q r Hq (K χ,s ; Q). COROLLARY (A.S. TREVISAN) The Betti numbers of the real toric manifold N P (χ) are given by b q (N P (χ)) = b q 1 (K χ,s ). S [n] ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

18 THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS EXAMPLE Hence: Again, let P be the square, K = K BP the 4-cycle. Let T 2 = N P (χ), χ = ( ) , and Kl = NP (χ 1 ), χ 1 = ( ) S H t1u t2u t1, 2u χ S ( ) ( ) ( ) ( ) K χ,s H tt1u, t3uu tt2u, t4uu K χ 1 S ( ) ( ) ( ) ( ) K χ1,s H tt1u, t3uu tt2, 3u, t3, 4uu tt1, 2u, t1, 4uu b 0 (T 2 ) = b 1 (H) = 1 b 0 (Kl) = b 1 (H) = 1 b 1 (T 2 ) = b 0 (K χ,t1u ) + b 0 (K χ,t2u ) = 2 b 1 (Kl) = b 0 (K χ1,t1u) + b 0 (K χ1,t2u) = 1 b 2 (T 2 ) = b 1 (K χ,t1,2u ) = 1 b 2 (Kl) = b 1 (K χ1,t1,2u) = 0 ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

19 THE HESSENBERG MANIFOLDS THE HESSENBERG MANIFOLDS Every Weyl group W determines a smooth, complex projective toric variety T W, with fan given by the reflecting hyperplanes of W. Its real locus, T W (R), is a smooth, connected, compact real toric variety of dimension equal to the rank of W. In W = S n, the manifold T n = T Sn is the Hessenberg variety. T n (R) is a smooth, real toric variety of dim n 1, with associated polytope the permutahedron P n. THEOREM (HENDERSON 2010) b i (T n (R)) = A 2i ( n 2i where A 2i is the Euler secant number, defined as the coefficient of x 2i /(2i)! in the Maclaurin expansion of sec(x), ), ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

20 THE HESSENBERG MANIFOLDS P n has 2 n 2 facets: each non-empty, proper subset Q [n] facet F Q with vertices in which all coordinates in positions in Q are smaller than all coordinates in positions not in Q. The corresponding column vectors of the characteristic matrix χ : Z 2n 2 2 Ñ Z n 1 2 are given by: χ i = i-th standard basis vector of R n 1 (1 i n); χ n = i n χi ; and χ Q = ipq χi. E.g.: ( ) 1 χ = ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

21 THE HESSENBERG MANIFOLDS The dual simplicial complex, K n = K BPn, is the barycentric subdivision of the boundary of the (n 1)-simplex. Given a subset S [n 1], the induced subcomplex on vertex set supp(χ S ) depends only on r := S, so denote it by K n,r. K n,r is the order complex associated to a rank-selected poset of a certain subposet of the Boolean lattice B n. Thus, K n,r is Cohen Macaulay; in fact, K n,2r 1 K n,2r ª A 2r S r 1. Hence, we recover Henderson s computation: n 1 b i (T n (R)) = b i 1 ((K n ) χ,s ( ) n 1 ) = b i 1 (K n,r ) r S [n 1] r=1 (( ) ( )) ( ) n 1 n 1 n = + A 2i = A 2i. 2i 1 2i 2i ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

22 CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES As before, let X ν Ñ X be a regular, finite abelian cover, corresponding to an epimorphism ν : π 1 (X ) Γ, and let k = C. The cellular cochains on X ν decompose as C q (X ν ; k) à ρphom(γ,k ) C q (X; k ρ ν ), The cup product map, C p (X ν, k) b k C q (X ν, k)! ÝÝÑ C p+q (X ν, k), restricts to those pieces, as follows: C p (X; k ρ ν ) b k C q (X; k ρ 1 ν) C p+q (X X; k ρ ν b k k ρ 1 ν)! C p+q (X; k (ρ ρ1 ) ν) µ C p+q (X X; k (ρbρ1 ) ν) where µ is induced by the multiplication map on coefficients, and is induced by a cellular approximation to the diagonal : X Ñ X X. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

23 CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES PROPOSITION (A.S. TREVISAN) Let Z K (BZ p ) ν be a regular abelian cover, with characteristic homomorphism χ : Z m p Ñ Γ. The cup product in H (Z K (BG) ν ; k) 8à q=0 à ρphom(γ;k ) rh q 1 (K ρ χ ; k) is induced by the following maps on simplicial cochains: rc p 1( K ρ χ ; k ) b r C q 1 ( K ρ 1 χ; k ) Ñ r C p+q 1 ( K (ρbρ1 ) χ; k ) ˆσ b ˆτ ÞÑ # {σ \ τ if σ X τ = H, 0 otherwise, where σ \ τ is the simplex with vertex set the union of the vertex sets of σ and τ, and ˆσ is the Kronecker dual of σ. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

