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Combinatorial formulae for the χy-genus of a multioriented quasitoric manifold

By T. E. Panov


In [1] the authors introduced the class of smooth manifolds with a compact torus action whose orbit space carries combinatorial structure of a simple polytope. Following [2], [3], we call these manifolds quasitoric. The name refers to the fact that topological and combinatorial properties of quasitoric manifolds are similar to that of non-singular algebraic toric varieties [4] (or toric manifolds). Any quasitoric manifold is defined by combinatorial data: the lattice of faces of a simple polytope and a characteristic function that assigns an integer primitive vector defined up to sign to each facet. Despite their simple and specific definition, quasitoric manifolds in many cases may serve as model examples (for instance, each complex cobordism class contains a quasitoric manifold [5]). All these facts enable to use quasitoric manifolds for solving topological problems by combinatorial methods and vice versa. Some applications were obtained in [2], [6], where quasitoric manifolds are studied in the general context of “manifolds defined by simple polytopes”. Unlike toric varieties, quasitoric manifolds may fail to be complex; however, they always admit a stably (or weakly almost) complex structure. As it was shown in [3]

Year: 1999
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