DDGA MODELS FOR MOMENT-ANGLE COMPLEXES
MATTHIAS FRANZ
Abstract.
A dga model for the integral singular cochains on a moment-anglecomplex is given by the twisted tensor product of the corresponding Stanley–Reisner ring and an exterior algebra. We present a short proof of this fact andextend it to real moment-angle complexes. We also compare various descrip-tions of the cohomology rings of these spaces, including one stated withoutproof by Gitler and López de Medrano. Introduction
Let Σ be a simplicial complex on the set [ m ] = { , . . . , m } , containing the emptysimplex ∅ and possibly having ghost vertices, and let(1.1) Z (Σ) = Z Σ ( D , S ) = [ σ ∈ Σ ( D , S ) σ ⊂ ( D ) m be the associated moment-angle complex, where the exponents in(1.2) ( D , S ) σ = ( D ) σ × ( S ) [ m ] (cid:114) σ indicate the factors of the m -fold Cartesian product.The moment-angle complex Z (Σ) is homotopy-equivalent to the complement ofa complex coordinate subspace arrangement, which is a smooth toric variety. Theintegral cohomology ring of Z (Σ) was computed by the author [9, Sec. 4] (usingthe language of toric varieties) and shortly afterwards by Baskakov–Buchstaber–Panov [3]. The result is an isomorphism of graded rings(1.3) H ∗ ( Z (Σ)) = Tor R ( Z , Z [Σ]) , where R = Z [ t , . . . , t m ] and Z [Σ] is the Stanley–Reisner ring of Σ, also withgenerators t , . . . , t m of degree 2. Taking the Koszul resolution of Z over R , onecan describe the ring (1.3) as the cohomology of the commutative differential gradedalgebra (cdga)(1.4) A (Σ) = Z [Σ] ⊗ ^ ( s , . . . , s m ) , d s i = t i , d t i = 0for i ∈ [ m ], where each s i has degree 1. Dividing out out all squares t i as well asall terms s i t i , one obtains a quasi-isomorphic dga B (Σ). As a cdga, it is generatedby the s i and t i = d s i and has the relations s i t i = t i t i = 0 for i ∈ [ m ] as well as t i · · · t i k = 0 whenever { i , . . . , i k } / ∈ Σ. Theorem 1.1.
The singular cochain algebra C ∗ ( Z (Σ)) is quasi-isomorphic to thedgas A (Σ) and B (Σ) , naturally with respect to inclusions of subcomplexes. Mathematics Subject Classification.
Primary 55U10; secondary 16E45, 55N10.The author was supported by an NSERC Discovery Grant. The argument appearing in [6, Thm. 7.7] and earlier publications by the same authors isincorrect, compare [10, Sec. 1]. a r X i v : . [ m a t h . A T ] J un MATTHIAS FRANZ
The quasi-isomorphism between C ∗ ( Z (Σ)) and A (Σ) is already implicit in theauthor’s computation of H ∗ ( Z (Σ)), see [9, Sec. 4]. A different proof has recentlybeen obtained by the author as a byproduct of his work on the cohomology ringsof partial quotients of moment-angle complexes [10, Prop. 6.1]. As remarked there,this result answers a question posed by Berglund [4, Question 5]. The aim of thepresent note is to give a much shorter proof for this dga model. Like Baskakov–Buchstaber–Panov’s calculation it is based on the dga B (Σ). The rational versionsof A (Σ) and B (Σ) are cdga models for the polynomial differential forms on Z (Σ)by a result of Panov–Ray [16, Thm. 6.2].The proof of Theorem 1.1 appears in the following section and an adaptationto real moment-angle complexes in Section 3. In the final section we relate theresulting cup product formulas for real and complex moment-angle complexes withothers appearing in the literature. We in particular provide a proof that has beenmissing so far for a product formula stated by Gitler and López de Medrano [11]. Acknowledgements.
I thank Don Stanley for his questions about dga models andSantiago López de Medrano for stimulating discussions.2.
