The cohomology rings of real toric spaces and smooth real toric varieties
TTHE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACESAND SMOOTH REAL TORIC VARIETIES
MATTHIAS FRANZ
Abstract.
We compute the mod 2 cohomology rings of smooth real toricvarieties and of real toric spaces, which are quotients of real moment-anglecomplexes by freely acting subgroups of the ambient 2-torus. The differentialgraded algebra we present is in fact a dga model for these coefficients. Wededuce from this description that smooth toric varieties are M-varieties. Introduction
Let Σ be a simplicial complex on the vertex set [ m ] = { , . . . , m } , possibly withghost vertices. The 2-torus G = ( Z ) m acts canonically on the real moment-anglecomplex(1.1) Z Σ = Z Σ ( D , S ) = [ σ ∈ Σ Z σ ⊂ ( D ) m where D = [ − ,
1] is the interval with boundary S = {± } , and the exponents in(1.2) Z σ = ( D ) σ × ( S ) [ m ] \ σ indicate the factors of the Cartesian products. The quotient X Σ = Z Σ /K by afreely acting subgroup K ⊂ G is called a real toric space [6], [8]. It is the topologicalcounterpart of a smooth real toric variety.More precisely, let Σ be a regular rational fan in R n with m rays, which we mayalso consider as a simplicial complex on the vertex set [ m ]. The associated realtoric variety X Σ = X Σ ( R ) is the fixed point set of the canonically defined complexconjugation on the complex toric variety X Σ ( C ). If the primitive generators of therays in Σ span the lattice Z n ⊂ R n , then there is a real toric space X Σ as definedabove that is a strong deformation retract of the real toric variety X Σ . For anarbitrary regular fan this statement has to be modified slightly, see Corollary 4.6.Jurkiewicz [21] has determined the mod 2 cohomology rings of smooth real pro-jective toric varieties, and Davis–Januszkiewicz [9] those of their topological ana-logues, small covers (over a polytope), compare Remark 3.10. The rational Bettinumbers of these spaces have been computed by Suciu–Trevisan [26], [28, Sec. VI.2],and their integral cohomology groups by Cai–Choi [5, Thm. 1.1] (more generallyfor shellable Σ). The cohomology ring of a real moment-angle complex has beencalculated by Cai [4, p. 514] for any coefficient ring k . Choi–Park [7, Thm. 4.5], [8,Main Thm.] have determined the cohomology rings of any real toric space underthe assumption that 2 is invertible in k . Mathematics Subject Classification.
Primary 14M25, 55M35; secondary 14F45, 55U10.The author was supported by an NSERC Discovery Grant. a r X i v : . [ m a t h . A T ] S e p MATTHIAS FRANZ
In this note we describe the cohomology rings of real toric spaces and smoothreal toric varieties with coefficients in k = F . The special role of the prime p = 2was already pointed out by Choi–Kaji–Theriault [6, p. 155] in their study of the p -local homotopy type of real toric spaces. The present work also complementsthe author’s computation [15] of the integer cohomology rings of smooth (complex)toric varieties as well as of quotients of (complex) moment-angle complexes by freelyacting closed subgroups of the ambient torus T = ( S ) m .To state our result, we define the following differential graded algebra (dga) A (Σ).As a graded vector space, it is the tensor product(1.3) A (Σ) = k [Σ] ⊗ F ( L )of the Stanley–Reisner ring of Σ with generators t , . . . , t m of degree 1 and thealgebra F ( L ) of function L → k with pointwise multiplication. We fix an iso-morphism L ∼ = ( Z ) n and write s , . . . , s n for the corresponding coordinate func-tions L → k , which generate F ( L ). The projection map p : G → L then is encodedby a characteristic matrix [ x ij ] ∈ ( Z ) n × m . In the case of a real toric variety, itscolumns [ x j , . . . , x nj ] are the coordinates mod 2 of the minimal integer represen-tatives of the rays of the fan.The differential on A (Σ) is given by(1.4) d s i = m X j =1 x ij t j and d t j = 0for 1 ≤ i ≤ n and 1 ≤ j ≤ m . Both k [Σ] and H ∗ ( G ) are subalgebras of A (Σ), sothat all s i ’s commute with each other and so do all t j ’s. We further impose therelations(1.5) s i t j = t j s i + x ij t j for all i and j .In the case of a real moment-angle complex we have L = G , and the characteristicmatrix [ x ij ] can be chosen to be the identity matrix. The relations (1.5) reduce to(1.6) s i t i = t i s i + t i and s i t j = t j s i for i = j . The dga B (Σ) given by Cai is the quotient of this specific A (Σ) by theadditional relations(1.7) s i t i = t i t i = 0for all 1 ≤ i ≤ m = n . In [16, Sec. 3] we remarked that the projection A (Σ) → B (Σ)is a quasi-isomorphism. More importantly, we showed [16, Thm. 3.1] that A (Σ)and B (Σ) are dga models for Z Σ ( D , S ), meaning that they are quasi-isomorphicto the singular cochain algebra of this real moment-angle complex. As far as A (Σ)is concerned, this generalizes to all real toric spaces, see Theorem 3.1. Theorem 1.1.
