Polyhedral products, toric manifolds, and twisted cohomology

POLYHEDRAL PRODUCTS, TORIC MANIFOLDS, AND

TWISTED COHOMOLOGY

Alex Suciu

Northeastern University

Princeton–Rider workshopHomotopy theory and toric spaces

February 23, 2012

ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 1 / 30

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 = KBP , ofdimension n� 1:

Vertex set [m] = t1, . . . ,mu.Add a simplex σ = (i1, . . . , ik ) whenever Fi1 , . . . ,Fik intersect.

FIGURE: A prism P and its dual simplicial complex K



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 X � � � X Fin , then-by-n minor given by the columns i1, . . . , in of χ is unimodular.

Let T = S1 if G = Z, and T = S0 = t�1u if G = Z2.

Given q P P, let F (q) = Fj1 X � � � X Fjk be the maximal face so thatq 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 thetoric 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): aclosed, orientable manifold of dimension 2n.

For G = Z2, this is a real toric manifold (or, small cover), denotedNP(χ): a closed, not necessarily orientable manifold of dim n.



EXAMPLE

Let P = ∆n be the n-simplex, and χ the n� (n + 1) matrix

( 1 �� 0 1...

...0 �� 1 1

).

ThenMP(χ) = CPn and NP(χ) = RPn.

P T�P T�P/ �

CP1

RP1



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

M = CP27CP2 is a toric manifold over the square, but it does notadmit 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 Garrisonand Scott.) Then, by a theorem of Delaunay, NP(χ) � X (R).

Davis and Januszkiewicz found presentations for the cohomologyrings H�(MP(χ),Z) and H�(NP(χ),Z2), similar to the ones givenby 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 onthe number of i-faces of P (0 ¤ i ¤ n).



In joint work with Alvise Trevisan, we compute H�(NP(χ),Q), bothadditively and multiplicatively.

The (rational) Betti numbers of NP(χ) no longer depend just onthe 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 00 1 0 1

).

The Klein bottle K` = NP(χ1), with χ1 =

(1 0 1 00 1 1 1

).

Then b1(T 2) = 2, yet b1(K`) = 1.

Idea: use finite covers involving (up to homotopy) certaingeneralized moment-angle complexes:

Zm�n2

// ZK (D1,S0) // NP(χ) ,

Zn2

// NP(χ) // ZK (RP8, �) .


GENERALIZED MOMENT-ANGLE COMPLEXES


Let (X ,A) be a pair of topological spaces, and K a simplicialcomplex on vertex set [m].The corresponding generalized moment-angle complex is

ZK (X ,A) =¤σPK

(X ,A)σ � X�m

where (X ,A)σ = tx P X�m | xi P A if i R σu.Construction interpolates between A�m and X�m.Homotopy invariance:(X ,A) � (X 1,A1) ùñ ZK (X ,A) � ZK (X 1,A1).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 , �).



Functoriality propertiesLet 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 , �) ãÑ (Y , �) be a cellular inclusion. Then,ZK (f )� : Cq(ZK (X )) ãÑ Cq(ZK (Y )) admits a retraction, @q ¥ 0.

Let φ : K ãÑ L be the inclusion of a full subcomplex. Then thereare induced maps Zφ : ZL(X ,A)� ZK (X ,A) andZφ : ZK (X ,A) ãÑ 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 thegraph Γ = K (1) = (V,E), where

ProdΓ(Gv ) = �vPV

Gv /t[gv ,gw ] = 1 if tv ,wu P E, gv P Gv , gw P Gwu.

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



Generalized Davis–Januszkiewicz spacesG 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, thenπ1(ZK (BG))ab = Gm, and thus ZK (EG,G) is the universalabelian cover of ZK (BG).

G = S1: Usual Davis–Januszkiewicz space, ZK (CP8).π1 = t1u.H�(ZK (CP8),Z) = S/IK , where S = Z[x1, . . . , xm], deg xi = 2.

G = Z2: Real Davis–Januszkiewicz space, ZK (RP8).π1 = WK , the right-angled Coxeter group associated to K (1).H�(ZK (RP8),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 =

�[e1, . . . ,em], deg ei = 1.



Standard moment-angle complexesComplex moment-angle complex, ZK (D2,S1) � ZK (ES1,S1).