Proof of Theorem 1.1
We will obtain Theorem 1.1 by dualizing the analogous homological result. Tostate the latter, we need to introduce some terminology. As already done in Theo-rem 1.1, we write C ( − ) and C ∗ ( − ) for normalized singular (co)chains with integralcoefficients, compare [12, Sec. VIII.6].Let Z h Σ i be the Stanley–Reisner coalgebra of Σ dual Z [Σ]. The canonical basisfor Z h Σ i , considered as a Z -module, are the monomials u α indexed by allowedmulti-indices α ∈ N m . A multi-index α is allowed if is supported on some simplexin Σ, that is, if(2.1) supp α := { i ∈ [ m ] | α i > } ∈ Σ . The degree of u α is 2( α + · · · + α m ). The structure maps are given by(2.2) ∆ u α = X β + γ = α u β ⊗ u γ , ε ( u α ) = ( α = 0,0 otherwise.We consider the tensor product of graded coalgebras(2.3) K (Σ) = Z h Σ i ⊗ ^ ( v , . . . , v m )where each v i is primitive of degree 1. We turn K (Σ) into a differential gradedcoalgebra (dgc) by defining(2.4) d ( u α ⊗ v τ ) = X α i > u α − i ⊗ v i ∧ v τ for allowed multi-indices α ∈ N m and τ ⊂ [ m ]. Here we have written α − i for themulti-index that is obtained from α by decreasing the i -th component by 1 as wellas v τ = v i ∧ · · · ∧ v i k if τ = { i < · · · < i k } . For σ ∈ Σ we also write u σ = u α where α is the indicator function of σ ⊂ [ m ],(2.5) α i = ( i ∈ σ ,0 if i / ∈ σ ,and we use the abbreviation u ∅ = v ∅ = u ∅ ⊗ v ∅ = 1. GA MODELS FOR MOMENT-ANGLE COMPLEXES 3
Let L (Σ) be the sub-dgc of K (Σ) spanned by all elements u σ ⊗ v τ with disjointsubsets σ ∈ Σ and τ ⊂ [ m ]. The dual of K (Σ) is the dga A (Σ), and that of L (Σ)is B (Σ). Theorem 2.1.
The dgcs C ( Z (Σ)) , K (Σ) and L (Σ) are quasi-isomorphic, natu-rally with respect to inclusions of subcomplexes. The proof is given in the remainder of this section. Applying the universalcoefficient theorem for cohomology then establishes Theorem 1.1.The following two observations are immediate. We write Σ | i for the restrictionof Σ to the single vertex i ∈ [ m ]. It contains either the empty simplex only oradditionally the 0-simplex { i } . Lemma 2.2.
For any σ ∈ Σ there are canonical isomorphisms of dgcs K ( σ ) = m O i =1 K ( σ | i ) , L ( σ ) = m O i =1 L ( σ | i ) . Lemma 2.3.
