Quasitoric manifold

In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth ${\displaystyle 2n}$-dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an ${\displaystyle n}$-dimensional torus, with orbit space an ${\displaystyle n}$-dimensional simple convex polytope.

Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,^[1] 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.^[2]

Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.^[3]