π1 = π2 = t1u.H�(ZK (D2,S1),Z) = TorS(S/IK ,Z).

Real moment-angle complex, ZK (D1,S0) � ZK (EZ2,Z2).π1 = W 1

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



THEOREM (BAHRI, BENDERSKY, COHEN, GITLER 2010)

Let K a simplicial complex on m vertices. There is a natural homotopyequivalence

Σ(ZK (X ,A)) � Σ( ª

I�[m]

pZKI (X ,A)),

where KI 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

Hk (ZK (X ,A);R) �à

I�[m]

(p�1)|I|à1

rHk�1(KI ;R).


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 decktransformations Γ. We then have a short exact sequence

1 // π1(Y , y0)p7 // π1(X , x0)

ν // Γ // 1 .

Conversely, every epimorphism ν : π � Γ defines a regular coverX 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

ÐÑ π1(X )

ν

%%

ab // π1(X )ab

χ

��Γ



Let Cq(X ν; k) be the group of cellular q-chains on X ν, withcoefficients in a field k. We then have natural isomorphisms

Cq(X ν;k) � Cq(X ;kΓ) � Cq(rX )bkπ kΓ.

Now suppose Γ is finite abelian, k = k, and chark = 0. Then, allk-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 thecharacter ρ � ν : π Ñ k�.

Thus, Hq(X ν; k) �À

ρPHom(Γ,k�) Hq(X ;kρ�ν).



Now let P be an n-dimensional, simple polytope with m facets,and let K = KBP be the simplicial complex dual to BP.

Let χ : Zm2 Ñ Zn

2 be a characteristic matrix for P.

Then ker(χ) � Zm�n2 acts freely on ZK (D1,S0), with quotient the

real toric manifold NP(χ).

NP(χ) comes equipped with an action of Zm2 / ker(χ) � Zn

2; theorbit space is P.

Furthermore, ZK (D1,S0) is homotopy equivalent to the maximalabelian cover of ZK (RP8), corresponding to the sequence

1 //W 1K

//WKab // Zm

2// 1 .

Thus, NP(χ) is, up to homotopy, a regular Zn2-cover of ZK (RP8),

corresponding to the sequence

1 // π1(NP(χ)) //WKν=χ�ab // Zn

2// 1 .


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] | λ(xi) � 1u.Let Kλ be the induced subcomplex on vertex set supp(λ).

PROPOSITION (A.S.–TREVISAN)

Hq(ZK (BZp);kλ) � rHq�1(Kλ;k).

Idea: The inclusion ι : (S1, �) ãÑ (BZp, �) induces a cellular inclusionZK (ι) : TK = ZK (S1) ãÑ ZK (BZp). Moreover, φ : Kλ ãÑ K induces a

cellular inclusion Zφ : TKλãÑ TK . Let λ̄ : Zm � Zm

pλÝÑ k�. We then get

(chain) retractionsCq(TK ;kλ̄)

��Cq(ZK (BZp);kλ)

44

// Cq(TKλ;kλ̄)

� // rCq�1(Kλ;k)ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 15 / 30


This shows that dimk Hq(ZK (BZp); kλ) ¥ dimk rHq�1(Kλ; k).For the reverse inequality, we use [BBCG], which, in this case, says

Hq(ZK (EZp,Zp); k) �à

I�[m]

(p�1)|I|à1

rHq�1(KI ; k),

and the fact that ZK (EZp,Zp) � (ZK (BZp))ab, which gives

Hq(ZK (EZp,Zp);k) �à

ρPHom(Zmp ,k�)

Hq(ZK (BZp); kρ).

THEOREM (A.S.–TREVISAN)

Let G be a prime-order cyclic group, and let ZK (BG)χ be the abeliancover defined by an epimorphism χ : Gm � Γ. Then

Hq(ZK (BG)χ;k) �à

ρPHom(Γ;k�)

rHq�1(Kρ�χ; k),

where Kρ�χ is the induced subcomplex of K on vertex set supp(ρ � χ).


THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS

THE Q-HOMOLOGY OF REAL TORIC MANIFOLDS

Let again P be a simple polytope, and set K = KBP .Let χ : Zm

2 Ñ Zn2 be a characteristic matrix for P.

