yusuf civan | Suleyman Demirel University (original) (raw)
Address: S�leyman, Turkey
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Papers by yusuf civan
We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes o... more 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 K-groups. Moreover, we compute the Sq 2 -homology of bounded flag manifolds to make use of relevant Atiyah-Hirzebruch spectral sequence of KO-theory.
Geometriae Dedicata, 2005
We study the geometry and topology of Bott towers in the context of toric geometry. We show that ... more 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 Danilov–Jurkiewicz 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.
K-theory, 2005
We describe Bott towers as sequences of toric manifolds M k , and identify the omniorientations w... more We describe Bott towers as sequences of toric manifolds M k , and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of M k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory 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 consider the enumeration of stably complex structures on M k , obtaining estimates for 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. Given a commutative ring spectrum E, we denote the reduced and unreduced cohomology algebras of any space X by E * (X) and E * (X + ) respectively. So E * (S n ) is a free module over the coefficient ring E * on a single n-dimensional generator s E n , defined by the unit of E. In particular, we use this notation for the integral Eilenberg-Mac Lane spectrum H and the complex K-theory spectrum K. Real K-theory requires the most detailed calculations, so we abbreviate s KO n to s n whenever possible. We require multiplicative maps f : E → F of
A vertex coloring of a simplicial complex Delta\DeltaDelta is called a linear coloring if it satisfies th... more A vertex coloring of a simplicial complex Delta\DeltaDelta is called a linear coloring if it satisfies the property that for every pair of facets (F1,F2)(F_1, F_2)(F1,F2) of Delta\DeltaDelta, there exists no pair of vertices (v1,v2)(v_1, v_2)(v1,v2) with the same color such that v1inF1backslashF_2v_1\in F_1\backslash F_2v1inF1backslashF2 and v2inF2backslashF1v_2\in F_2\backslash F_1v_2inF_2backslashF_1. We show that every simplicial complex Delta\DeltaDelta which is linearly colored with kkk colors includes a subcomplex Delta′\Delta'Delta′ with kkk vertices such that Delta′\Delta'Delta′ is a strong deformation retract of Delta\DeltaDelta. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.
We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes o... more 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 K-groups. Moreover, we compute the Sq 2 -homology of bounded flag manifolds to make use of relevant Atiyah-Hirzebruch spectral sequence of KO-theory.
Geometriae Dedicata, 2005
We study the geometry and topology of Bott towers in the context of toric geometry. We show that ... more 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 Danilov–Jurkiewicz 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.
K-theory, 2005
We describe Bott towers as sequences of toric manifolds M k , and identify the omniorientations w... more We describe Bott towers as sequences of toric manifolds M k , and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of M k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory 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 consider the enumeration of stably complex structures on M k , obtaining estimates for 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. Given a commutative ring spectrum E, we denote the reduced and unreduced cohomology algebras of any space X by E * (X) and E * (X + ) respectively. So E * (S n ) is a free module over the coefficient ring E * on a single n-dimensional generator s E n , defined by the unit of E. In particular, we use this notation for the integral Eilenberg-Mac Lane spectrum H and the complex K-theory spectrum K. Real K-theory requires the most detailed calculations, so we abbreviate s KO n to s n whenever possible. We require multiplicative maps f : E → F of
A vertex coloring of a simplicial complex Delta\DeltaDelta is called a linear coloring if it satisfies th... more A vertex coloring of a simplicial complex Delta\DeltaDelta is called a linear coloring if it satisfies the property that for every pair of facets (F1,F2)(F_1, F_2)(F1,F2) of Delta\DeltaDelta, there exists no pair of vertices (v1,v2)(v_1, v_2)(v1,v2) with the same color such that v1inF1backslashF_2v_1\in F_1\backslash F_2v1inF1backslashF2 and v2inF2backslashF1v_2\in F_2\backslash F_1v_2inF_2backslashF_1. We show that every simplicial complex Delta\DeltaDelta which is linearly colored with kkk colors includes a subcomplex Delta′\Delta'Delta′ with kkk vertices such that Delta′\Delta'Delta′ is a strong deformation retract of Delta\DeltaDelta. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.