P OLYHEDRAL PRODUCTS , TORIC MANIFOLDS , AND TWISTED COHOMOLOGY Alex Suciu Northeastern University Princeton–Rider workshop Homotopy theory and toric spaces February 23, 2012 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 1 / 30
T ORIC MANIFOLDS AND SMALL COVERS T ORIC 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: Vertex set [ m ] = t 1 , . . . , m ✉ . Add a simplex σ = ( i 1 , . . . , i k ) whenever F i 1 , . . . , F i k intersect. F IGURE : A prism P and its dual simplicial complex K A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 2 / 30
T ORIC 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 yields a subtorus T F ( q ) ✕ T k inside T n . To the pair ( P , χ ) , M. Davis and T. Januszkiewicz associate the 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 toric manifold, denoted M P ( χ ) : a closed, orientable manifold of dimension 2 n . For G = Z 2 , this is a real toric manifold (or, small cover ), denoted N P ( χ ) : a closed, not necessarily orientable manifold of dim n . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 3 / 30
T ORIC MANIFOLDS AND SMALL COVERS E XAMPLE � 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 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 4 / 30
T ORIC 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: M = CP 2 ✼ CP 2 is a toric manifold over the square, but it does not admit 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, 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 ). A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 5 / 30
T ORIC MANIFOLDS AND SMALL COVERS In joint work with Alvise Trevisan, we compute H ✝ ( N P ( χ ) , Q ) , both additively and multiplicatively. The (rational) Betti numbers of N P ( χ ) no longer depend just on the h -vector of P , but also on the characteristic matrix χ . E XAMPLE There are precisely two small covers over a square 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. Idea: use finite covers involving (up to homotopy) certain generalized moment-angle complexes: � Z K ( D 1 , S 0 ) � N P ( χ ) , Z m ✁ n 2 � N P ( χ ) � Z K ( RP ✽ , ✝ ) . Z n 2 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 6 / 30
G ENERALIZED MOMENT - ANGLE COMPLEXES G ENERALIZED 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 , ✝ ) . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 7 / 30
G ENERALIZED 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 (M. Davis) π 1 ( Z K ( X , ✝ )) is the graph product of G v = π 1 ( X , ✝ ) along the graph Γ = K ( 1 ) = ( V , E ) , 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. A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 8 / 30
G ENERALIZED 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 ✽ ) . π 1 = t 1 ✉ . 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 ✽ ) . π 1 = W K , the right-angled Coxeter group associated to K ( 1 ) . 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 ) . π 1 = G K , the right-angled Artin group associated to Γ = K ( 1 ) . H ✝ ( Z K ( S 1 ) , Z ) = E / J K , where E = ➍ [ e 1 , . . . , e m ] , deg e i = 1. A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 9 / 30
G ENERALIZED MOMENT - ANGLE COMPLEXES Standard moment-angle complexes Complex moment-angle complex, Z K ( D 2 , S 1 ) ✔ Z K ( ES 1 , S 1 ) . π 1 = π 2 = t 1 ✉ . 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 ) . π 1 = W ✶ K , the derived subgroup of W K . H ✝ ( Z K ( D 1 , S 0 ) , Z 2 ) = Tor R ( R / I K , Z 2 ) — only additively! E XAMPLE Let K be a circuit on 4 vertices. Then: Z K ( D 2 , S 1 ) = S 3 ✂ S 3 Z K ( D 1 , S 0 ) = S 1 ✂ S 1 A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 10 / 30
G ENERALIZED MOMENT - ANGLE COMPLEXES T HEOREM (B AHRI , B ENDERSKY , C OHEN , G ITLER 2010) 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 ] . C OROLLARY 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 ] A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 11 / 30
� � � � � F INITE ABELIAN COVERS F INITE ABELIAN COVERS Let X be a connected, finite-type CW-complex, 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 � 1 . � π 1 ( Y , y 0 ) � π 1 ( X , x 0 ) � 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 π 1 ( X ) Ð Ñ χ p ν X Γ A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 12 / 30
F INITE 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 ; k Γ ) ✕ C q ( r X ) ❜ k π k Γ . Now suppose Γ is finite abelian, k = k , and char k = 0. Then, all k -irreps of Γ are 1-dimensional, and so à C q ( X ν ; k ) ✕ C q ( X ; k ρ ✆ ν ) , ρ P Hom ( Γ , k ✂ ) where k ρ ✆ ν denotes the field k , viewed as a k π -module via the character ρ ✆ ν : π Ñ k ✂ . Thus, H q ( X ν ; k ) ✕ ➚ ρ P Hom ( Γ , k ✂ ) H q ( X ; k ρ ✆ ν ) . A LEX S UCIU (N ORTHEASTERN ) P OLYHEDRAL PRODUCTS P RINCETON –R IDER 2012 13 / 30
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