In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth 2 n {\displaystyle 2n} -dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an n {\displaystyle n} -dimensional torus, with orbit space an n {\displaystyle n} -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 i {\displaystyle i} -th subcircle of the n {\displaystyle n} -torus T n {\displaystyle T^{n}} by T i {\displaystyle T_{i}} so that T 1 × … × T n = T n {\displaystyle T_{1}\times \ldots \times T_{n}=T^{n}} . Then coordinate-wise multiplication of T n {\displaystyle T^{n}} on C n {\displaystyle \mathbb {C} ^{n}} is called the standard representation. Given open sets X {\displaystyle X} in M 2 n {\displaystyle M^{2n}} and Y {\displaystyle Y} in C n {\displaystyle \mathbb {C} ^{n}} , that are closed under the action of T n {\displaystyle T^{n}} , a T n {\displaystyle T^{n}} -action on M 2 n {\displaystyle M^{2n}} is defined to be locally isomorphic to the standard representation if h ( t x ) = α ( t ) h ( x ) {\displaystyle h(tx)=\alpha (t)h(x)} , for all t {\displaystyle t} in T n {\displaystyle T^{n}} , x {\displaystyle x} in X {\displaystyle X} , where h {\displaystyle h} is a homeomorphism X → Y {\displaystyle X\rightarrow Y} , and α {\displaystyle \alpha } is an automorphism of T n {\displaystyle T^{n}} . Given a simple convex polytope P n {\displaystyle P^{n}} with m {\displaystyle m} facets, a T n {\displaystyle T^{n}} -manifold M 2 n {\displaystyle M^{2n}} is a quasitoric manifold over P n {\displaystyle P^{n}} if,
the T n {\displaystyle T^{n}} -action is locally isomorphic to the standard representation, there is a projection π : M 2 n → P n {\displaystyle \pi :M^{2n}\rightarrow P^{n}} that maps each l {\displaystyle l} -dimensional orbit to a point in the interior of an l {\displaystyle l} -dimensional face of P n {\displaystyle P^{n}} , for l = 0 , {\displaystyle l=0,} . . . , {\displaystyle ...,} n {\displaystyle n} . The definition implies that the fixed points of M 2 n {\displaystyle M^{2n}} under the T n {\displaystyle T^{n}} -action are mapped to the vertices of P n {\displaystyle P^{n}} by π {\displaystyle \pi } , while points where the action is free project to the interior of the polytope.
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