Alex Suciu Northeastern University PrincetonRider workshop - - PowerPoint PPT Presentation

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 2012 1 / 30

TORIC MANIFOLDS AND SMALL COVERS

TORIC MANIFOLDS AND SMALL COVERS

Let P be an n-dimensional convex polytope; facets F1, . . . , Fm. Assume P is simple (each vertex is the intersection of n facets). Then P determines a dual simplicial complex, K = K❇P, of dimension n ✁ 1:

Vertex set [m] = t1, . . . , m✉. Add a simplex σ = (i1, . . . , ik) whenever Fi1, . . . , Fik intersect. FIGURE: A prism P and its dual simplicial complex K

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 2 / 30

TORIC MANIFOLDS AND SMALL COVERS

Let χ be an n-by-m matrix with coefficients in G = Z or Z2. χ is characteristic for P if, for each vertex v = Fi1 ❳ ☎ ☎ ☎ ❳ Fin, the n-by-n minor given by the columns i1, . . . , in of χ is unimodular. Let T = S1 if G = Z, and T = S0 = t✟1✉ if G = Z2. Given q P P, let F(q) = Fj1 ❳ ☎ ☎ ☎ ❳ Fjk be the maximal face so that q P F(q)✆. The map χ yields a subtorus TF(q) ✕ Tk inside Tn. To the pair (P, χ), M. Davis and T. Januszkiewicz associate the toric manifold Tn ✂ P/ ✒, where (t, p) ✒ (u, q) if p = q and t ☎ u✁1 P TF(q). For G = Z, this is a complex toric manifold, denoted MP(χ): a closed, orientable manifold of dimension 2n. For G = Z2, this is a real toric manifold (or, small cover), denoted NP(χ): a closed, not necessarily orientable manifold of dim n.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 3 / 30

TORIC MANIFOLDS AND SMALL COVERS

EXAMPLE Let P = ∆n be the n-simplex, and χ the n ✂ (n + 1) matrix 1 ☎☎☎ 0 1 ... . . .

0 ☎☎☎ 1 1

• .

Then MP(χ) = CPn and NP(χ) = RPn. P T ✂ P T ✂ P/ ✒ CP1 RP1

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 4 / 30

TORIC MANIFOLDS AND SMALL COVERS

More generally, if X is a smooth, projective toric variety, then X(C) = MP(χ) and X(R) = NP(χ mod 2Z). But the converse does not hold:

M = CP2✼CP2 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, NP(χ) is a hyperbolic 3-manifold. (Characteristic χ exist for P = dodecahedron, by work of Garrison and Scott.) Then, by a theorem of Delaunay, NP(χ) ✢ X(R).

Davis and Januszkiewicz found presentations for the cohomology rings H✝(MP(χ), Z) and H✝(NP(χ), Z2), similar to the ones given by Danilov and Jurkiewicz for toric varieties. In particular, dimQ H2i(MP(χ), Q) = dimZ2 Hi(NP(χ), Z2) = hi(P), where (h0(P), . . . , hn(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 2012 5 / 30

TORIC MANIFOLDS AND SMALL COVERS

In joint work with Alvise Trevisan, we compute H✝(NP(χ), Q), both additively and multiplicatively. The (rational) Betti numbers of NP(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 = NP(χ), with χ = 1 0 1 0

0 1 0 1

• .

The Klein bottle Kℓ = NP(χ✶), with χ✶ = 1 0 1 0

0 1 1 1

• .

Then b1(T 2) = 2, yet b1(Kℓ) = 1. Idea: use finite covers involving (up to homotopy) certain generalized moment-angle complexes: Zm✁n

2

ZK (D1, S0) NP(χ) ,

Zn

2

NP(χ) ZK (RP✽, ✝) .

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 6 / 30

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 ZK (X, A) = ↕

σPK

(X, A)σ ⑨ X ✂m where (X, A)σ = tx P X ✂m ⑤ xi P A if i ❘ σ✉. Construction interpolates between A✂m and X ✂m. Homotopy invariance: (X, A) ✔ (X ✶, A✶) ù ñ ZK (X, A) ✔ ZK (X ✶, A✶). Converts simplicial joins to direct products: ZK✝L(X, A) ✕ ZK (X, A) ✂ ZL(X, A). Takes a cellular pair (X, A) to a cellular subcomplex of X ✂m. Particular case: ZK (X) := ZK (X, ✝).