24 FORMALITY PROPERTIES FORMALITY PROPERTIES A finite-type CW-complex X is formal if its Sullivan minimal model is quasi-isomorphic to (H (X, Q), 0) roughly speaking, H (X, Q) determines the rational homotopy type of X. (Notbohm Ray) If X is formal, then Z K (X ) is formal. In particular, toric complexes T K = Z K (S 1 ) and generalized Davis Januszkiewicz spaces Z K (BG) are always formal. (Félix, Tanré) More generally, if both X and A are formal, and the inclusion i : A ãñ X induces a surjection i : H (X, Q) Ñ H (A, Q), then Z K (X, A) is formal. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

25 FORMALITY PROPERTIES (Baskakov, Denham A.S.) Moment angle complexes Z K (D 2, S 1 ) are not always formal: they can have non-trivial triple Massey products. For instance, K = (Denham A.S.) There exist polytopes P and dual triangulations K = K BP for which Z K (D 2, S 1 ) is not formal. Thus, there are real moment-angle complexes (even manifolds) Z K (D 1, S 0 ) which are not formal. (Panov Ray) Complex toric manifolds M P (χ) are always formal. Question: are the real toric manifolds N P (χ) always formal? ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

26 ABELIAN DUALITY AND PROPAGATION OF RESONANCE ABELIAN DUALITY & PROPAGATION OF RESONANCE Let X be a connected, finite-type CW-complex, with G = π 1 (X ). In the background for much of these computations lie the jump loci for cohomology with coefficients in rank 1 local systems, V i (X ) = tρ P Hom(G, C ) H i (X, C ρ ) 0u. Also, the closely related resonance varieties", R i (X ) = ta P H 1 (X, C) H i (H (X, C), a) 0u. Question: How do the duality properties of a space X affect the nature of its cohomology jump loci? Recall that X is a duality space of dimension n if H p (X, ZG) = 0 for p n and H n (X, ZG) 0 and torsion-free. By analogy, we say X is an abelian duality space of dimension n if H p (X, ZG ab ) = 0 for p n and H n (X, ZG ab ) 0 and torsion-free. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

27 ABELIAN DUALITY AND PROPAGATION OF RESONANCE THEOREM (GRAHAM DENHAM, A. S., SERGEY YUZVINSKY) Let X be an abelian duality space of dim n. For any character ρ : G Ñ C, if H p (X, C ρ ) 0, then H q (X, C ρ ) 0 for all p q n. Thus, the characteristic varieties of X propagate": V 1 (X ) V 2 (X ) V n (X ). COROLLARY If X admits a minimal cell structure, and X is an abelian duality space of dim n, then resonance propagates: R 1 (X ) R 2 (X ) R n (X ). REMARK Propagation of V i s does not imply propagation of R i s. Eg, let M = H R /H Z be the 3-dim Heisenberg manifold. Then V 1 = V 2 = V 3 = t1u, but R 1 = R 2 = C 2, and R 3 = t0u. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

28 TORIC COMPLEXES TORIC COMPLEXES Let K be a simplicial complex of dimension d, on vertex set V, and let T K = Z K (S 1, ) be the respective toric complex. T K is a connected, minimal CW-complex, with dim T K = d + 1. π 1 (T K ) = G Γ := xv P V(Γ) vw = wv if tv, wu P E(Γ)y, the right-angled Artin group associated to the graph Γ = K (1). K (G Γ, 1) = T Γ, where Γ is the flag complex of Γ. THEOREM (S. PAPADIMA A.S. 2009) V i (T K ) = W (C ) W and R i (T K ) = W C W where the union is taken over all W V for which there is a simplex σ P L VzW and an index j i such that r Hj 1 σ (lk LW (σ), C) 0. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

29 TORIC COMPLEXES K is Cohen Macaulay if for each simplex σ P K, the cohomology rh (lk(σ), Z) is concentrated in degree n σ and is torsion-free. THEOREM (BRADY MEIER 2001, JENSEN MEIER 2005) G Γ is a duality group if and only if Γ is Cohen Macaulay. Moreover, G Γ is a Poincaré duality group if and only if Γ is a complete graph. THEOREM (DSY) T K is an abelian duality space (of dimension d + 1) if and only if K is Cohen Macaulay, in which case both V i (T K ) and R i (T K ) propagate. EXAMPLE Let Γ =. Then resonance does not propagate: R 1 (G Γ ) = C 4, but R 2 (G Γ ) = C 2 t0u Y t0u C 2. ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30

30 REFERENCES REFERENCES A. Suciu, A. Trevisan, Real toric varieties and abelian covers of generalized Davis Januszkiewicz spaces, preprint G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality and propagation of resonance, preprint Further references: G. Denham, A. Suciu, Moment-angle complexes, monomial ideals, and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, S. Papadima, A. Suciu, Toric complexes and Artin kernels, Adv. Math. 220 (2009), no. 2, ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON RIDER / 30