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 2011 1 / 22
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 ❇ P , of dimension n ✁ 1: ➓ Vertex set [ m ] = t 1 , . . . , m ✉ . ➓ Add a simplex σ = ( i 1 , . . . , i k ) whenever F i 1 , . . . , F i k intersect. Figure: A prism P and its dual simplicial complex K Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 2 / 22
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 i 1 ❳ ☎ ☎ ☎ ❳ F i n , 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✟ 1 ✉ if G = Z 2 . Given q P P , let F ( q ) = F j 1 ❳ ☎ ☎ ☎ ❳ F j k 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 ) ➓ a closed, orientable manifold of dimension 2 n . For G = Z 2 , this is a real q-tm (or, small cover ), denoted N P ( χ ) ➓ a closed, not necessarily orientable manifold of dimension n . Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 3 / 22
Quasi-toric manifolds and small covers Example � 1 ☎☎☎ 0 1 � . Let P = ∆ n be the n -simplex, and χ the n ✂ ( n + 1 ) matrix ... . . . 0 ☎☎☎ 1 1 Then M P ( χ ) = CP n N P ( χ ) = RP n . and P T ✂ P T ✂ P / ✒ CP 1 RP 1 Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 4 / 22
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 2 Z ) . But the converse does not hold: ➓ M = CP 2 ✼ CP 2 is a quasi-toric manifold over the square, but it does not admit any complex structure. Thus, M ✢ X ( C ) . ➓ 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 2 i ( M P ( χ ) , Q ) = dim Z 2 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 2011 5 / 22
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 : � 1 0 1 0 The torus T 2 = N P ( χ ) , with χ = � . 0 1 0 1 � 1 0 1 0 The Klein bottle K ℓ = N P ( χ ✶ ) , with χ ✶ = � . 0 1 1 1 Then b 1 ( T 2 ) = 2, yet b 1 ( K ℓ ) = 1. Key ingredient in our approach: use finite covers involving (up to homotopy) certain generalized moment-angle complexes: Z m ✁ n � Z K ( S 1 , S 0 ) � N P ( χ ) , 2 � N P ( χ ) Z n � Z K ( RP ✽ , ✝ ) . 2 Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 6 / 22
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 ↕ ( X , A ) σ ⑨ X ✂ m Z K ( X , A ) = σ P K where ( X , A ) σ = t x P X ✂ m ⑤ x i P A if i ❘ σ ✉ . Construction interpolates between A ✂ m and X ✂ m . Homotopy invariance: ( X , A ) ✔ ( X ✶ , A ✶ ) ù ñ Z K ( X , A ) ✔ Z K ( X ✶ , A ✶ ) . 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 2011 7 / 22
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 , ✝ ) ã Ñ ( Y , ✝ ) be a cellular inclusion. Then, Z K ( f ) ✝ : C q ( Z K ( X )) ã Ñ C q ( Z K ( Y )) admits a retraction, ❅ q ➙ 0. Let φ : K ã Ñ L be the inclusion of a full subcomplex. Then there are induced maps Z φ : Z L ( X , A ) ։ Z K ( X , A ) and Ñ Z L ( X , A ) , such that Z φ ✆ Z φ = id. Z φ : Z K ( X , A ) ã 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 ) = ✝ v P V G v / t [ g v , g w ] = 1 if t v , w ✉ P E , g v P G v , g w P G w ✉ . 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 2011 8 / 22
Generalized moment-angle complexes Generalized Davis–Januszkiewicz spaces GDJ space Z K ( BG ) . G abelian topological group G � 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 ✽ ) . ➓ π 1 = t 1 ✉ . ➓ H ✝ ( Z K ( CP ✽ ) , 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 ✽ ) . ➓ π 1 = W K , the right-angled Coxeter group associated to K ( 1 ) . ➓ H ✝ ( Z K ( RP ✽ ) , 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 ) . ➓ π 1 = A K , the right-angled Artin group associated to K ( 1 ) . ➓ H ✝ ( Z K ( S 1 ) , Z ) = E / J K , where E = ➍ [ e 1 , . . . , e m ] , deg e i = 1. Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 9 / 22
Generalized moment-angle complexes Standard moment-angle complexes Complex moment-angle complex, Z K ( D 2 , S 1 ) ✔ Z K ( ES 1 , S 1 ) . ➓ π 1 = π 2 = t 1 ✉ . ➓ 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 ( E Z 2 , Z 2 ) . ➓ π 1 = W ✶ K , the derived subgroup of W K . ➓ 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 2011 10 / 22
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 Z K I ( X , A ) , I ⑨ [ m ] where K I 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 ( p ✁ 1 ) ⑤ I ⑤ à à r H k ( Z K ( X , A ) ; R ) ✕ H k ✁ 1 ( K I ; R ) . 1 I ⑨ [ m ] Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 11 / 22
� � � � � 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 p ✼ ν � π 1 ( Y , y 0 ) � π 1 ( X , x 0 ) � 1 . � 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 : ab � π 1 ( X ) ab X ab X ν Ð Ñ π 1 ( X ) χ p ν X Γ Alex Suciu (Northeastern University) Rational cohomology of real toric manifolds Toric Methods, July 2011 12 / 22
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