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 7 / 30

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 ZK (f): ZK (X, A) Ñ ZK (Y, B). Let f : (X, ✝) ã Ñ (Y, ✝) be a cellular inclusion. Then, ZK (f)✝ : Cq(ZK (X)) ã Ñ Cq(ZK (Y)) admits a retraction, ❅q ➙ 0. Let φ: K ã Ñ L be the inclusion of a full subcomplex. Then there are induced maps Zφ : ZL(X, A) ։ ZK (X, A) and Zφ : ZK (X, A) ã Ñ ZL(X, A), such that Zφ ✆ Zφ = id. Fundamental group and asphericity (M. Davis) π1(ZK (X, ✝)) is the graph product of Gv = π1(X, ✝) along the graph Γ = K (1) = (V, E), where ProdΓ(Gv) = ✝

vPV Gv/t[gv, gw] = 1 if tv, w✉ P E, gv P Gv, gw P Gw✉.

Suppose X is aspherical. Then ZK (X) is aspherical iff K is a flag complex.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 8 / 30

GENERALIZED MOMENT-ANGLE COMPLEXES

Generalized Davis–Januszkiewicz spaces G abelian topological group G

• GDJ space ZK (BG).

We have a bundle Gm Ñ ZK (EG, G) Ñ ZK (BG). If G is a finitely generated (discrete) abelian group, then π1(ZK (BG))ab = Gm, and thus ZK (EG, G) is the universal abelian cover of ZK (BG). G = S1: Usual Davis–Januszkiewicz space, ZK (CP✽).

π1 = t1✉. H✝(ZK (CP✽), Z) = S/IK , where S = Z[x1, . . . , xm], deg xi = 2.

G = Z2: Real Davis–Januszkiewicz space, ZK (RP✽).

π1 = WK , the right-angled Coxeter group associated to K (1). H✝(ZK (RP✽), Z2) = R/IK , where R = Z2[x1, . . . , xm], deg xi = 1.

G = Z: Toric complex, ZK (S1).

π1 = GK , the right-angled Artin group associated to Γ = K (1). H✝(ZK (S1), Z) = E/JK , where E = ➍[e1, . . . , em], deg ei = 1.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 9 / 30

GENERALIZED MOMENT-ANGLE COMPLEXES

Standard moment-angle complexes Complex moment-angle complex, ZK (D2, S1) ✔ ZK (ES1, S1).

π1 = π2 = t1✉. H✝(ZK (D2, S1), Z) = TorS(S/IK , Z).

Real moment-angle complex, ZK (D1, S0) ✔ ZK (EZ2, Z2).

π1 = W ✶

K , the derived subgroup of WK .

H✝(ZK (D1, S0), Z2) = TorR(R/IK , Z2) — only additively!

EXAMPLE Let K be a circuit on 4 vertices. Then: ZK (D2, S1) = S3 ✂ S3 ZK (D1, S0) = S1 ✂ S1

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 10 / 30

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(ZK (X, A)) ✔ Σ ➟

I⑨[m]

♣ ZKI(X, A)

• ,

where KI is the induced subcomplex of K on the subset I ⑨ [m]. COROLLARY If X is contractible and A is a discrete subspace consisting of p points, then Hk(ZK (X, A); R) ✕ à

I⑨[m] (p✁1)⑤I⑤

à

1

r Hk✁1(KI; R).

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 11 / 30

FINITE ABELIAN COVERS

FINITE ABELIAN COVERS

Let X be a connected, finite-type CW-complex, π = π1(X, x0). Let p : Y Ñ X a (connected) regular cover, with group of deck transformations Γ. We then have a short exact sequence 1

π1(Y, y0)

p✼

π1(X, x0)

ν

Γ 1 .

Conversely, every epimorphism ν: π ։ Γ defines a regular cover X ν Ñ X (unique up to equivalence), with π1(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

p

• X

Ð Ñ π1(X)

ν

• ab π1(X)ab

χ

• Γ

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 12 / 30

FINITE ABELIAN COVERS

Let Cq(X ν; k) be the group of cellular q-chains on X ν, with coefficients in a field k. We then have natural isomorphisms Cq(X ν; k) ✕ Cq(X; kΓ) ✕ Cq(r X) ❜kπ kΓ. Now suppose Γ is finite abelian, k = k, and char k = 0. Then, all k-irreps of Γ are 1-dimensional, and so Cq(X ν; k) ✕ à

ρPHom(Γ,k✂)

Cq(X; kρ✆ν), where kρ✆ν denotes the field k, viewed as a kπ-module via the character ρ ✆ ν: π Ñ k✂. Thus, Hq(X ν; k) ✕ ➚

ρPHom(Γ,k✂) Hq(X; kρ✆ν).