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

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] | the j-th entry of χS is non-zerou.

Compute the reduced simplicial Betti numbersb̃q(Kχ,S) = dimQ

rHq(Kχ,S ;Q).

COROLLARY (A.S.–TREVISAN)

The Betti numbers of the real toric manifold NP(χ) are given by

bq(NP(χ)) =¸

S�[n]

b̃q�1(Kχ,S).


THE RATIONAL HOMOLOGY OF REAL TORIC MANIFOLDS

EXAMPLE

Again, let P be the square, K = KBP the 4-cycle.

Let T 2 = NP(χ), χ =(

1 0 1 00 1 0 1

), and K` = NP(χ

1), χ1 =(

1 0 1 00 1 1 1

).

S H t1u t2u t1,2u

χS ( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 0 1 ) ( 1 1 1 1 )

Kχ,S H tt1u, t3uu tt2u, t4uu K

χ1S ( 0 0 0 0 ) ( 1 0 1 0 ) ( 0 1 1 1 ) ( 1 1 0 1 )

Kχ1,S H tt1u, t3uu tt2,3u, t3,4uu tt1,2u, t1,4uu

Hence:

b0(T 2) = b̃�1(H) = 1 b0(K`) = b̃�1(H) = 1

b1(T 2) = b̃0(Kχ,t1u) + b̃0(Kχ,t2u) = 2 b1(K`) = b̃0(Kχ1,t1u) + b̃0(Kχ1,t2u) = 1

b2(T 2) = b̃1(Kχ,t1,2u) = 1 b2(K`) = b̃1(Kχ1,t1,2u) = 0


THE HESSENBERG MANIFOLDS


Every Weyl group W determines a smooth, complex projectivetoric variety TW , with fan given by the reflecting hyperplanes of W .

Its real locus, TW (R), is a smooth, connected, compact real toricvariety 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 associatedpolytope the permutahedron Pn.

THEOREM (HENDERSON 2010)

bi(Tn(R)) = A2i

(n2i

),

where A2i is the Euler secant number, defined as the coefficient ofx2i /(2i)! in the Maclaurin expansion of sec(x),



Pn has 2n � 2 facets: each non-empty, proper subset Q � [n] facet F Q with vertices in which all coordinates in positions in Q aresmaller than all coordinates in positions not in Q.

The corresponding column vectors of the characteristic matrixχ : Z2n�2

2 Ñ Zn�12 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 0 0 1 1 1 0 0 1 1 1 0 0 10 1 0 1 1 0 1 1 0 1 1 0 1 00 0 1 1 0 1 1 1 1 0 1 1 0 0

)ALEX SUCIU (NORTHEASTERN) POLYHEDRAL PRODUCTS PRINCETON–RIDER 2012 20 / 30


The dual simplicial complex, Kn = KBPn , is the barycentricsubdivision of the boundary of the (n� 1)-simplex.

Given a subset S � [n� 1], the induced subcomplex on vertex setsupp(χS) depends only on r := |S|, so denote it by Kn,r .

Kn,r is the order complex associated to a rank-selected poset of acertain subposet of the Boolean lattice Bn. Thus, Kn,r isCohen–Macaulay; in fact,

Kn,2r�1 � Kn,2r �A2rª

Sr�1.

Hence, we recover Henderson’s computation:

bi(Tn(R)) =¸

S�[n�1]

b̃i�1((Kn)χ,S) =n�1̧

r=1

(n� 1

r

)b̃i�1(Kn,r )

=

((n� 12i � 1

)+

(n� 1

2i

))A2i =

(n2i

)A2i .


CUP PRODUCTS IN ABELIAN COVERS OF GDJ-SPACES


As before, let X ν Ñ X be a regular, finite abelian cover, correspondingto an epimorphism ν : π1(X )� Γ, and let k = C. The cellular cochainson X ν decompose as

Cq(X ν;k) �à

ρPHom(Γ,k�)

Cq(X ; kρ�ν),

The cup product map, Cp(X ν, k)bk Cq(X ν, k) !ÝÝÑ Cp+q(X ν, k),restricts to those pieces, as follows:

Cp(X ; kρ�ν)bk Cq(X ; kρ1�ν)! //

�

��

Cp+q(X ;k(ρ�ρ1)�ν)

