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

Abstract

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], a stably complex structure (i.e. a complex structure in the stable tangent bundle) on a quasitoric manifold is also defined combinatorially, namely, by specifying an orientation of the simple polytope and choosing signs for vectors given by the characteristic function. A quasitoric manifold with such additional structure was called in [3] multioriented. Hence, a multioriented quasitoric manifold determines a complex cobordism class, for which one can define characteristic numbers and complex Hirzebruch genera. We calculate the χy-genus (in particular, the signature and the Todd genus) in terms of the combinatorial data by applying the Atiyah–Hirzebruch formula to one specific circle action with isolated fixed points. A convex n-dimensional polytope P is called simple if the number of codimension-one faces (or facets) meeting at each vertex is n. Any simple polytope is a manifold with corners. Let M be a compact 2n-dimensional manifold with an action of the torus T = {(e2πiφ1 , . . . , e2πiφn) ∈ Cn}, φi ∈ R. Then M is called a quasitoric manifold if the T-action is locally isomorphic to the standard action of T on C by diagonal matrices, while the orbit space is diffeomorphic, as manifold with corners, to a simple polytope P (see [1], [3] for the details). Let M be a quasitoric manifold with orbit space P, and let F = {F1, . . . , Fm} be the set of facets of P, m = ]F . The interior of facet Fi consists of orbits with the same one-dimensional isotropy subgroup GFi = { ( e1i, . . . , eni ) ∈ Tn}, φ ∈ R. In this way one defines a characteristic function λ : F → Z that takes facet Fi to the primitive vector λi = (λ1i, . . . , λni)> ∈ Z (which is defined only up to sign), and a (n×m)-matrix Λ with columns λ(Fi). Each vertex p of P can be represented as the intersection of n facets: p = Fi1 ∩ · · · ∩ Fin ; we denote by Λ(p) = Λ(i1,...,in) the minor matrix of Λ formed by the columns i1, . . . , in. Then detΛ(p) = det(λi1 , . . . , λin) = ±1. In what follows we refer to the vectors λi = λ(Fi) as facet vectors.