Let Σ , Σ be subcomplexes of Σ . There are short exact sequences −→ K (Σ ∩ Σ ) −→ K (Σ ) ⊕ K (Σ ) −→ K (Σ ∪ Σ ) −→ , −→ L (Σ ∩ Σ ) −→ L (Σ ) ⊕ L (Σ ) −→ L (Σ ∪ Σ ) −→ . Let y be the usual parametrization of S , considered as a singular 1-simplex.Choose a singular 2-simplex x in D that restricts to y on the edge (12) and mapsthe other two edges (01) and (02) to the point 1 ∈ S . Then d y = 0 , ∆ y = y ⊗ ⊗ y, (2.6) d x = x (12) − x (02) + x (01) ∆ x = x ⊗ x (01) ⊗ x (12) + 1 ⊗ x (2.7) = y, = x ⊗ ⊗ x. Note that for the last line to hold it is crucial that we work with normalized chains.We use the singular simplices x and y to define a dgc map(2.8) Ψ(Σ) : L (Σ) → C ( Z (Σ)) . For m = 1 we map u x , v y and 1 e ∈ S , the identity element; this iswell-defined by (2.6) and (2.7). For m > σ ⊂ [ m ] we set(2.9) Ψ( σ ) : L ( σ ) = m O i =1 L ( σ | i ) N Ψ( σ | i ) −−−−−−→ m O i =1 C ( Z ( σ | i )) ∇ −−−→ C (cid:0) Z ( σ | ) × · · · × Z ( σ | m ) (cid:1) = C ( Z ( σ )) , using Lemma 2.2. (Recall that the shuffle map ∇ is a morphism of dgcs, see [8,(17.6)].) In the general case Ψ(Σ) is determined by imposing naturality with respectto inclusions of subcomplexes.We claim that both dgc maps in the zigzag(2.10) K (Σ) ←−− - L (Σ) Ψ(Σ) −−−→ C ( Z (Σ))are quasi-isomorphisms. (For the inclusion map, compare [6, Lemma 7.10].) Thecase m = 1 is settled by a direct verification. Now assume m >
1. If Σ has a singlemaximal simplex σ , then our claim is a consequence of the Eilenberg–Zilber andKünneth theorems. The general case now follows by induction on the size of Σ from MATTHIAS FRANZ
Lemma 2.3 and the Mayer–Vietoris theorem together with the five lemma. Thiscompletes the proof.
Remark 2.4.
Theorems 1.1 and 2.1 remain valid for all generalized moment-anglecomplexes Z Σ ( D n , S n − ) with even n ≥
2, up to the obvious degree shifts. Forexample, the generators s i and t i in (1.4) are now of degrees | s i | = n − | t i | = n .The singular n -simplex x is obtained by collapsing all but the last facet of thestandard n -simplex to a point, and y is this last facet.If n ≥ | y | is even and | x | is odd. Proceeding as before, we geta quasi-isomorphism between C ∗ ( Z Σ ( D n , S n − )) and the cdga ˜ B (Σ) with genera-tors s i of degree n − t i = d s i of degree n as well as relations(2.11) s i s i = s i t i = 0 , and t i · · · t i k = 0 if { i , . . . , i k } / ∈ Σ.Note that the Stanley–Reisner relations are monomial and therefore independentof the order of the anticommuting variables t i .In general, such a quasi-isomorphism does not hold for the case n = 1, which wetreat in the following section.3. Real moment-angle complexes
It is not difficult to adapt our approach to real moment-angle complexes(3.1) Z R (Σ) = Z Σ ( D , S ) ⊂ ( D ) m . We start with the homological setting and there with the case m = 1.As a complex, we define the analogue L (Σ) of L (Σ) as before, except that nowthe degrees are | u | = 1 and | v | = 0. We turn L (Σ) into a dgc via the diagonal∆ v = v ⊗ ⊗ v + v ⊗ v, (3.2) ∆ u = u ⊗ ⊗ u + u ⊗ v. (3.3)Let x be the canonical path from e = 1 to g = − ∈ S , considered as a singular1-simplex in D = [ − , y = g − e . Then d x = y, d y = 0 , (3.4) ∆ y = g ⊗ g − e ⊗ e = y ⊗ e + e ⊗ y + y ⊗ y, (3.5) ∆ x = x ⊗ g + e ⊗ x = x ⊗ e + e ⊗ x + x ⊗ y, (3.6)which shows that the map(3.7) L (Σ) → C ( Z R (Σ)) , e, v y, u x is a morphism of dgcs. (Since it is injective, one can also use it to justify that L (Σ)is actually a dgc.) As before, one verifies easily that (3.7) is a quasi-isomorphism.For m > L (Σ). GA MODELS FOR MOMENT-ANGLE COMPLEXES 5
We now turn to cohomology. The dual of the dgc L (Σ) is the dga B (Σ) withgenerators s i of degree 0 and t i of degree 1 satisfying the relations d s i = − t i , d t i = 0 , (3.8) s i s i = s i , t i s i = t i , s i t i = 0 , t i t i = 0 , Y j ∈ σ t j = 0(3.9)for any i ∈ [ m ] and σ / ∈ Σ. plus the rule that variables corresponding to distinct subscripts commute in the graded sense.We can sum up our discussion as follows.
