Complex 2-plane bundles over complex projective space

Switzer, Robert M.

doi:10.1007/bf01214517

Cited by 11 publications

(5 citation statements)

References 7 publications

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“…On the other hand, as stated in [3], the construction for the Postnikov decomposition is much more difficult than that of CW-decomposition. In some special cases, one may give a detail description of the Postnikov decomposition [14]. However, Eckmann and Hilton [5] showed that the cohomology obstructions for Postnikov and CW-decompositions are equivalent.…”

Section: Resultsmentioning

confidence: 99%

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The vector bundle decomposition

Leslie¹,

Yue²

2003

Pacific J. Math.

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In this paper, we studied the real vector bundle decomposition problem. We first give a general decomposition result which relates a given vector bundle to some cohomology classes with local coefficients in the homotopy group of a Grassmann manifold; it is those classes that obstruct the decomposition. Those classes are natural with respect to the induced vector bundle by a map. For some special decompositions, we gave a relationship between those classes and the well-known characteristic classes such as Stiefel-Whitney classes and Chern classes. We determined the local coefficients in the cohomology group which contain the decomposition obstruction classes. We find applications in the study of subbundles of low codimension. In particular, codimension 1 decomposition classes are investigated in which we find that one of the two decomposition classes for the universal bundle over BO(2n + 1) is in H 2n+1 (BO(2n + 1), Z). This result gives rise to a new geometric interpretation for the order two elements in the integral cohomology group in odd dimension. We further make use of the cellular structure of the classifying space BO(n) to see the 'local' structure for the restriction of the universal bundle to each cell. In this way, we can construct the obstruction classes for the codimension 1 vector bundle decomposition. We gave an example to calculate the decomposition obstruction for the tangent bundle of RP 2n , which turns out to be the generator in the cohomology of RP 2n with twisted integer coefficients. On the other hand, we exhibit a trivial summand in the tangent bundle for any odd dimensional cobordism classes.

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Section: Resultsmentioning

confidence: 99%

“…In [2], Barth and Van de Ven gave a decomposability criterion for complex algebraic 2-bundles over CP n . Switzer [14] contributed in the classification of complex topological 2-vector bundle over CP n for n = 4, 5 and 6.…”

Section: Introductionmentioning

confidence: 99%

The vector bundle decomposition

Leslie¹,

Yue²

2003

Pacific J. Math.

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“…It is later proved, both by Rees [16] and by Smith [17], that for every l ≥ 5 there exists some nontrivial rank 2 bundle over CP l with vainishing Chern classes, so that vanishing Chern enumeration is nontrivial. Using obstruction-theoretic techniques, Switzer [18] gives alternative proofs of the Atiyah-Rees theorems and goes further to resolve Chern enumeration for rank two bundles over CP 5 and CP 6 .…”

Section: Introductionmentioning

confidence: 99%

Metastable complex vector bundles over complex projective spaces

Yang¹

2022

Preprint

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“…In this paper, the classification of complex vector bundles over 8-dimensional spin c manifolds will be investigated. Since the classification of rank 2 complex vector bundles have been studied by Switzer, see [12] and [11] for instance, we will mainly focus on the classification of rank 3 and 4 complex vector bundles. Our main results can be stated as follows.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the facts of Corollary 1.6 have been got by Thomas [13,Theorem A]. Recall that the classification of rank 2 complex vector bundles over CP 4 have been done by Switzer [11,Theorem 2]. Therefore, combing their results with Corollary 1.7, yields that complex vector bundles over CP 4 are all classified.…”

Section: Introductionmentioning

confidence: 99%

Topological classification of complex vector bundles over $8$-dimensional spin$^{c}$ manifolds

Yang¹

2020

Preprint

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In this paper, complex vector bundles of rank r over 8-dimensional spin c manifolds are classified in terms of the Chern classes of the complex vector bundles and the cohomology ring of the manifolds, where r = 3 or 4. As an application, we got that two rank 3 complex vector bundles over 4-dimensional complex projective spaces CP 4 are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank 3 complex vector bundles over CP 4 are determined. Combing Thomas's and Switzer's results with our work, we can assert that complex vector bundles over CP 4 are all classified.

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Complex 2-plane bundles over complex projective space

Cited by 11 publications

References 7 publications

The vector bundle decomposition

The vector bundle decomposition

Metastable complex vector bundles over complex projective spaces

Topological classification of complex vector bundles over $8$-dimensional spin$^{c}$ manifolds

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