The dga A (Σ) is quasi-isomorphic to the singular cochain alge-bra C ∗ ( X Σ ; F ) , naturally with respect to the inclusion of subcomplexes of Σ . Inparticular, there is an isomorphism of graded algebras H ∗ ( X Σ ; F ) = H ∗ ( A (Σ)) . The analogous statement holds for all smooth real toric varieties (Theorem 4.7),and it implies that the cohomology of these spaces can additively be expressed asa torsion product involving the Stanley–Reisner ring k [Σ], see Proposition 3.9. HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 3
A crucial ingredient in our proof of Theorem 1.1 is the differential graded coal-gebra model for a fibre bundle recently obtained in [17] as a consequence of thecomultiplicativity of Szczarba’s twisting cochain.Recall that a complex variety X with an antiholomorphic involution τ is calledmaximal (or an M-variety) if the mod 2 Betti sum of the real locus X τ is as largeas allowed by Smith theory, that is, equal to the corresponding sum for X , com-pare [19, Prop. 7.3.7]. This generalizes the classical notion of M-curves introducedby Harnack. Bihan–Franz–McCrory–van Hamel [2, Thm. 1.1] have shown that atoric variety is maximal if it is smooth and compact, or (if Borel–Moore homology isused) possibly singular and of dimension at most 3. As a consequence of our resultswe obtain the following substantial enlargement of this list, see Corollary 4.8. Theorem 1.2.
Every smooth toric variety is maximal.
The example of a 6-dimensional non-maximal singular projective toric varietydue to Hower [20] shows that the smoothness assumption cannot be left out.The paper is organized as follows. In Section 2 we review necessary back-ground material including homotopy Gerstenhaber structures and Szczarba’s twist-ing cochain. Real toric spaces are considered in Section 3 and toric varieties inSection 4. 2.
Preliminaries
We work over the field k = F . Unless indicated otherwise, all tensor products aretaken over k , and all (co)chains complexes and (co)homologies are with coefficientsin k . All algebras and coalgebras are non-negatively graded. We often use theSweedler notation(2.1) ∆ c = X ( c ) c (1) ⊗ c (2) when we apply the diagonal to an element of a coalgebra.2.1. Normalized chains and cochains.
It will be convenient to use simplicialsets. A good reference for our purposes is [24]. All simplicial sets we consider willbe Kan complexes.The normalized chain complex C ( X ) of a simplicial set X is obtained from non-normalized chains by dividing out the subcomplex of degenerate chains, cf. [23,Sec. VIII.6]. The projection map is a homotopy equivalence. For X = pt a pointwe have a canonical isomorphism C ( pt ) = k .We will use that C ( X ) is a differential graded coalgebra (dgc). The augmen-tation ε : C ( X ) → k is induced by the unique map X → pt , and the Alexander–Whitney diagonal is given by(2.2) ∆ x = n X k =0 x (0 , . . . , k ) ⊗ x ( k, . . . , n )for an n -simplex x ∈ X , where the terms on the right indicate faces of x . Theshuffle map(2.3) ∇ : C ( X ) ⊗ C ( Y ) → C ( X × Y )is a quasi-isomorphism of dgcs for any simplicial sets X and Y , see [10, (17.6)].It induces an associative product on the chain complex C ( G ) of any simplicial MATTHIAS FRANZ group G . In case of a discrete group G , the dga C ( G ) is isomorphic to the groupalgebra k [ G ] (with trivial differential).The dual of the dgc C ( X ) is the dga C ∗ ( X ) of normalized singular cochainson X with the usual cup product. If X is discrete, then C ∗ ( X ) = H ∗ ( X ) is thealgebra F ( X ) of functions X → k with pointwise operations.2.2. Homotopy Gerstenhaber structures.
The dgc C ( X ) is in fact a homo-topy Gerstenhaber coalgebra, and dually the dga C ∗ ( X ) a homotopy Gerstenhaberalgebra. In the former case, this essentially means that there are cooperations(2.4) E k : C ( X ) → C ( X ) ⊗ k ⊗ C ( X )of degree k for any k ≥ C ( X ) compatible with the multiplication, compare [22, p. 223]. In our case themaps E k are interval cut operations for the surjections(2.5) e k = ( k + 1 , , k + 1 , , k + 1 , . . . , k + 1 , k, k + 1) , see [1, §2.2] and [17, Sec. 3]. We only need to know the following two facts: Thecooperation E k ( x ) vanishes for k > | x | , and for any n -simplex x ∈ X we have(2.6) E k ( x ) ≡ X ≤ i < ···
Twisted Cartesian products are simplicialversions of fibre bundles, see [24, §18]. A twisted Cartesian product with base X ,structure group G and a G -space F as fibre is written as X × τ F , where(2.7) τ : X > → G is the twisting function. The twisted Cartesian product differs from the usualCartesian product of simplicial sets only by a modified zeroeth face map.If we think of X × τ F as associated to the principal G -bundle P = X × τ G , wewrite is as(2.8) X × τ F = P × G F. Similarly, the pull-back of P along a map q : Y → X is written as(2.9) Y × τ ◦ q G = Y X × P. Classifying spaces and universal bundles.