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 13 / 30

FINITE ABELIAN COVERS

Now let P be an n-dimensional, simple polytope with m facets, and let K = K❇P be the simplicial complex dual to ❇P. Let χ: Zm

2 Ñ Zn 2 be a characteristic matrix for P.

Then ker(χ) ✕ Zm✁n

2

acts freely on ZK (D1, S0), with quotient the real toric manifold NP(χ). NP(χ) comes equipped with an action of Zm

2 / ker(χ) ✕ Zn 2; the

• rbit space is P.

Furthermore, ZK (D1, S0) is homotopy equivalent to the maximal abelian cover of ZK (RP✽), corresponding to the sequence 1

W ✶

K

WK

ab Zm 2

1 .

Thus, NP(χ) is, up to homotopy, a regular Zn

2-cover of ZK (RP✽),

corresponding to the sequence 1

π1(NP(χ)) WK

ν=χ✆ab Zn 2

1 .

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 14 / 30

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 π1(ZK (BZp))ab = Zm

p , with generators x1, . . . , xm.

Let λ: Zm

p Ñ k✂ be a character; supp(λ) := ti P [m] ⑤ λ(xi) ✘ 1✉.

Let Kλ be the induced subcomplex on vertex set supp(λ). PROPOSITION (A.S.–TREVISAN) Hq(ZK (BZp); kλ) ✕ r Hq✁1(Kλ; k). Idea: The inclusion ι: (S1, ✝) ã Ñ (BZp, ✝) induces a cellular inclusion ZK (ι): TK = ZK (S1) ã Ñ ZK (BZp). Moreover, φ: Kλ ã Ñ K induces a cellular inclusion Zφ : TKλ ã Ñ TK. Let ¯ λ: Zm ։ Zm

p λ

Ý Ñ k✂. We then get (chain) retractions Cq(TK; k¯

λ)

• Cq(ZK (BZp); kλ)
• Cq(TKλ; k¯

λ) ✕

r

Cq✁1(Kλ; k)

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 15 / 30

THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES

This shows that dimk Hq(ZK (BZp); kλ) ➙ dimk r Hq✁1(Kλ; k). For the reverse inequality, we use [BBCG], which, in this case, says Hq(ZK (EZp, Zp); k) ✕ à

I⑨[m] (p✁1)⑤I⑤

à

1

r Hq✁1(KI; k), and the fact that ZK (EZp, Zp) ✔ (ZK (BZp))ab, which gives Hq(ZK (EZp, Zp); k) ✕ à

ρPHom(Zm

p ,k✂)

Hq(ZK (BZp); kρ). THEOREM (A.S.–TREVISAN) Let G be a prime-order cyclic group, and let ZK (BG)χ be the abelian cover defined by an epimorphism χ: Gm ։ Γ. Then Hq(ZK (BG)χ; k) ✕ à

ρPHom(Γ;k✂)

r Hq✁1(Kρ✆χ; k), where Kρ✆χ is the induced subcomplex of K on vertex set supp(ρ ✆ χ).

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 16 / 30

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❇P. Let χ: Zm

2 Ñ Zn 2 be a characteristic matrix for P.

For each subset S of [n] = t1, . . . , n✉:

Compute χS = ➦

iPS χi, where χi is the i-th row of χ.

Find the induced subcomplex Kχ,S of K on vertex set supp(χS) = tj P [m] ⑤ the j-th entry of χS is non-zero✉. Compute the reduced simplicial Betti numbers ˜ bq(Kχ,S) = dimQ r Hq(Kχ,S; Q).

COROLLARY (A.S.–TREVISAN) The Betti numbers of the real toric manifold NP(χ) are given by bq(NP(χ)) = ➳

S❸[n]

˜ bq✁1(Kχ,S).

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 17 / 30

THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS

EXAMPLE Again, let P be the square, K = K❇P the 4-cycle. Let T 2 = NP(χ), χ = 1 0 1 0

0 1 0 1

• , and Kℓ = NP(χ✶), χ✶ =

1 0 1 0

0 1 1 1

• .

S ❍ t1✉ t2✉ t1, 2✉ χS ( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 0 1 ) ( 1 1 1 1 ) Kχ,S ❍ tt1✉, t3✉✉ tt2✉, t4✉✉ K χ✶

S

( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 1 1 ) ( 1 1 0 1 ) Kχ✶,S ❍ tt1✉, t3✉✉ tt2, 3✉, t3, 4✉✉ tt1, 2✉, t1, 4✉✉ Hence:

b0(T 2) = ˜ b✁1(❍) = 1 b0(Kℓ) = ˜ b✁1(❍) = 1 b1(T 2) = ˜ b0(Kχ,t1✉) + ˜ b0(Kχ,t2✉) = 2 b1(Kℓ) = ˜ b0(Kχ✶,t1✉) + ˜ b0(Kχ✶,t2✉) = 1 b2(T 2) = ˜ b1(Kχ,t1,2✉) = 1 b2(Kℓ) = ˜ b1(Kχ✶,t1,2✉) = 0

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 18 / 30

THE HESSENBERG MANIFOLDS

THE HESSENBERG MANIFOLDS

Every Weyl group W determines a smooth, complex projective toric variety TW, with fan given by the reflecting hyperplanes of W. Its real locus, TW (R), is a smooth, connected, compact real toric variety of dimension equal to the rank of W. In W = Sn, the manifold Tn = TSn is the Hessenberg variety. Tn(R) is a smooth, real toric variety of dim n ✁ 1, with associated polytope the permutahedron Pn. THEOREM (HENDERSON 2010) bi(Tn(R)) = A2i n 2i

• ,

where A2i is the Euler secant number, defined as the coefficient of x2i/(2i)! in the Maclaurin expansion of sec(x),

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 19 / 30

THE HESSENBERG MANIFOLDS

Pn has 2n ✁ 2 facets: each non-empty, proper subset Q ⑨ [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: Z2n✁2

2

Ñ Zn✁1

2

are given by: χi = i-th standard basis vector of Rn✁1 (1 ↕ i ➔ n); χn = ➦

i➔n χi; and χQ = ➦ iPQ χi. E.g.:

χ = 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• ALEX SUCIU (NORTHEASTERN)

POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 20 / 30

THE HESSENBERG MANIFOLDS

The dual simplicial complex, Kn = K❇Pn, is the barycentric subdivision of the boundary of the (n ✁ 1)-simplex. Given a subset S ❸ [n ✁ 1], the induced subcomplex on vertex set supp(χS) depends only on r := ⑤S⑤, so denote it by Kn,r. Kn,r is the order complex associated to a rank-selected poset of a certain subposet of the Boolean lattice Bn. Thus, Kn,r is Cohen–Macaulay; in fact, Kn,2r✁1 ✔ Kn,2r ✔

A2r

➟ Sr✁1. Hence, we recover Henderson’s computation: bi(Tn(R)) = ➳

S⑨[n✁1]

˜ bi✁1((Kn)χ,S) =

n✁1

➳

r=1

n ✁ 1 r

• ˜

bi✁1(Kn,r) = n ✁ 1 2i ✁ 1

• +

n ✁ 1 2i

• A2i =

n 2i

• A2i.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 21 / 30

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: π1(X) ։ Γ, and let k = C. The cellular cochains

• n X ν decompose as

Cq(X ν; k) ✕ à

ρPHom(Γ,k✂)

Cq(X; kρ✆ν), The cup product map, Cp(X ν, k) ❜k Cq(X ν, k)

✦

Ý Ý Ñ Cp+q(X ν, k), restricts to those pieces, as follows: Cp(X; kρ✆ν) ❜k Cq(X; kρ✶✆ν)

✦

• ✕
• Cp+q(X; k(ρ☎ρ✶)✆ν)

Cp+q(X ✂ X; kρ✆ν ❜k kρ✶✆ν)

µ✝

Cp+q(X ✂ X; k(ρ❜ρ✶)✆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 2012 22 / 30

CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES

PROPOSITION (A.S.–TREVISAN) Let ZK (BZp)ν be a regular abelian cover, with characteristic homomorphism χ: Zm

p Ñ Γ. The cup product in

H✝(ZK (BG)ν; k) ✕

✽

à

q=0

  à

ρPHom(Γ;k✂)

r Hq✁1(Kρ✆χ; k)   is induced by the following maps on simplicial cochains: r Cp✁1 Kρ✆χ; k✂ ❜ r Cq✁1 Kρ✶✆χ; k✂ Ñ r Cp+q✁1 K(ρ❜ρ✶)✆χ; k✂ ˆ σ ❜ ˆ τ ÞÑ ★ ✟④ σ ❭ τ if σ ❳ τ = ❍,

• therwise,

where σ ❭ τ is the simplex with vertex set the union of the vertex sets

• f σ and τ, and ˆ

σ is the Kronecker dual of σ.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 23 / 30

