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Tangential structures on toric manifolds, and connected sums of polytopes
 MATH. RES. NOTICES
, 2001
"... We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension2 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 elem ..."
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Cited by 30 (7 self)
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We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension2 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 ndimensional polytopes; when P n is a product of simplices, we describe P n #Q n by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Q n.
HOMOTOPY DECOMPOSITIONS AND K–THEORY Of Bott Towers
, 2004
"... We describe Bott towers as sequences of toric manifolds M k, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M k is homotopy equivalent to a wedge of Thom complexes, and display its complex Ktheory as an algebra o ..."
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Cited by 7 (1 self)
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We describe Bott towers as sequences of toric manifolds M k, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M k is homotopy equivalent to a wedge of Thom complexes, and display its complex Ktheory as an algebra over the coefficient ring. We extend the results to KOtheory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky’s analysis of the Adams Spectral Sequence [2]. By way of application we investigate stably complex structures on M k, identifying those which arise from omniorientations and those which are almost complex. We conclude with observations on the rôle of Bott towers in complex cobordism theory.
Operations And Quantum Doubles In Complex Oriented Cohomology Theory
, 1999
"... We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the LandweberNovikov algebra, as represented by a subalg ..."
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We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the LandweberNovikov algebra, as represented by a subalgebra of operations in double complex cobordism. We work in the context of Boardman's eightfold way, which offers an important framework for clarifying the relationship between quantum doubles and the standard machinery of Hopf algebroids of homology cooperations. These considerations give rise to novel structures in double cohomology theory, and we explore the twist operation and extensions of the quantum antipode by way of example.
BOTT TOWERS, CROSSPOLYTOPES AND TORUS ACTIONS
, 2003
"... Abstract. We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained fro ..."
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Abstract. We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying DanilovJurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties. 1.
KOgroups of Bounded Flag Manifolds
 TURK J MATH
, 2002
"... We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes of associated complex line bundles. We use the existence of such a splitting to assist our computation of real and complex Kgroups. Moreover, we compute the Sq2homology of bounded flag manifolds to make ..."
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We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes of associated complex line bundles. We use the existence of such a splitting to assist our computation of real and complex Kgroups. Moreover, we compute the Sq2homology of bounded flag manifolds to make use of relevant AtiyahHirzebruch spectral sequence of KOtheory.
Toric genera
, 2009
"... Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from ana ..."
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Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact Φ is a version of tom Dieck’s bundling transformation of 1970, defined on T kequivariant complex cobordism classes and taking values in the complex cobordism algebra Ω ∗ U (BT k +) of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever’s generalised elliptic genus as universal amongst genera that are rigid on SUmanifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.