Given a simplicial group G ,there is a universal G -bundle EG → BG whose total space is the twisted Cartesianproduct EG = BG × τ G G for a canonical twisting function(2.10) τ G : BG > → G, see [24, §21].If G is discrete, then the n -simplices of BG are of the form b = [ g , . . . , g n ]with g , . . . , g n ∈ G . Such a simplex is degenerate if and only if g i = 1 for some i .The twisting function is given by τ G ( b ) = g for n ≥
1. The differential of b is(2.11) d b = [ g , . . . , g n ] + n − X i =1 [ g , . . . , g i g i +1 , . . . , g n ] + [ g , . . . , g n − ] HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 5 for n ≥
1, and for any k ≤ l the face of b with vertices k , k + 1, . . . , l is given by(2.12) b ( k, . . . , l ) = [ g k +1 , . . . , g l ] . The simplicial Borel construction of a G -space X is X G = EG × G X . Lemma 2.1.
Let G be a simplicial group, and let X be a G -space. There is ahomotopy equivalence (cid:0) EG × G X (cid:1) BG × EG → X, natural in X .Proof. See [11, Thm. 2.8.2] or [12, Thm. 3.11]. The homotopy equivalence is actu-ally G -equivariant. (cid:3) Twisted tensor products.
A twisting cochain t : C → A is a map of de-gree − ε t = 0 and such that the map d ( c ⊗ a ) = d c ⊗ a + c ⊗ d a + X ( c ) c (1) ⊗ t ( c (2) ) a (2.13)is a differential on C ⊗ A . The resulting complex is denoted by C ⊗ t A .Given a twisted Cartesian product X × τ F with structure group G , Szczarba [27,Sec. 2] has constructed a twisting cochain t : C ( X ) → C ( G ) and a quasi-isomor-phism of complexes(2.14) ψ : C ( X ) ⊗ t C ( F ) → C ( X × τ F ) , both natural in the given data. For discrete G the twisting cochain is of the form(2.15) t ( x ) = ( τ ( x ) − − n = 1,0 otherwisefor any n -simplex x ∈ X .In [17, Sec. 8 & 9] we defined a coalgebra structure on the twisted tensor productand showed that the map ψ is a morphism of dgcs. The diagonal is given by(2.16) ∆( c ⊗ a ) = X ( c ) , ( a ) X k ≥ X r (cid:0) c (1) ⊗ t ( c r, ) · · · t ( c r,k ) a (1) (cid:1) ⊗ (cid:0) c r,k +1 ⊗ a (2) (cid:1) for c ∈ C ( X ) and a ∈ C ( F ), where(2.17) E k ( c (2) ) = X r c r, ⊗ · · · ⊗ c r,k ⊗ c r,k +1 . (Recall that we have E k ( c ) = 0 for k > | c | , so that the sum over k in (2.16) isactually finite.)2.6. Face coalgebras.
Let Σ be a simplicial complex on the vertex set [ m ] = { , . . . , m } , containing the empty simplex ∅ and possibly ghost vertices. Any sim-plicial fan canonically defines a simplicial complex with the rays as vertex set. Wewrite Σ | j for the restriction of Σ to the vertex j ∈ [ m ]. It contains either the emptysimplex only or additionally the 0-simplex { j } .The face coalgebra (or Stanley–Reisner coalgebra) k h Σ i is the graded coalgebradual to the Stanley–Reisner ring k [Σ], compare [3, p. 335]. Its canonical k -basis isgiven by cogenerators u α for all multi-indices α ∈ N m whose supports satisfy(2.18) supp α := { j ∈ [ m ] | α j = 0 } ∈ Σ . MATTHIAS FRANZ
The degree of u α is | α | = α + · · · + α m . The structure maps are given by(2.19) ∆ u α = X β + γ = α u β ⊗ u γ , and ε ( u α ) = ( α = 0,0 otherwise.For any two subcomplexes Σ , Σ ⊂ Σ there is a short exact sequence(2.20) 0 −→ k h Σ ∩ Σ i −→ k h Σ i ⊕ k h Σ i −→ k h Σ ∪ Σ i −→ . Assume that Σ has a unique maximal simplex σ . By abuse of notation we writethis as Σ = σ . In this case there is an isomorphism of graded coalgebras(2.21) k h σ i = m M j =1 k h σ | j i . Real toric spaces
Let Σ be a simplicial complex on the set [ m ], and let(3.1) Z Σ = Z Σ ( D , S )be the associated real moment-angle complex. The 2-torus G = ( Z ) m acts on Z Σ ina canonical fashion. Let K ⊂ G be a freely acting subgroup. The projection Z Σ → X Σ = Z Σ /K is equivariant with respect to the quotient map p : G → L = G/K .3.1.
