Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

By V. Buchstaber, Nigel Ray. 2000

84 citations recorded by the source corpus.

We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn#Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/80bd72ac9df34a907de77534ff20cd9792273c91

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Categorical aspects of toric topology (2007, background): “pis a ﬁbration, so there is a weak equivalence M[l] →holimPl by Proposition 4.8(2). In 2-dimensional integral homology, linduces l∗: ZV →Zn, which extends to a dicharacteristic homomorphism ℓ: TV →Tn [9]. The kernel of ℓis an (m−n)- dimensional subtorus T[l] < TV , which acts freely on ZK with qu”
Torus Actions and their Applications in Topology and Combinatorics (2002, background): “In the consequent definitions we follow [48], taking into account adjustments and specifications from [38].”
On the cohomology of torus manifolds (2003, background): “wise by a codimension ksubtorus. In particular, an intersection of ncharacteristic submanifolds consists of ﬁnitely many T-ﬁxed points. 6 MIKIYA MASUDA AND TARAS PANOV Each Miis orientable. Following [4], we say that Mis omnioriented if an orientation is speciﬁed for Mand for every characteristic su”
Projectivities in simplicial complexes and colorings of simple polytopes (2001, methodology): “This operation on simple polytopes has been introduced by Barnette [1] as joining; it is called connected sum in [ 5 ].”
Torus graphs and simplicial posets (2005, background): “ seen to consist of ﬁnite number of isolated points. A characteristic submanifold of M is a codimension-two connected component of the set ﬁxed pointwise by a circle subgroup of T. An omniorientation [5] of M consists of a choice of orientation for M and for each characteristic submanifold. A choice”