Quasitoric manifold

In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth formula_1-dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an formula_2-dimensional torus, with orbit space an formula_2-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 formula_4-th subcircle of the formula_2-torus formula_6 by formula_7 so that formula_8. Then coordinate-wise multiplication of formula_6 on formula_10 is called the standard representation.
Given open sets formula_11 in formula_12 and formula_13 in formula_10, that are closed under the action of formula_6, a formula_16-action on formula_12 is defined to be locally isomorphic to the standard representation if formula_18, for all formula_19 in formula_6, formula_21 in formula_11, where formula_23 is a homeomorphism formula_24, and formula_25 is an automorphism of formula_6.
Given a simple convex polytope formula_27 with formula_28 facets, a formula_6-manifold formula_12 is a quasitoric manifold over formula_27 if,
The definition implies that the fixed points of formula_12 under the formula_6-action are mapped to the vertices of formula_27 by formula_43, 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 formula_44 of formula_27 are ordered so that the intersection formula_46 is a vertex formula_47 of formula_48, called the initial vertex.
A dicharacteristic function is a homomorphism formula_49, such that if formula_50 is a codimension-formula_51 face of formula_27, then formula_53 is a monomorphism on restriction to the subtorus formula_54 in formula_55.
The restriction of λ to the subtorus formula_56 corresponding to the initial vertex formula_47 is an isomorphism, and so formula_58 can be taken to be a basis for the Lie algebra of formula_6. The epimorphism of Lie algebras associated to λ may be described as a linear transformation formula_60, represented by the formula_61 dicharacteristic matrix formula_62 given by
The formula_4th column of formula_62 is a primitive vector formula_66 in formula_67, called the facet vector. As each facet vector is primitive, whenever the facets formula_68 meet in a vertex, the corresponding columns formula_69 form a basis of formula_67, with determinant equal to formula_71. The isotropy subgroup associated to each facet formula_72 is described by
for some formula_74 in formula_75.
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 formula_76 be oriented, forcing a choice of sign for each vector formula_77. 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 formula_62 as formula_79, where formula_80 is the identity matrix and formula_81 is an formula_82 submatrix.
Relation to the moment-angle complex.
The kernel formula_83 of the dicharacteristic function acts freely on the moment angle complex formula_84, and so defines a principal formula_83-bundle formula_86 over the resulting quotient space formula_12. This quotient space can be viewed as
where pairs formula_89, formula_90 of formula_91 are identified if and only if formula_92 and formula_93 is in the image of formula_53 on restriction to the subtorus formula_95 that corresponds to the unique face formula_96 of formula_27 containing the point formula_98, for some formula_99.
It can be shown that any quasitoric manifold formula_12 over formula_27 is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.
Examples.
The moment angle complex formula_109 is the formula_110-sphere formula_111, the kernel formula_83 is the diagonal subgroup formula_113, so the quotient of formula_109 under the action of formula_83 is formula_103.
for integers formula_124.
The moment angle complex formula_125 is a product of formula_2 copies of 3-sphere embedded in formula_127, the kernel formula_83 is given by
so that the quotient of formula_125 under the action of formula_83 is the formula_2-th stage of a Bott tower. The integer values formula_124 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 formula_134 over formula_12 given by
can be associated with each facet formula_72 of formula_27, for formula_139, where formula_83 acts on formula_141, by the restriction of formula_83 to the formula_4-th subcircle of formula_55 embedded in formula_145. These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of formula_12, the preimage of a facet formula_147 is a formula_148-dimensional quasitoric facial submanifold formula_149 over formula_72, whose isotropy subgroup is the restriction of formula_53 on the subcircle formula_7 of formula_55. Restriction of formula_134 to formula_149 gives the normal 2-plane bundle of the embedding of formula_149 in formula_12.
Let formula_158 in formula_159 denote the first Chern class of formula_134. The integral cohomology ring formula_161 is generated by formula_158, for formula_139, subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of formula_27; linear relations determined by the dicharacterstic function comprise the second set:
Therefore only formula_166 are required to generate formula_161 multiplicatively.