A dga model for real toric spaces.
Recall that the dga A (Σ) has beendefined in the introduction. Theorem 3.1.
The dga A (Σ) is naturally quasi-isomorphic to C ∗ ( X Σ ) . Here naturality refers to the inclusion of subcomplexes Σ , → Σ.To prove Theorem 3.1 we construct a natural zigzag of quasi-isomorphisms con-necting the two dgas. This occupies the remainder of this section. We use theabbreviation C = Z = { , o } , and we write o , . . . , o m for the canonical genera-tors of G = ( Z ) m .3.1.1. First step.
Lemma 3.2.
The map ( Z Σ ) G → ( X Σ ) L is a homotopy equivalence and naturalin Σ .Proof. Naturality is clear. Because K acts freely on Z Σ with quotient X Σ , the map(3.2) ( Z Σ ) G = EG × G Z Σ → EL × L X Σ = ( X Σ ) L is a bundle with fibre EK and therefore a homotopy equivalence. (cid:3) Let(3.3) DJ Σ = Z Σ ( BC, ∗ ) ⊂ BG be the real Davis–Januszkiewicz space associated to Σ. Depending on the context,we think of it as a space over BG or over BL , that is, together with the canonicalmap DJ Σ → BG or its prolongation to BL determined by the map Bp : BG → BL . Lemma 3.3.
There is a zigzag of homotopy equivalences between ( Z Σ ) G and DJ Σ .The zigzag consists of maps over BG and is natural in Σ . HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 7
Proof.
The zigzag is(3.4) ( Z Σ ) G = Z Σ (cid:0) EC × C D , EC × C S (cid:1) ← Z Σ (cid:0) EC × C D , ∗ (cid:1) → Z Σ ( BC, ∗ ) , compare [15, Lemma 5.1]. All maps are compatible with the maps to BG = Z Σ ( BC, BC ) and natural in Σ.By induction on the size of Σ it follows from the Künneth and Mayer–Vietoristheorems that each map is a quasi-isomorphism. An analogous argument based onthe Seifert–van Kampen theorem shows that each map induces an isomorphism onfundamental groups and therefore is a homotopy equivalence. (cid:3)
Lemma 3.4.
The space X Σ is homotopy equivalent to Y Σ = DJ Σ × BL EL , naturallyin Σ .Proof. This is a consequence of Lemma 2.1, the two previous lemmas and the longexact sequence of homotopy groups of a bundle, cf. [24, Thm. 7.6, Prop. 18.4]. (cid:3)
Suppressing the map DJ Σ → BL , we have Y Σ = DJ Σ × τ L L , see Section 2.3.Hence the dgc C ( X Σ ) is quasi-isomorphic to the twisted tensor product(3.5) C ( DJ Σ ) ⊗ t C ( L )with the dgc structure discussed in Section 2.5. (We again suppress the canonicalmap C ( DJ Σ ) → C ( BG ).) All quasi-isomorphisms involved are natural in Σ.3.1.2. Second step.
Recall from Section 2.6 that k h Σ i is the face coalgebra of Σwith coefficients in k . We construct a map(3.6) Ψ : k h Σ i → C ( DJ Σ )as follows. For m = 1 we set(3.7) Ψ( u k ) = w k := [ o, . . . , o | {z } k ]where o is the generator of C . Note that the counit u = 1 is mapped to the unique0-simplex w = ∗ ∈ DJ Σ , which is still well-defined in the case Σ = { ∅ } . Forgeneral m we define(3.8) Ψ( u α ) = w α := ∇ (cid:0) w α ⊗ · · · ⊗ w α m (cid:1) for α ∈ N m with support σ ∈ Σ, where(3.9) ∇ : m O j =1 C ( Y j ) → C (cid:16) m Y j =1 Y j (cid:17) = C ( DJ σ ) ⊂ C ( DJ Σ )is the shuffle map and Y j = BC for j ∈ σ and the point ∗ otherwise. Lemma 3.5.
The map Ψ is a quasi-isomorphism of dgcs. Hence DJ Σ is formal,naturally in Σ .Proof. What we have said in Section 2.4 implies that Ψ is an isomorphism of dgcsfor m = 1. (Note that d w k = w k − + w k − = 0 for k > Z .) If m > σ is a simplex, then the claim follows from the isomorphism (2.21),the Künneth theorem and the fact that the shuffle map is a quasi-isomorphism ofdgcs. Based on this and the short exact sequence (2.20), the general case is doneby induction on the size of Σ and a Mayer–Vietoris argument. (cid:3) MATTHIAS FRANZ
Remark 3.6.
The formality of DJ Σ over k = Z is of course analogous to the inte-gral formality of the (complex) Davis–Januszkiewicz space Z Σ ( BS , ∗ ) establishedby the author [11, Thm. 3.3.2], [13, Thm. 1.4] and Notbohm–Ray [25, Thm. 4.8].As a consequence we get a chain mapΦ : M (Σ) := k h Σ i ⊗ t k [ L ] → C ( DJ Σ ) ⊗ t C ( L ) , (3.10) u α ⊗ g Ψ( u α ) ⊗ g, natural in Σ. Note that on the left-hand side we simply write t for the composition(3.11) k h Σ i Ψ −→ C ( DJ Σ ) t −→ C ( G ) p ∗ −→ C ( L ) = k [ L ] . Proposition 3.7.
