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THE RATIONAL COHOMOLOGY OF SMOOTH,

REAL TORIC VARIETIES

Alex Suciu

Northeastern University Joint work with Alvise Trevisan

Workshop on Arrangements and Configuration Spaces ALTA, University of Bremen

May 25, 2012

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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

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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 (quasi-) 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.

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TORIC MANIFOLDS AND SMALL COVERS

EXAMPLE (TORIC MANIFOLDS OVER THE n-SIMPLEX) 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

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TORIC MANIFOLDS AND SMALL COVERS

EXAMPLE (TORIC MANIFOLDS OVER THE SQUARE) ✌

1

• 1
• ✌

1

• ✌

1 ✌

✌

1

• 1
• ✌

1

• ✌

1 1 ✌

✌

1

• 1
• ✌

✁2 1

• ✌ 1

✁1 ✌

CP1 ✂ CP1 CP2# CP2 CP2# CP2 ✌

1

• 1
• ✌

1

• ✌

1 ✌

✌

1

• 1
• ✌

1

• ✌

1 1 ✌

S1 ✂ S1 RP2#RP2

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TORIC MANIFOLDS AND SMALL COVERS

If X is a smooth, projective toric variety, then X(C) = MP(χ), for some P and χ, and X(R) = NP(χ mod 2Z). On the other hand: 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. Hence, by a theorem of Delaunay, NP(χ) ✢ X(R). Concrete example: P = dodecahedron. (Characteristic matrices χ do exist for P, by work of Garrison and Scott.)

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TORIC MANIFOLDS AND SMALL COVERS

Davis and Januszkiewicz showed that: MP(χ) admits a perfect Morse function with only critical points of even index. Moreover, rank H2i(MP(χ), Z) = hi(P), where (h0(P), . . . , hn(P)) is the h-vector of P, which depends only

• n the number of i-faces of P (0 ↕ i ↕ n).

NP(χ) admits a perfect Morse function over Z2. Moreover, dimZ2 Hi(NP(χ), Z2) = hi(P). They also gave 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.

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TORIC MANIFOLDS AND SMALL COVERS

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 Recall there are precisely two small covers over the 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✽, ✝) .

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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 (or, polyhedral product) is the space 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.

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GENERALIZED MOMENT-ANGLE COMPLEXES

Usual moment-angle complexes: Complex moment-angle complex, ZK (D2, S1).

π1 = π2 = t1✉.

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

π1 = W ✶

K , the derived subgroup of WK , the right-angled Coxeter

group associated to K (1).

EXAMPLE Let K = two points. Then: ZK (D2, S1) = D2 ✂ S1 ❨ S1 ✂ D2 = S3 ZK (D1, S0) = D1 ✂ S0 ❨ S0 ✂ D1 = S1

D1 S0 D1 × S0 S0 × D1 ZK(D1, S0) S0 × S0

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GENERALIZED MOMENT-ANGLE COMPLEXES

EXAMPLE Let K be a circuit on 4 vertices. Then: ZK (D2, S1) = S3 ✂ S3 ZK (D1, S0) = S1 ✂ S1 EXAMPLE More generally, let K be an n-gon. Then: ZK (D2, S1) = #

n✁3 r=1 r ☎

n ✁ 2 r + 1

• Sr+2 ✂ Sn✁r

ZK (D1, S0) = an orientable surface of genus 1 + 2n✁3(n ✁ 4)

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GENERALIZED MOMENT-ANGLE COMPLEXES

The GMAC construction enjoys nice functoriality properties in both

• arguments. E.g:

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). Also, much is known about the fundamental group and the asphericity problem for ZK (X) = ZK (X, ✝) (work of Davis et al). E.g.: 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.

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GENERALIZED MOMENT-ANGLE COMPLEXES

Generalized Davis–Januszkiewicz spaces G abelian topological group G

• GDJ space 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.

In general, 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).

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GENERALIZED MOMENT-ANGLE COMPLEXES

THEOREM (BAHRI, BENDERSKY, COHEN, GITLER) 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).

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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

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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).

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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 .

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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). LEMMA Hq(ZK (BZp); kλ) ✕ r Hq✁1(Kλ; k).

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THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES

Sketch of proof: The inclusion (S1, ✝) ã Ñ (BZp, ✝) induces a cellular inclusion TK = ZK (S1) ã Ñ ZK (BZp). The inclusion φ: Kλ ã Ñ K induces a cellular inclusion 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) Hence: dimk Hq(ZK (BZp); kλ) ➙ dimk r Hq✁1(Kλ; k).

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THE HOMOLOGY OF ABELIAN COVERS OF GDJ SPACES

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 Let ZK (BZp)χ be the abelian cover defined by an epimorphism χ: (Zp)m ։ Γ. Then Hq(ZK (BZp)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).

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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).

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THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS

THEOREM The Betti numbers of the real toric manifold NP(χ) are given by bq(NP(χ)) = ➳

S❸[n]

˜ bq✁1(Kχ,S). As an application, we recover a result of Nakayama and Nishimura. COROLLARY A real, n-dimensional toric manifold NP(χ) is orientable if and only if there is a subset S ❸ [n] such that Kχ,S = K. Reason: NP(χ) is orientable iff bn(NP(χ)) = 1

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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

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THE HESSENBERG MANIFOLDS

THE HESSENBERG MANIFOLDS

Every Weyl group W determines a smooth, complex projective toric variety TW.

Fan given by the reflecting hyperplanes of W. Polytope PW is the convex hull of a regular orbit W ☎ x0. dimC TW = rank W.

Tn = TSn is the Hessenberg variety, of cx dim n ✁ 1; polytope is the permutahedron Pn (the iterated truncation of the simplex ∆n✁1). Tn is isomorphic to the De Concini–Procesi wonderful model YG, where G is the building set in (Cn)✝ which consists of all subspaces spanned by txi ⑤ i P I✉, where ❍ ✘ I ❸ [n]. Thus, Tn can be obtained by iterated blow-ups:

1

Blow up CPn✁1 at the n coordinate points.

2

Blow up along the proper transforms of the (n

2) coordinate lines.

3

Blow up along the proper transforms of the (n

3) coordinate planes...

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THE HESSENBERG MANIFOLDS

Remark: There is another De Concini–Procesi model, YH, isomorphic to the moduli space M0,n+2, and a surjective, Sn-equivariant birational morphism M0,n+2 ։ Tn. The real locus of TW, denoted TW (R), is a smooth, connected, compact real toric variety of dimension equal to the rank of W. 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),

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THE HESSENBERG MANIFOLDS

We may recover Henderson’s computation, using our general

• approach. To start with, note that:

Pn has 2n ✁ 2 facets: each subset ❍ ✘ Q ⑨ [n] determines a 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,

χQ = ➦

iPQ χi.

EXAMPLE P3 is a truncated triangle, that is, a hexagon. Characteristic matrix χ =

• 1

1 1 1 1 1 1 1

• .

T3(R) is obtained from RP2 by blowing up 3 points.

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THE HESSENBERG MANIFOLDS

EXAMPLE P4 is a truncated octahedron; it has 14 facets (6 squares and 8 hexagons). Characteristic matrix: 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)

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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: 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.

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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.

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CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES

PROPOSITION 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.

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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) COHOMOLOGY OF REAL TORIC MANIFOLDS BREMEN WORKSHOP 31 / 32

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) ZL(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) COHOMOLOGY OF REAL TORIC MANIFOLDS BREMEN WORKSHOP 32 / 32