Cp+q(X �X ; kρ�ν bk kρ1�ν)µ� // Cp+q(X �X ;k(ρbρ1)�ν)

∆�

OO

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


PROPOSITION (A.S.–TREVISAN)

Let ZK (BZp)ν be a regular abelian cover, with characteristichomomorphism χ : Zm

p Ñ Γ. The cup product in

H�(ZK (BG)ν; k) �8à

q=0

àρPHom(Γ;k�)

rHq�1(Kρ�χ; k)

is induced by the following maps on simplicial cochains:

rCp�1(Kρ�χ; k�)b rCq�1(Kρ1�χ; k�

)Ñ rCp+q�1(K(ρbρ1)�χ;k�

)σ̂b τ̂ ÞÑ

#�{σ\ τ if σX τ = H,

0 otherwise,

where σ\ τ is the simplex with vertex set the union of the vertex setsof σ and τ, and σ̂ is the Kronecker dual of σ.


FORMALITY PROPERTIES


A finite-type CW-complex X is formal if its Sullivan minimal modelis 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 generalizedDavis–Januszkiewicz spaces ZK (BG) are always formal.

(Félix, Tanré) More generally, if both X and A are formal, and theinclusion i : A ãÑ X induces a surjection i� : H�(X ,Q)Ñ H�(A,Q),then ZK (X ,A) is formal.



(Baskakov, Denham–A.S.) Moment angle complexes ZK (D2,S1)are not always formal: they can have non-trivial triple Masseyproducts. For instance, K =

(Denham–A.S.) There exist polytopes P and dual triangulationsK = KBP 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?


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 locifor cohomology with coefficients in rank 1 local systems,

V i(X ) = tρ P Hom(G,C�) | H i(X ,Cρ) � 0u.

Also, the closely related “resonance varieties",

R i(X ) = ta P H1(X ,C) | H i(H�(X ,C), �a) � 0u.

Question: How do the duality properties of a space X affect thenature of its cohomology jump loci?

Recall that X is a duality space of dimension n if Hp(X ,ZG) = 0for p � n and Hn(X ,ZG) � 0 and torsion-free.

By analogy, we say X is an abelian duality space of dimension n ifHp(X ,ZGab) = 0 for p � n and Hn(X ,ZGab) � 0 and torsion-free.


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 characterρ : 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 ) � V2(X ) � � � � � Vn(X ).

COROLLARY

If X admits a minimal cell structure, and X is an abelian duality spaceof dim n, then resonance propagates:

R1(X ) � R2(X ) � � � � � Rn(X ).

REMARK

Propagation of V i ’s does not imply propagation of Ri ’s.Eg, let M = HR/HZ be the 3-dim Heisenberg manifold. ThenV1 = V2 = V3 = t1u, but R1 = R2 = C2, and R3 = t0u.


TORIC COMPLEXES

TORIC COMPLEXES

Let K be a simplicial complex of dimension d , on vertex set V, andlet TK = ZK (S1, �) be the respective toric complex.

TK is a connected, minimal CW-complex, with dim TK = d + 1.

π1(TK ) = GΓ := 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 (GΓ,1) = T∆Γ , where ∆Γ is the flag complex of Γ.

THEOREM (S. PAPADIMA–A.S. 2009)

V i(TK ) =¤W

(C�)W and Ri(TK ) =¤W

CW

where the union is taken over all W � V for which there is a simplexσ P LVzW and an index j ¤ i such that rHj�1�|σ|(lkLW(σ),C) � 0.


TORIC COMPLEXES

K is Cohen–Macaulay if for each simplex σ P K , the cohomologyrH�(lk(σ),Z) is concentrated in degree n� |σ| 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 isCohen–Macaulay, in which case both V i(TK ) and Ri(TK ) propagate.

EXAMPLE

Let Γ = . Then resonance does not propagate:R1(GΓ) = C4, but R2(GΓ) = C2 � t0u Y t0u �C2.


REFERENCES

REFERENCES

A. Suciu, A. Trevisan, Real toric varieties and abelian covers ofgeneralized Davis–Januszkiewicz spaces, preprint 2012.

G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality andpropagation 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.


Documents

Polyhedral products, toric manifolds, and twisted cohomology