The map Φ is a quasi-isomorphism of complexes.Proof. Let m (cid:67) k [ L ] be the augmentation ideal. It is nilpotent as m n +1 = 0. Thefiltrations of both the domain and the target of Φ by the powers of the augmentationideal therefore are finite.We consider the associated spectral sequence. The map between the first pages(3.12) k h Σ i ⊗ t Ψ m q / m q +1 → C ( DJ Σ ) ⊗ t m q / m q +1 is of the form Ψ ⊗
1. Moreover, the differentials on both sides are of the form d ⊗ t maps to m . Hence the map (3.12) is a quasi-isomorphismby Lemma 3.5, and so is Φ. (cid:3) Third step.
As discussed in Section 2.5, the differential on C ( DJ Σ ) ⊗ t C ( L )induced by Szczarba’s twisting cochain is(3.13) d ( c ⊗ g ) = X ( c ) c (1) ⊗ t ( c (2) ) g, for c ∈ C ( DJ Σ ) and g ∈ L , and the diagonal is given by(3.14) ∆( c ⊗ g ) = X ( c ) X r (cid:0) c (1) ⊗ t ( c r, ) · · · t ( c r,k ) g (cid:1) ⊗ (cid:0) c r,k +1 ⊗ g (cid:1) , where(3.15) E k ( c (2) ) = X r c r, ⊗ · · · ⊗ c r,k ⊗ c r,k +1 . By the formula (2.15), all c r,i with i ≤ k have to be of degree 1 for the correspondingsummand in (3.14) to be non-zero, and the same applies to the element c (2) in (3.13).Hence for an element w α with α ∈ N m we have(3.16) d ( w α ⊗ g ) = X α j > w α | j ⊗ p ∗ ( o j + 1) g, where we have written α | j for the multi-index obtained from α by decreasing the j -th component by 1.Now consider an l -simplex c = [ g , . . . , g l ] ∈ DJ σ for some σ ⊂ [ m ]. Accordingto (2.12) and (2.16) we have(3.17) E k ( c ) ≡ X ≤ i < ···
2. Moreover, if twoof the elements g i s are equal, then t ( g i ) · · · t ( g i k ) = 0 because each term t ( g i s ) = g i s + 1 squares to 0 in C ( G ) = k [ G ]. (Remember that we work in characteristic 2.) HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 9
By the definition of the shuffle map we have(3.18) w α = X π [ o π (1) , . . . , o π ( l ) ]for any α ∈ N m of degree | α | = l , where the sum is over all shuffles π of the sequence(3.19) (cid:0) , . . . , | {z } α , . . . , m, . . . , m | {z } α m (cid:1) . Hence(3.20) E k ( w α ) ≡ X π X ≤ j < ··· We finally show that the differential d and the product ∗ on thegraded vector space A (Σ) = k [Σ] ⊗ F ( L ) dual to the differential(3.26) d ( u α ⊗ g ) = X α j > u α | j ⊗ p ∗ ( o j + 1) g and the diagonal (3.24) on M (Σ) are those described in the introduction. We beginwith the product. Recall that the canonical pairing between A (Σ) and M (Σ) = k h Σ i ⊗ t k [ G ] is(3.27) h f ⊗ a, u ⊗ g i = h f, u i h a, g i for u ∈ k h Σ i , f ∈ k [Σ], g ∈ L and a ∈ F ( L ), and that s i : L ∼ = ( Z ) n → k is the i -th coordinate function. Like all functions in F ( L ), it is linearly extended to k [ L ]. Lemma 3.8. For any ≤ i ≤ n , any ≤ j ≤ m and g ∈ L we have (cid:10) , p ∗ ( o j + 1) g (cid:11) = 0 and (cid:10) s i , p ∗ ( o j + 1) g (cid:11) = x ij . Proof. The linear extension of the constant function 1 ∈ F ( L ) counts the numberof group elements appearing in a linear combination in k [ L ]. It therefore evaluatesto 0 on an even number of terms, which proves the first claim.By definition, we have x ij = h s i , p ( o j ) i . If this value equals 1, then the i -thcoordinate of either g or p ( o j ) g is equal to 1 (but not both), which implies that theleft-hand side of the second identity above is 1. If x ij = 0, then the i -th coordinatesof g and p ( o j ) g agree, so that the left-hand side vanishes, again as desired. (cid:3) Let f , h ∈ k [Σ] and a , b ∈ F ( L ). Formula (3.24) implies(3.28) ( f ⊗ a ) ∗ (1 ⊗ b ) = f ⊗ a b and also(3.29) ( f ⊗ ∗ ( h ⊗ b ) = f h ⊗ b because we have h , p ∗ ( b γ,σ ) i = 0 for any γ ∈ N m and any ∅ = σ ⊂ supp γ byLemma 3.8. In particular, we get(3.30) f ∗ h = f h, a ∗ b = a b, f ∗ a = f ⊗ a = f a, where we have identified f with f ⊗ ∈ A (Σ) and a with 1 ⊗ a ∈ A (Σ). Sinceboth products on A (Σ) are associative, it only remains to verify the commutationrule (1.5) for ∗ , that is, to establish the identity(3.31) (1 ⊗ s i ) ∗ ( t j ⊗ 1) = t j ⊗ s i + x ij t j ⊗ . for all 1 ≤ i ≤ n and 1 ≤ j ≤ m .Let u r ∈ k h Σ i be the cogenerator dual to the generator t r ∈ k [Σ]. (We areassuming here that r is not a ghost vertex of Σ.) Evaluated on the