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 ZK (X) is formal. In particular, toric complexes TK = ZK (S1) and generalized Davis–Januszkiewicz spaces ZK (BG) are always formal. (Félix, Tanré) More generally, if both X and A are formal, and the inclusion i : A ã Ñ X induces a surjection i✝ : H✝(X, Q) Ñ H✝(A, Q), then ZK (X, A) is formal.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 24 / 30

FORMALITY PROPERTIES

(Baskakov, Denham–A.S.) Moment angle complexes ZK (D2, S1) 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❇P for which ZK (D2, S1) is not formal. Thus, there are real moment-angle complexes (even manifolds) ZK (D1, S0) which are not formal. (Panov–Ray) Complex toric manifolds MP(χ) are always formal. Question: are the real toric manifolds NP(χ) always formal?

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 25 / 30

ABELIAN DUALITY AND PROPAGATION OF RESONANCE

ABELIAN DUALITY & PROPAGATION OF RESONANCE

Let X be a connected, finite-type CW-complex, with G = π1(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✂) ⑤ Hi(X, Cρ) ✘ 0✉. Also, the closely related “resonance varieties", Ri(X) = ta P H1(X, C) ⑤ Hi(H✝(X, C), ☎a) ✘ 0✉. 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 Hp(X, ZG) = 0 for p ✘ n and Hn(X, ZG) ✘ 0 and torsion-free. By analogy, we say X is an abelian duality space of dimension n if Hp(X, ZGab) = 0 for p ✘ n and Hn(X, ZGab) ✘ 0 and torsion-free.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 26 / 30

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 Hp(X, Cρ) ✘ 0, then Hq(X, Cρ) ✘ 0 for all p ↕ q ↕ n. Thus, the characteristic varieties of X “propagate": V1(X) ❸ V2(X) ❸ ☎ ☎ ☎ ❸ Vn(X). COROLLARY If X admits a minimal cell structure, and X is an abelian duality space

• f dim n, then resonance propagates:

R1(X) ❸ R2(X) ❸ ☎ ☎ ☎ ❸ Rn(X). REMARK Propagation of Vi’s does not imply propagation of Ri’s. Eg, let M = HR/HZ be the 3-dim Heisenberg manifold. Then V1 = V2 = V3 = t1✉, but R1 = R2 = C2, and R3 = t0✉.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 27 / 30

TORIC COMPLEXES

TORIC COMPLEXES

Let K be a simplicial complex of dimension d, on vertex set V, and let TK = ZK (S1, ✝) be the respective toric complex. TK is a connected, minimal CW-complex, with dim TK = d + 1. π1(TK ) = GΓ := ①v P V(Γ) ⑤ vw = wv if tv, w✉ P E(Γ)②, the right-angled Artin group associated to the graph Γ = K (1). K(GΓ, 1) = T∆Γ, where ∆Γ is the flag complex of Γ. THEOREM (S. PAPADIMA–A.S. 2009) Vi(TK ) = ↕

W

(C✂)W and Ri(TK ) = ↕

W

CW where the union is taken over all W ❸ V for which there is a simplex σ P LV③W and an index j ↕ i such that r Hj✁1✁⑤σ⑤(lkLW(σ), C) ✘ 0.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 28 / 30

TORIC COMPLEXES

K is Cohen–Macaulay if for each simplex σ P K, the cohomology r H✝(lk(σ), Z) is concentrated in degree n ✁ ⑤σ⑤ and is torsion-free. THEOREM (BRADY–MEIER 2001, JENSEN–MEIER 2005) GΓ is a duality group if and only if ∆Γ is Cohen–Macaulay. Moreover, GΓ is a Poincaré duality group if and only if Γ is a complete graph. THEOREM (DSY) TK is an abelian duality space (of dimension d + 1) if and only if K is Cohen–Macaulay, in which case both Vi(TK ) and Ri(TK ) propagate. EXAMPLE Let Γ = . Then resonance does not propagate: R1(GΓ) = C4, but R2(GΓ) = C2 ✂ t0✉ ❨ t0✉ ✂ C2.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 29 / 30

REFERENCES

REFERENCES

• A. Suciu, A. Trevisan, Real toric varieties and abelian covers of

generalized Davis–Januszkiewicz spaces, preprint 2012.

• G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality and

propagation of resonance, preprint 2012. Further references:

• G. Denham, A. Suciu, Moment-angle complexes, monomial ideals,

and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, 25–60.

• S. Papadima, A. Suciu, Toric complexes and Artin kernels, Adv.
• Math. 220 (2009), no. 2, 441–477.

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 30 / 30