Quasitoric manifold

Definitions

Denote the i -th subcircle of the n -torus T n by T i so that T 1 × … × T n = T n . Then coordinate-wise multiplication of T n on C n is called the standard representation.

Given open sets X in M 2 n and Y in C n , that are closed under the action of T n , a T n -action on M 2 n is defined to be locally isomorphic to the standard representation if h ( t x ) = α ( t ) h ( x ) , for all t in T n , x in X , where h is a homeomorphism X → Y , and α is an automorphism of T n .

Given a simple convex polytope P n with m facets, a T n -manifold M 2 n is a quasitoric manifold over P n if,

1. the T n -action is locally isomorphic to the standard representation,
2. there is a projection π : M 2 n → P n that maps each l -dimensional orbit to a point in the interior of an l -dimensional face of P n , for l = 0 , . . . , n .

The definition implies that the fixed points of M 2 n under the T n -action are mapped to the vertices of P n by π , while points where the action is free project to the interior of the polytope.

The dicharacteristic function

A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix. In this setting it is useful to assume that the facets F 1 , … , F m of P n are ordered so that the intersection F 1 ∩ ⋯ ∩ F n is a vertex v of P n , called the initial vertex.

A dicharacteristic function is a homomorphism λ : T m → T n , such that if F i 1 ∩ ⋯ ∩ F i k is a codimension- k face of P n , then λ is a monomorphism on restriction to the subtorus T i 1 × ⋯ × T i k in T m .

The restriction of λ to the subtorus T 1 × … × T n corresponding to the initial vertex v is an isomorphism, and so λ ( T 1 ) , … , λ ( T n ) can be taken to be a basis for the Lie algebra of T n . The epimorphism of Lie algebras associated to λ may be described as a linear transformation Z m → Z n , represented by the n × m dicharacteristic matrix Λ given by

The i th column of Λ is a primitive vector λ i = ( λ 1 , i , … , λ n , i ) in Z n , called the facet vector. As each facet vector is primitive, whenever the facets F i 1 ∩ ⋯ ∩ F i n meet in a vertex, the corresponding columns λ i 1 , … λ i n form a basis of Z n , with determinant equal to ± 1 . The isotropy subgroup associated to each facet F i is described by

for some θ in R .

In their original treatment of quasitoric manifolds, Davis and Januskiewicz introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle λ ( T i ) be oriented, forcing a choice of sign for each vector λ i . The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix Λ as ( I n ∣ S ) , where I n is the identity matrix and S is an n × ( m − n ) submatrix.

Relation to the moment-angle complex

The kernel K ( λ ) of the dicharacteristic function acts freely on the moment angle complex Z P n , and so defines a principal K ( λ ) -bundle Z P n → M 2 n over the resulting quotient space M 2 n . This quotient space can be viewed as

where pairs ( t 1 , p 1 ) , ( t 2 , p 2 ) of T n × P n are identified if and only if p 1 = p 2 and t 1 − 1 t 2 is in the image of λ on restriction to the subtorus T i 1 × ⋯ × T i k that corresponds to the unique face F i 1 ∩ ⋯ ∩ F i k of P n containing the point p 1 , for some 1 ≤ k ≤ n .

It can be shown that any quasitoric manifold M 2 n over P n is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.

Examples

The moment angle complex Z Δ n is the ( 2 n + 1 ) -sphere S 2 n + 1 , the kernel K ( λ ) is the diagonal subgroup { ( t , … , t ) } < T n + 1 , so the quotient of Z Δ n under the action of K ( λ ) is C P n .

for integers a ( i , j ) .

The moment angle complex Z I n is a product of n copies of 3-sphere embedded in C 2 n , the kernel K ( λ ) is given by

so that the quotient of Z I n under the action of K ( λ ) is the n -th stage of a Bott tower. The integer values a ( i , j ) are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.

The cohomology ring of a quasitoric manifold

Canonical complex line bundles ρ i over M 2 n given by

can be associated with each facet F i of P n , for 1 ≤ i ≤ m , where K ( λ ) acts on C i , by the restriction of K ( λ ) to the i -th subcircle of T m embedded in C . These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of M 2 n , the preimage of a facet π − 1 ( F i ) is a 2 ( n − 1 ) -dimensional quasitoric facial submanifold M i over F i , whose isotropy subgroup is the restriction of λ on the subcircle T i of T m . Restriction of ρ i to M i gives the normal 2-plane bundle of the embedding of M i in M 2 n .

Let x i in H 2 ( M 2 n ; Z ) denote the first Chern class of ρ i . The integral cohomology ring H ∗ ( M 2 n ; Z ) is generated by x i , for 1 ≤ i ≤ m , subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of P n ; linear relations determined by the dicharacterstic function comprise the second set:

Therefore only x n + 1 , … , x m are required to generate H ∗ ( M 2 n ; Z ) multiplicatively.

Comparison with toric manifolds