element u r ⊗ g ,the right-hand side above vanishes unless r = j , in which case it is equal to(3.32) h s i , g i + x ij . The left-hand side of (3.31) gives(3.33) h s i , g i h t j , u r i + h s i , p ∗ (˜ b ) g i h t j , u r i where ˜ b = b γ,σ for u γ = u r and σ = { r } . This expression again vanishes unless r = j , and in this case it equals(3.34) h s i , g i + h s i , p ∗ ( o j + 1) g i . This is the same as (3.32), again by Lemma 3.8.We now turn to the differential. Because we already know the products to agree,we only have to look at the generators t j and s i . For any multi-index α we have h d t j , u α ⊗ g i = (cid:10) t j ⊗ , d ( u α ⊗ g ) (cid:11) = X α r > (cid:10) t j ⊗ , u α | r ⊗ p ∗ ( o r + 1) g (cid:11) (3.35) = X α r > h t j , u α | r i h , p ∗ ( o r + 1) g i = 0 , once again by Lemma 3.8. By the same token, we also have h d s i , u j ⊗ g i = h s i , d ( u j ⊗ g ) i = (cid:10) s i ⊗ , ⊗ p ∗ ( o j + 1) g (cid:11) (3.36) = (cid:10) s i , p ∗ ( o j + 1) g (cid:11) = x ij HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 11 for any j such that there is a corresponding cogenerator u j of degree 1. Thisconfirms the identities (1.4) and completes the proof.3.2. Expressing the cohomology as a torsion product. The chosen decom-position L ∼ = ( Z ) n gives rise to an isomorphism R = H ∗ ( BL ) ∼ = k [¯ t , . . . , ¯ t n ]. Themap Bp ∗ : H ∗ ( BL ) → H ∗ ( BG ) is of the form(3.37) ¯ t i m X j =1 x ij t j . The Stanley–Reisner ring k [Σ] is an algebra over R via the map Bp ∗ . Proposition 3.9. There is an isomorphism of graded vector spaces H ∗ ( X Σ ) = Tor R ( k , k [Σ]) , natural in Σ .Proof. Let ¯Σ = [ n ] be the full simplex on n = m vertices. The differential graded R -module K = A ( ¯Σ) is free over R as a graded module with basis(3.38) s σ = Y i ∈ σ s i for σ ⊂ [ n ]. This together with the identity d s i = ¯ t i shows that K is in fact theKoszul resolution of k over R . (Since X ¯Σ = ( D ) n is contractible, Theorem 3.1confirms that K is acyclic.)The differential (1.4) of s i in A (Σ) is the image of ˜ t i under the map Bp ∗ , whichmeans that we have an isomorphism of chain complexes(3.39) A (Σ) = K ⊗ R k [Σ] , natural in Σ. Taking cohomology concludes the proof. (cid:3) Remark 3.10. Note that in general the canonical product on the Tor term doesnot correspond to the cup product. (This fails already for Σ = { ∅ } .) It holds,however, if k [Σ] is free over R because H ∗ G ( X Σ ) = H ∗ ( DJ Σ ) = k [Σ] surjects onto H ∗ ( X Σ ) = Tor R ( k , k [Σ]) = k ⊗ R k [Σ] in this case, compare [19, Prop. 7.1.6]. Thishappens for smooth real projective toric varieties or, more generally, for small coverswith shellable Σ. We thus recover the description of their cohomology rings as givenby Jurkiewicz [21, Thm. 4.3.1] and Davis–Januszkiewicz [9, Thm. 4.14].4. Toric varieties Let N ∼ = Z n be a lattice, and let Σ be a regular rational fan in N R = N ⊗ Z R . Asremarked in the introduction, the associated real toric variety X Σ can be definedas the fixed point set of the complex conjugation on the corresponding (complex)toric variety. It can also be described intrinsically as toric varieties are defined overthe integers, compare [18, p. 78] and [14, Sec. 2]. In this final part of the paper westudy X Σ from a topological point of view. Recall that it comes with an action ofthe group L = T N ∼ = ( R × ) n where R × = R \ { } .Let ˜ N ⊂ N be the sublattice spanned by the primitive generators of the raysin Σ. We abbreviate N = N ⊗ Z Z , and we write N / ˜ N for the quotient of N by the image of the map ˜ N → N induced by the inclusion. Recall that N iscanonically isomorphic to the group L ∼ = ( Z ) n contained in L and therefore alsoin X Σ . The following two results have been established by Uma [29, Thm. 2.5 (2),Rem. 7.4] in the special case ˜ N = N . Lemma 4.1. The connected components of X Σ are in bijection with the set N / ˜ N .Proof. We show that each component W ⊂ X Σ contains a point g ∈ L and thatthe assignment W [ g ] ∈ Q = N / ˜ N is well-defined and bijective.This is certainly true if Σ = σ is a single cone, say of dimension m , since X Σ ∼ = R m × ( R × ) n − m and N / ˜ N ∼ = ( Z ) n − m in