The rational cohomology of real quasi-toric manifolds

Transcription

1 The rational cohomology of real quasi-toric manifolds Alex Suciu Northeastern University Joint work with Alvise Trevisan (VU Amsterdam) Toric Methods in Homotopy Theory Queen s University Belfast July 20, 2011 Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

2 Quasi-toric manifolds and small covers Quasi-toric manifolds and small covers Let P be an n-dimensional convex polytope; facets F 1,..., F m. Assume P is simple (each vertex is the intersection of n facets). Then P determines a dual simplicial complex, K = K BP, of dimension n 1: Ÿ Vertex set [m] = t1,..., mu. Ÿ Add a simplex σ = (i 1,..., i k ) whenever F i1,..., F ik intersect. Figure: A prism P and its dual simplicial complex K Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

3 Quasi-toric manifolds and small covers Let χ be an n-by-m matrix with coefficients in G = Z or Z 2. χ is characteristic for P if, for each vertex v = F i1 X X F in, the n-by-n minor given by the columns i 1,..., i n of χ is unimodular. Let T = S 1 if G = Z, and T = S 0 = t 1u if G = Z 2. Given q P P, let F (q) = F j1 X X F jk be the maximal face so that q P F (q). The map χ associates to F (q) a subtorus T F (q) T k inside T n. To the pair (P, χ), Davis and Januszkiewicz associate the quasi-toric manifold T n P/, where (t, p) (u, q) if p = q and t u 1 P T F (q). For G = Z, this is a complex q-tm, denoted M P (χ) Ÿ a closed, orientable manifold of dimension 2n. For G = Z 2, this is a real q-tm (or, small cover), denoted N P (χ) Ÿ a closed, not necessarily orientable manifold of dimension n. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

4 Quasi-toric manifolds and small covers Example Let P = n be the n-simplex, and χ the n (n + 1) matrix Then M P (χ) = CP n and N P (χ) = RP n. ( ) P T P T P/ CP 1 RP 1 Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

5 Quasi-toric manifolds and small covers More generally, if X is a smooth, projective toric variety, then X (C) = M P (χ) and X (R) = N P (χ mod 2Z). But the converse does not hold: Ÿ Ÿ M = CP 2 7CP 2 is a quasi-toric manifold over the square, but it does not admit any 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, N P (χ) is a hyperbolic 3-manifold. (Characteristic χ exist for P = dodecahedron, by work of Garrison and Scott.) Then, by a theorem of Delaunay, N P (χ) X (R). Davis and Januszkiewicz found presentations for the cohomology rings H (M P (χ), Z) and H (N P (χ), Z 2 ), similar to the ones given by Danilov and Jurkiewicz for toric varieties. In particular, dim Q H 2i (M P (χ), Q) = dim Z2 H i (N P (χ), Z 2 ) = h i (P), where (h 0 (P),..., h n (P)) is the h-vector of P, depending only on the number of i-faces of P (0 i n). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

6 Quasi-toric manifolds and small covers Our goal is to compute H (N P (χ), Q), both additively and multiplicatively. The Betti numbers of N P (χ) no longer depend just on the h-vector of P, but also on the characteristic matrix χ. Example Let P be the square (with n = 2, m = 4). There are precisely two small covers over P: The torus T 2 = N P (χ), with χ = ( ). The Klein bottle Kl = N P (χ 1 ), with χ 1 = ( ). Then b 1 (T 2 ) = 2, yet b 1 (Kl) = 1. Key ingredient in our approach: use finite covers involving (up to homotopy) certain generalized moment-angle complexes: Z m n 2 Z n 2 Z K (S 1, S 0 ) N P (χ), N P (χ) Z K (RP 8, ). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

7 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 Z K (X, A) = σpk(x, A) σ X m where (X, A) σ = tx P X m x i P A if i R σu. Construction interpolates between A m and X m. Homotopy invariance: (X, A) (X 1, A 1 ) ùñ Z K (X, A) Z K (X 1, A 1 ). Converts simplicial joins to direct products: Z K L (X, A) Z K (X, A) Z L (X, A). Takes a cellular pair (X, A) to a cellular subcomplex of X m. Particular case: Z K (X ) := Z K (X, ). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

8 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 Z K (f ) : Z K (X, A) Ñ Z K (Y, B). Let f : (X, ) ãñ (Y, ) be a cellular inclusion. Then, Z K (f ) : C q (Z K (X )) ãñ C q (Z K (Y )) admits a 0. Let φ : K ãñ L be the inclusion of a full subcomplex. Then there are induced maps Z φ : Z L (X, A) Z K (X, A) and Z φ : Z K (X, A) ãñ Z L (X, A), such that Z φ Z φ = id. Fundamental group and asphericity (Davis) π 1 (Z K (X, )) is the graph product of G v = π 1 (X, ) along the graph Γ = K (1), where Prod Γ (G v ) = G v /t[g v, g w ] = 1 if tv, wu P E, g v P G v, g w P G w u. vpv Suppose X is aspherical. Then Z K (X ) is aspherical iff K is a flag complex. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