Theorem 3.1.
There is a quasi-isomorphisms of dgas C ∗ ( Z R (Σ)) → B (Σ) , natural with respect to inclusions of subcomplexes. We in particular recover Cai’s isomorphism of graded rings [7, Secs. 3 & 4](3.10) H ∗ ( Z R (Σ)) = H ∗ ( B (Σ)) . In fact, our proof shares some similarities with Cai’s. This would be even more soif we worked with cubical singular chains, compare [14]. We also remark that inthe case of real moment-angle complexes it is not necessary to pass to normalized(singular) chains. (The shuffle map is a morphism of dgcs for non-normalized chainsalready, and the formulas (3.4)–(3.6) do not need normalization, either.)We discuss the dga A (Σ) analogous to A (Σ) only for coefficients in Z . It hasthe same generators s i and t i as B (Σ) and the relations d s i = t i , d t i = 0 , (3.11) s i s i = s i , t i s i = s i t i + t i , Y j ∈ σ t j = 0(3.12)for i ∈ [ m ] and σ / ∈ Σ, again with the additional rule that variables correspondingto different subscripts commute. Observe that the ideal generated by the rela-tions (3.12) is closed under the differential, so that A (Σ) is a well-defined dga. Theprojection map A (Σ) → B (Σ) ⊗ Z is again obtained by dividing out the idealgenerated by the products s i t i and t i for all i ∈ [ m ], and it can be seen to bea quasi-isomorphism by an argument analogous to the one given before or to [6,Lemma 7.10].The Stanley–Reisner ring Z [Σ], now with generators of degree 1, is containedin A (Σ) as a sub-dga (with trivial differential). Moreover, if Σ = [ m ] is the fullsimplex, then A (Σ) is the Koszul resolution of Z over R = Z [ t , . . . , t m ]. Ingeneral, A (Σ) is the tensor product of this resolution and Z [Σ] over R , which givesthe additive isomorphism(3.13) H ∗ ( Z R (Σ); Z ) = Tor R ( Z , Z [Σ]) . It is not multiplicative for the canonical product on the torsion product, as can beseen for Σ = { ∅ } already, cf. [10, Sec. 10.3]. The minus sign in d s i comes from the general definition of the differential on the dual of acomplex, cf. [12, eq. (II.3.1)]. It could be removed by replacing t i with − t i , that is, by mapping u to − x . The minus sign does not appear in [7, p. 512] because of a different sign convention forthe dual complex. MATTHIAS FRANZ Comparison of several product formulas
The aim of this section is to relate the product formula in the cohomology ofa (complex) moment-angle complex with Baskakov’s formula [2] and also the for-mula for real moment-angle complexes with one claimed by Gitler and López deMedrano [11] as well as the one given by Bahri–Bendersky–Cohen–Gitler [1] forarbitrary polyhedral products. We note that another description for a class ofpolyhedral products including all Z Σ ( D n , S n − ) has been given by Zheng [17, Ex-ample 7.12].We start with a variant of the generalized smash moment-angle complexes intro-duced in [1, Def. 2.2]. For a closed subset A of a compact Hausdorff space X anda basepoint ∗ ∈ A we define the space(4.1) S ( X, A ) = (cid:8) x ∈ Z ( X, A ) (cid:12)(cid:12) x i = ∗ for some i ∈ [ m ] (cid:9) and based on it the pair(4.2) ˆ Z Σ ( X, A ) = (cid:0) Z Σ ( X, A ) , S Σ ( X, A ) (cid:1) . We then have an isomorphism(4.3) H ∗ ( ˆ Z Σ ( X, A )) = H ∗ c (cid:0) Z Σ ( X, A ) (cid:114) S Σ ( X, A ) (cid:1) = H ∗ c (cid:0) Z Σ ( X (cid:114) ∗ , A (cid:114) ∗ ) (cid:1) where H ∗ c ( − ) denotes cohomology with compact supports, cf. [13].We now specialize to(4.4) ˆ Z R (Σ) = ˆ Z Σ ( D , S )(where the basepoint is e = 1 ∈ S ) and observe that(4.5) Z Σ (cid:0) D (cid:114) { e } , S (cid:114) { e } (cid:1) = Z Σ (cid:0) [ − , , {− } (cid:1) ≈ C Σis the unbounded cone over the simplicial complex Σ.The analysis of Z R (Σ) in the preceding section carries over to the present case.One simply ignores the element e ∈ S and the counit 1 in the cochain algebra. (Re-call that the cohomology with compact supports is a ring without unit in general.)The result is as quasi-isomorphism between the relative cochain algebra C ∗ ( ˆ Z R (Σ))and the multiplicatively closed subcomplex ˆ B (Σ) ⊂ B (Σ) spanned by all m -foldproducts(4.6) a · · · a m where each a i = s i or t i .In particular, there is a multiplicative isomorphism(4.7) H ∗ c ( C Σ) = H ∗ ( C Σ , Σ) ∼ = H ∗ ( ˆ B (Σ))where C Σ denotes the bounded cone over Σ with base Σ. Not surprisingly, ˆ B (Σ)does not have a unit unless Σ = { ∅ } .We now compare ˆ B (Σ) to the dgas B (Σ) and B (Σ) for complex and real moment-angle complexes, respectively. In the complex case, we have a direct sum decom-position of complexes(4.8) B ∗ (Σ) = M α ⊂ [ m ] ˆ B ∗−| α | (Σ α ) GA MODELS FOR MOMENT-ANGLE COMPLEXES 7 where Σ α is the full subcomplex of Σ on the vertex set α . This gives Hochster’sformula(4.9) H ∗ ( Z (Σ)) = M α ⊂ [ m ] H ∗−| α | c ( C Σ α ) = M α ⊂ [ m ] ˜ H ∗−| α |− (Σ α ) , cf. [6, Thm. 3.2.7], where we have used the additive isomorphism(4.10) H ∗ c ( C Σ) = ˜ H ∗− (Σ)between the reduced cohomology of the simplicial complex Σ and the cohomologywith compact supports of the unbounded cone over it. (Recall that ˜ H − ( ∅ ) = Z .)The additive isomorphism (4.9) can be made multiplicative in the following way:For α ∩ β = ∅ , the product(4.11) H ∗ c ( C Σ α ) ⊗ H ∗ c ( C Σ β ) → H ∗ c ( C Σ α ∪ β ) , vanishes. For disjoint α , β we use the cross product(4.12) a ⊗ b a ∗ b := π ∗ α ( a ) ∪ π ∗ β ( b )where π α : C Σ α ∪ β → C Σ α is the (well-defined) restriction of the canonical projec-tion R α ∪ β → R α , and analogously for π β . This is Baskakov’s formula [2], expressedin terms of Cartesian products of cones and cohomology with compact supportsinstead of joins of simplices and reduced cohomology.For a real moment-angle complex we have a direct sum decomposition(4.13) B (Σ) = M α ⊂ [ m ] ˆ B ∗ (Σ α ) , hence also a Hochster formula(4.14) H ∗ ( Z R (Σ)) = M α ⊂ [ m ] H ∗ c ( C Σ α ) = M α ⊂ [ m ] ˜ H ∗− (Σ α ) . Note that there are no degree shifts by | α | this time. The isomorphism becomesmultiplicative if one uses the product (4.12) for all subsets α , β ⊂ [ m ]. (Thisproduct is still well-defined for compact supports.) We obtain a product that isvisibly graded commutative, something that was not obvious from the multiplica-tion rules (3.9). Looking back, we can see that these asymmetric formulas arosefrom the non-commutativity of the Alexander–Whitney map and the fact that onlyone the two vertices of the singular 1-simplex x in X = D can be the basepoint e .The multiplication we have defined on (4.14) coincides with the ∗ -product givenby Bahri–Bendersky–Cohen–Gitler [1, Thm. 1.4], as can be seen by tracing throughtheir definitions in [1, Sec. 1].We finally consider another description of H ∗ ( Z (Σ)) in the polytopal case. Let P be a simple polytope with m facets, and let Σ be the boundary complex of thedual simplicial polytope. For any subset α ⊂ [ m ], let P α ⊂ P be the union of thecorresponding facets. Lemma 4.1.
