Quasitoric manifold
In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an -dimensional torus, with orbit space an -dimensional simple convex polytope.

Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz, who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.

Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.

Definitions
Denote the -th subcircle of the -torus  by  so that . Then coordinate-wise multiplication of  on  is called the standard representation.

Given open sets  in  and  in , that are closed under the action of , a -action on  is defined to be locally isomorphic to the standard representation if , for all  in ,  in , where  is a homeomorphism , and  is an automorphism of .

Given a simple convex polytope  with  facets, a -manifold  is a quasitoric manifold over  if,

 the -action is locally isomorphic to the standard representation,
 there is a projection  that maps each -dimensional orbit to a point in the interior of an -dimensional face of , for    .

The definition implies that the fixed points of  under the -action are mapped to the vertices of  by , 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  of  are ordered so that the intersection  is a vertex  of , called the initial vertex.

A dicharacteristic function is a homomorphism , such that if  is a codimension- face of , then  is a monomorphism on restriction to the subtorus  in .

The restriction of λ to the subtorus  corresponding to the initial vertex  is an isomorphism, and so  can be taken to be a basis for the Lie algebra of . The epimorphism of Lie algebras associated to λ may be described as a linear transformation , represented by the  dicharacteristic matrix  given by

The th column of  is a primitive vector  in , called the facet vector.  As each facet vector is primitive, whenever the facets  meet in a vertex, the corresponding columns  form a basis of , with determinant equal to .  The isotropy subgroup associated to each facet  is described by

for some  in .

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  be oriented, forcing a choice of sign for each vector . 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  as , where  is the identity matrix and  is an  submatrix.

Relation to the moment-angle complex
The kernel  of the dicharacteristic function acts freely on the moment angle complex , and so defines a principal -bundle  over the resulting quotient space .  This quotient space can be viewed as

 

where pairs ,  of  are identified if and only if  and  is in the image of  on restriction to the subtorus  that corresponds to the unique face  of  containing the point , for some .

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

Examples
 The -dimensional complex projective space  is a quasitoric manifold over the -simplex . If  is embedded in  so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is

The moment angle complex  is the -sphere , the kernel  is the diagonal subgroup , so the quotient of  under the action of  is .

 The Bott manifolds that form the stages in a Bott tower are quasitoric manifolds over -cubes. The -cube  is embedded in  so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix  has  given by

for integers .

The moment angle complex  is a product of  copies of 3-sphere embedded in , the kernel  is given by

,

so that the quotient of  under the action of  is the -th stage of a Bott tower. The integer values  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  over  given by

 ,

can be associated with each facet  of , for , where  acts on , by the restriction of  to the -th subcircle of  embedded in .  These bundles are known as the facial bundles associated to the quasitoric manifold.  By the definition of , the preimage of a facet  is a -dimensional quasitoric facial submanifold  over , whose isotropy subgroup is the restriction of  on the subcircle  of .  Restriction of  to  gives the normal 2-plane bundle of the embedding of  in .

Let  in  denote the first Chern class of . The integral cohomology ring  is generated by , for , subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of ; linear relations determined by the dicharacterstic function comprise the second set:

 .

Therefore only  are required to generate  multiplicatively.

Comparison with toric manifolds
 Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.
 Not all quasitoric manifolds are toric manifolds. For example, the connected sum  can be constructed as a quasitoric manifold, but it is not a toric manifold.

Notes

References

Algebraic topology
Topology
Manifolds