9 Generalized moment-angle complexes Generalized Davis Januszkiewicz spaces G abelian topological group G GDJ space Z K (BG). We have a bundle G m Ñ Z K (EG, G) Ñ Z K (BG). If G is a finitely generated (discrete) abelian group, then π 1 (Z K (BG)) ab = G m, and thus Z K (EG, G) is the universal abelian cover of Z K (BG). G = S 1 : Usual Davis Januszkiewicz space, Z K (CP 8 ). Ÿ π 1 = t1u. Ÿ H (Z K (CP 8 ), Z) = S/I K, where S = Z[x 1,..., x m ], deg x i = 2. G = Z 2 : Real Davis Januszkiewicz space, Z K (RP 8 ). Ÿ π 1 = W K, the right-angled Coxeter group associated to K (1). Ÿ H (Z K (RP 8 ), Z 2 ) = R/I K, where R = Z 2 [x 1,..., x m ], deg x i = 1. G = Z: Toric complex, Z K (S 1 ). Ÿ π 1 = A K, the right-angled Artin group associated to K (1). Ÿ H (Z K (S 1 ), Z) = E/J K, where E = [e 1,..., e m ], deg e i = 1. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

10 Generalized moment-angle complexes Standard moment-angle complexes Complex moment-angle complex, Z K (D 2, S 1 ) Z K (ES 1, S 1 ). Ÿ π 1 = π 2 = t1u. Ÿ H (Z K (D 2, S 1 ), Z) = Tor S (S/I K, Z). Real moment-angle complex, Z K (D 1, S 0 ) Z K (EZ 2, Z 2 ). Ÿ π 1 = WK 1, the derived subgroup of W K. Ÿ H (Z K (D 1, S 0 ), Z 2 ) = Tor R (R/I K, Z 2 ) only additively! Example Let K be a circuit on 4 vertices. Then Z K (D 2, S 1 ) = S 3 S 3, while Z K (D 1, S 0 ) = S 1 S 1 (embedded in the 4-cube). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

11 Generalized moment-angle complexes Theorem (Bahri, Bendersky, Cohen, Gitler) Let K a simplicial complex on m vertices. There is a natural homotopy equivalence ( Σ(Z K (X, A)) Σ ª ) pz KI (X, A), I [m] where K I 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 à (p 1) H k (Z K à I (X, A); R) rh k 1 (K I ; R). I [m] 1 Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

12 Finite abelian covers Finite abelian covers Let X be a connected, finite-type CW-complex, with π = π 1 (X, x 0 ). Let p : Y Ñ X a (connected) regular cover, with group of deck transformations Γ. We then have a short exact sequence 1 π 1 (Y, y 0 ) p 7 π 1 (X, x 0 ) ν Γ 1. Conversely, every epimorphism ν : π Γ defines a regular cover X ν Ñ X (unique up to equivalence), with π 1 (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 ÐÑ π 1 (X ) ab π 1 (X ) ab p X ν χ Γ Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

13 Finite abelian covers Let C q (X ν ; k) be the group of cellular q-chains on X ν, with coefficients in a field k. We then have natural isomorphisms C q (X ν ; k) C q (X; kγ) C q (r X ) bkπ kγ. Now suppose Γ is finite abelian, k = k, and char k = 0. Then, all k-irreps of Γ are 1-dimensional, and so C q (X ν ; k) à ρphom(γ,k ) C q (X; k ρ ν ), where k ρ ν denotes the field k, viewed as a kπ-module via the character ρ ν : π Ñ k. Thus, H q (X ν ; k) À ρphom(γ,k ) H q(x; k ρ ν ). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

14 Finite abelian covers Now let P be an n-dimensional, simple polytope with m facets, and set K = K BP. Let χ : Z m 2 Ñ Zn 2 be a characteristic matrix for P. Then ker(χ) Z m n 2 acts freely on Z K (D 1, S 0 ), with quotient the real quasi-toric manifold N P (χ). N P (χ) comes equipped with an action of Z m 2 / ker(χ) Zn 2 ; the orbit space is P. Furthermore, Z K (D 1, S 0 ) is homotopy equivalent to the maximal abelian cover of Z K (RP 8 ), corresponding to the sequence 1 W 1 K W K ab Z m 2 1. Thus, N P (χ) is, up to homotopy, a regular Z n 2 -cover of Z K (RP 8 ), corresponding to the sequence 1 π 1 (N P (χ)) W K ν=χ ab Z n 2 1. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

15 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 π 1 (Z K (BZ p )) ab = Z m p, generated by x 1,..., x m. Let λ : Z m p Ñ k be a character, supp(λ) = ti P [m] λ(x i ) 1u. Let K λ be the induced subcomplex on vertex set supp(λ). Proposition H q (Z K (BZ p ); k λ ) r Hq 1 (K λ ; k). Idea: The inclusion ι : (S 1, ) ãñ (BZ p, ) induces a cellular inclusion Z K (ι) : T K = Z K (S 1 ) ãñ Z K (BZ p ). Moreover, φ : K λ ãñ K induces a cellular inclusion Z φ : T Kλ ãñ T K. Let λ : Z m Z m λ p ÝÑ k. We then get (chain) retractions C q (T K ; k λ ) C q (Z K (BZ p ); k λ) C q (T Kλ ; k λ ) r Cq 1 (K λ ; k) Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