There is a ring isomorphism Θ α : H ∗ ( P, P α ) → H ∗ c ( C Σ α ) MATTHIAS FRANZ for any α ⊂ [ m ] . Moreover, the diagram H ∗ ( P, P α ) ⊗ H ∗ ( P, P β ) H ∗ ( P, P α ∪ β ) H ∗ c ( C Σ α ) ⊗ H ∗ c ( C Σ β ) H ∗ c ( C Σ α ∪ β ) Θ α ⊗ Θ β ∪ Θ α ∪ β ∗ commutes for all α , β ⊂ [ m ] .Proof. Let Σ be the barycentric subdivision of Σ, considered as a triangulationof ∂P . As a topological space, Σ α can be identified with a subcomplex of Σ , hence C Σ α with a subcomplex of C Σ ≈ P . We can also identify P α with the union ofthe closed blocks (or cells) in Σ dual to the vertices in α , cf. [15, §64].We claim that the canonical inclusion of pairs(4.15) ( C Σ α , Σ α ) → ( C Σ , P α )is a strong deformation retract: Similar to the proof of [15, Lemma 70.1], we candefine a strong deformation retraction that moves the vertex v σ ∈ C Σ correspond-ing to a simplex σ ∈ Σ to the vertex v σ ∩ α ∈ C Σ α along a straight line, which isinside σ if σ ∩ α = ∅ . If σ has no vertex in α , then v σ is moved to the apex v ∅ of the cone, and v ∅ is mapped to itself. We extend the map linearly to each sim-plex τ ∈ C Σ . If τ is contained in σ ∈ Σ, then it is mapped to the cone over thesimplex σ ∩ α ∈ Σ α (with the empty simplex ∅ giving the apex). The deformationretraction restricts to one from P α onto Σ α . We therefore get an isomorphism(4.16) Θ α : H ∗ ( P, P α ) → H ∗ ( C Σ α , Σ α ) = H ∗ c ( C Σ α )in cohomology.To show that the above diagram commutes, we work on the chain level. We usesimplicial chains for the left-hand side of (4.15), which canonically map to singularchains on the right. We choose a vertex ordering for C Σ α ∪ β such that all verticessmaller than the apex v ∅ are in α and all greater ones in β . (Some may be in both.)To a simplex σ ∈ C Σ α ∪ β we have to apply the Alexander–Whitney diagonal andpossibly the projections from C Σ α ∪ β to C Σ α and C Σ β , which send “superfluous”vertices to v ∅ . Afterwards we evaluate the resulting tensor product on a ⊗ b where a , b ∈ C ( P ) are cocycles vanishing on P α and P β , respectively.Because of the way we have ordered the simplices, the following happens: If σ does not contain v ∅ , then the result is 0 for both ways of going through thediagram. Otherwise we obtain ( − | b || σ | a ( σ ) b ( σ ) for both ways where σ is thefront face of σ ending in v ∅ and σ the back face starting there. Hence the diagramcommutes in either case. (cid:3) As a consequence, we get a ring isomorphism(4.17) H ∗ ( Z R ( P )) = M α ⊂ [ m ] H ∗ ( P, P α )where the multiplication on the right-hand side is given by the cup products(4.18) H ∗ ( P, P α ) ⊗ H ∗ ( P, P β ) → H ∗ ( P, P α ∪ β )for all α , β ⊂ [ m ]. This description of the cohomology ring of a real moment-anglemanifold was stated without proof by Gitler and López de Medrano [11, p. 1526]. GA MODELS FOR MOMENT-ANGLE COMPLEXES 9
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Department of Mathematics, University of Western Ontario, London, Ont. N6A 5B7,Canada
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