this case. Note that if τ ⊂ σ isa face and W ⊂ X σ a component, then all points in W ∩ X σ ∩ L have the sameimage in Q .We now turn to the general case. Let σ , σ ∈ Σ, and let W and W be componentsof X σ and X σ , respectively. Assume that W and W intersect and let τ = σ ∩ σ .Then W ∩ W is a union of components of X τ . By what we have said above, allpoints in W ∩ W ∩ L have the same image in Q , hence the same is true for allpoints in W ∩ L and W ∩ L . This proves that the map from the components of X Σ to Q is well-defined. It is also surjective by construction.Let g ∈ L , and let ¯ y ∈ ˜ N ⊂ L be the image of the primitive generator y ∈ ˜ N of some ray ρ in Σ. Then g and g + ¯ y lie in the same component of X ρ , hence alsoin the same component W of X Σ . By repeating this argument, we see that themap W [ g ] is injective. (cid:3) Remark 4.2. Let N = N ∩ ˜ N R be the smallest reduced sublattice of N containingall rays of Σ, and let Σ be the fan Σ, considered as lying in N . Writing Y Σ = X Σ and L Σ = T N/N , we have a canonical inclusion Y Σ , → X Σ and a non-canonicalisomorphism X Σ ∼ = Y Σ × L Σ .Fix an ordering of the, say, m rays in Σ, and define a linear map ˆ N = Z m → N be sending each basis vector to the minimal representative of the corresponding rayin Σ. Let ˆΣ be the subfan of the positive orthant in ˆ N R that is combinatoriallyequivalent to Σ under this map. The associated real toric variety Z Σ = X ˆΣ isthe “real Cox construction”; it is the complement of the real coordinate subspacearrangement defined by Σ, considered as a simplicial complex. (See [3, Thm. 5.4.5]or [15, Sec. 4] for the analogous construction for toric varieties.) By construction,the Cox construction comes with a map of fans ( ˆΣ , ˆ N ) → (Σ , N ), hence with amorphism of real toric varieties Z Σ → X Σ . Let K be the kernel of the associatedmap of real algebraic tori G = T ˆ N → L = T N .The following example illustrates that the real Cox construction behaves differ-ently from the complex case in that it may fail to surject onto X Σ even if the raysin Σ span the vector space N R . Example 4.3. Consider the fan in Z with rays spanned by the vectors y = [1 , y = [ − , 2] (and no 2-dimensional cones). As discussed in [15, Sec. 9], thecorresponding toric variety is X Σ ( C ) = ( C \ { } ) / ± ’ S / ± RP . Thisdescription in fact is the quotient of the Cox construction.The real Cox construction therefore is R \ { } , which consists of two copiesof R × R × ’ D × S that are glued together such that the compact retracts formthe edges of a square. The real toric variety X Σ = X Σ ( R ) itself is also covered bytwo copies of R × R × . Because y and y agree modulo 2, the gluing map betweenthe two copies of R × × R × is the identity, so that the compact retracts form two HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 13 circles. The map Z Σ → X Σ is again the quotient by K = Z , but this time it isnot surjective.The relation between a real toric variety and its Cox construction is as follows.(Our formulation is in fact equally true in the complex setting.) Proposition 4.4. As a topological space, Z Σ is a principal K -bundle, and thereis an L -equivariant homeomorphism X Σ ≈ L × G / K Z Σ / K . Proof. Using Remark 4.2, we can reduce the claim to the case where the rays of Σspan N R , so that N and ˜ N have the same rank. Moreover, as mentioned earlier,the case ˜ N = N of our claim appears in [29, Lemma 7.3, Rem. 7.4]; we have K ∼ = ( R × ) n − m and L = G / K in this case. It therefore suffices to consider the map(4.1) π : ˜ X Σ → X Σ where ˜ X Σ = Z Σ / ˜ K is the real toric variety associated to Σ, considered as a fanin ˜ N . Here ˜ K is the kernel of the quotient map G → ˜ L = T ˜ N .Both the kernel and cokernel of the map ˜ L → L are isomorphic to Q = N / ˜ N ,and its image is ¯ G := L /Q = G / K . Let q be the size of Q . Let us consider therestriction(4.2) ˜ Y σ ˜ X σ ˜ Y σ × ˜ L σ Y σ X σ Y σ × L σ of π for σ ∈ Σ. It is equivariant with respect to the map ˜ L σ → L σ , which againhas kernel Q since the regular cone σ spans the same sublattice in N and ˜ N . Thisimplies that Q freely permutes the connected components of ˜ X σ , and that its orbitsare the non-empty fibres of the map π .Lemma 4.1 tells us that X Σ has q components, while ˜ X Σ is connected, so thatits image lies in the component W ⊂ X Σ containing 1 ∈ L . Fix a σ ∈ Σ, and let r be the number of components of X σ , which is equal to the corresponding numberfor ˜ X σ . By equivariance, each component of X Σ contains the same number r/q of components of X σ . On the other hand, the image of the map π σ also has r/q components. This implies that W is the image of π and therefore the quotientof ˜ X Σ by Q .As a result, we see that Z Σ is a principal K -bundle with base W . The projectionmap Z Σ → W is equivariant with respect to the quotient map G → ¯ G . Since eachof the q components of X Σ is of the form g · W for some g ∈ L , the surjectivemap L × W → X Σ induces the desired L -equivariant homeomorphism. (cid:3) Remark 4.5. Let σ ∈ Σ with corresponding cone ˆ σ ∈ ˆΣ. The toric morphism Z Σ → X Σ sends the G -orbit of Z Σ indexed by ˆ σ to the L -orbit of X Σ indexedby σ . The homeomorphism from Proposition 4.4 therefore is natural in the followingsense: If we form the fan ˆΣ in ˆ N for a given fan Σ, we can use subfans of it in thesame lattice ˆ N to construct the Cox constructions Z Σ for all subfans Σ ⊂ Σ. Bywhat we have just said, the homeomorphism X Σ ≈ L × G / K Z Σ / K then restrictsto a homeomorphism X Σ ≈ L × G / K Z Σ / K . Corollary 4.6. We keep the notation and set K = K ∩ G . Then there is an L -equivariant strong deformation retract L × G/K Z Σ /K , → X Σ , natural in Σ .Proof. We write ¯ G = G/K . As mentioned in the introduction already, the inclusion Z Σ → Z Σ is a G -equivariant strong deformation retract, see [14, Thm. 2.1]. By anargument analogous to [15, Prop. 4.1] this implies that the map Z Σ /K → Z Σ / K is a ¯ G -equivariant strong deformation retract. Hence(4.3) L × ¯ G Z Σ /K , → L × ¯ G Z Σ / K = L × G / K Z Σ / K ≈ X Σ , is an L -equivariant strong deformation retract. In the last step we have used Propo-sition 4.4 and before that the fact that G / K contains the connected component ofthe identity in L . Naturality holds as explained in Remark 4.5. (cid:3) The dga A (Σ) from the introduction is still well-defined for the real toric vari-ety X Σ instead of a real toric space. We keep the ordering of the rays in Σ chosenfor the Cox construction. The coefficient x ij of the characteristic matrix then isthe i -th coordinate (modulo 2) of the primitive generator y j of the j -th ray. Theorem 4.7. The dga A (Σ) is naturally quasi-isomorphic to C ∗ ( X Σ ) .Proof. As in Section 3.1, we replace X Σ by the homotopy-equivalent space(4.4) Y Σ = ( X Σ ) L BL × EL. Writing ¯ G = G/K as in the previous proof, we have(4.5) ( X Σ ) L = EL × L X Σ ’ E ¯ G × ¯ G Z Σ /K ’ DJ Σ by Corollary 4.6 and Lemma 3.3, and we can complete the proof as from Sec-tion 3.1.2 on. (cid:3) Corollary 4.8. Smooth toric varieties are maximal.Proof. As a consequence of Theorem 4.7, Proposition 3.9 carries over to real toricvarieties. Moreover, an analogous result is valid for smooth (complex) toric va-rieties, see [11, Thm. 3.3.2] or [13, Thm. 1.2]. Of course, R and k [Σ] must begraded evenly in this case. Nevertheless, as ungraded vector spaces we have thesame torsion product both for the real and the complex points of a smooth toricvariety. (cid:3) Remark 4.9. The notion of maximality makes sense for any space with an in-volution, in particular for the partial quotient Z Σ ( D , S ) / K of a moment-anglecomplex by a freely acting closed subgroup K of the ambient torus G = ( S ) m .Corollary 4.8 extends to this case if and only if the induced map G → G / K = L restricts to a surjection G → L on the 2-torsion elements. This happens if and onlyif K is of the form ˜ K × Q where ˜ K is a subtorus and Q finite of odd order. HE MOD 2 COHOMOLOGY RINGS OF REAL TORIC SPACES 15 References [1] C. Berger, B. Fresse, Combinatorial operad actions on cochains, Math. Proc. Camb. Philos.Soc. (2004), 135–174; doi:10.1017/S0305004103007138[2] F. Bihan, M. Franz, C. McCrory, J. van Hamel, Is every toric variety an M-variety? Manuscripta Math. 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