16 The homology of abelian covers of GDJ spaces This shows that dim k H q (Z K (BZ p ); k λ ) dim k Hq 1 r (K λ ; k). For the reverse inequality, we use [BBCG], à which, in this case, says (p 1) H q (Z K (EZ p, Z p à I ); k) rh q 1 (K I ; k), I [m] and the fact that Z K (EZ p, Z p ) (Z K (BZ p )) ab, which gives H q (Z K (EZ p, Z p ); k) à 1 ρphom(z m p,k ) H q (Z K (BZ p ); k ρ ). Theorem Let G be a prime-order cyclic group, and let Z K (BG) χ be the abelian cover defined by an epimorphism χ : G m Γ. Then H q (Z K (BG) χ ; k) à ρphom(γ;k ) rh q 1 (K ρ χ ; k), where K ρ χ is the induced subcomplex of K on vertex set supp(ρ χ). Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

17 The homology of real quasi-toric manifolds The homology of real quasi-toric manifolds Let again P be a simple polytope, and set K = K BP. Let χ : Z m 2 Ñ Zn 2 be a characteristic matrix for P. Denote by χ i P Z m 2 the i-th row of χ. For each subset S [n], write χ S = ips χ i P Z m 2. S also determines a character ρ S : Z n 2 Ñ k, taking the i-th generator to 1 if i P S, and to 1 if i R S. Every ρ P Hom(Z n 2, C ) arises as ρ = ρ S, where S = supp(ρ). supp(ρ S χ) consists of those j P [m] for which the j-th entry of χ S is non-zero. Let K χ,s be the induced subcomplex on this vertex set. Corollary The Betti numbers of the real, quasi-toric manifold N P (χ) are given by b q (N P (χ)) = b q 1 (K χ,s ). S [n] Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

18 The homology of real quasi-toric manifolds Example Again, let P be the square, and K = K BP the 4-cycle. Let T 2 = N P (χ), where χ = ( ). Compute: S H t1u t2u t1, 2u χ S ( ) ( ) ( ) ( ) supp(χ S ) H t1, 3u t2, 4u t1, 2, 3, 4u K χ,s H tt1u, t3uu tt2u, t4uu K Thus: b 0 (T 2 ) = b 1 (H) = 1, b 1 (T 2 ) = b 0 (K χ,t1u ) + b 0 (K χ,t2u ) = = 2, b 2 (T 2 ) = b 1 (K ) = 1. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

19 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 : π 1 (X ) Γ, and let k = C. The cellular cochains on X ν decompose as C q (X ν ; k) à ρphom(γ,k ) C q (X; k ρ ν ), The cup product map, C p (X ν, k) b k C q (X ν, k)! ÝÝÑ C p+q (X ν, k), restricts to those pieces, as follows: C p (X; k ρ ν ) b k C q (X; k ρ 1 ν) C p+q (X X; k ρ ν b k k ρ 1 ν)! C p+q (X; k (ρ ρ1 ) ν) µ C p+q (X X; k (ρbρ1 ) 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 University) Rational cohomology of real toric manifolds Toric Methods, July / 22

20 Cup products in abelian covers of GDJ-spaces Proposition Let Z K (BZ p ) ν be a regular abelian cover, with characteristic homomorphism χ : Z m p Ñ Γ. The cup product in H (Z K (BG) ν ; k) 8à q=0 à ρphom(γ;k ) rh q 1 (K ρ χ ; k) is induced by the following maps on simplicial cochains: rc p 1( K ρ χ ; k ) b r C q 1 ( K ρ 1 χ; k ) Ñ r C p+q 1 ( K (ρbρ1 ) χ; k ) ˆσ b ˆτ ÞÑ # {σ \ τ if σ X τ = H, 0 otherwise, where σ \ τ is the simplex with vertex set the union of the vertex sets of σ and τ, and ˆσ is the Kronecker dual of σ. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

21 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 Z K (X ) is formal. In particular, toric complexes T K = Z K (S 1 ) and generalized Davis Januszkiewicz spaces Z K (BG) are always formal. (Félix, Tanré) More generally, if both X and A are formal, and the inclusion i : A ãñ X induces a surjection i : H (X, Q) Ñ H (A, Q), then Z K (X, A) is formal. (Panov Ray) Complex quasi-toric manifolds M P (χ) are always formal. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22

22 Formality properties (Baskakov, Denham A.S.) Moment angle complexes Z K (D 2, S 1 ) are not always formal: they can have non-zero Massey products. Example (D-S) A simplicial complex K for which Z K (D 2, S 1 ) carries a non-trivial triple Massey product. It follows that real moment-angle complexes Z K (D 1, S 0 ) are not always formal. Question: are real quasi-toric manifolds N P (χ) formal? Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July / 22