Vector bundles and homogeneous spaces

Atiyah, Michael; Hirzebruch, Friedrich

doi:10.1090/pspum/003/0139181

Cited by 318 publications

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“…since the Chern character is natural and commutes with coboundaries, [8]. The result follovlS from theorem (1.…”

mentioning

confidence: 79%

“…Let G be a compact 1-connected Lie group, then K*(B G ) and K*(G) are Mell-known, [8] and [31J. If the rank of G is r, K* (B G ) .…”

Section: Introductionmentioning

confidence: 99%

“…sequences for the three spaces collapse, [8]. The exterior product induces a map of the Gpectral sequences which on theE 2 -terms is the isomorphism induced by the cohomology exterior product, and on the ~-terms is compatible with the K-theory exterior product.…”

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confidence: 99%

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On the K-theory of the loop space of a Lie group

Clarke

1974

Math. Proc. Camb. Phil. Soc.

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“…since the Chern character is natural and commutes with coboundaries, [8]. The result follovlS from theorem (1.…”

mentioning

confidence: 79%

“…Let G be a compact 1-connected Lie group, then K*(B G ) and K*(G) are Mell-known, [8] and [31J. If the rank of G is r, K* (B G ) .…”

Section: Introductionmentioning

confidence: 99%

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On the K-theory of the loop space of a Lie group

Clarke

1974

Math. Proc. Camb. Phil. Soc.

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“…The spectral sequence of [4], [11] shows that K^(X), F^(A'), FH(A) are finitely generated. By [42], these are the same as K0(B), F0(C ®R B), F0(H ®R B).…”

Section: Ifk Is a Field And K C A C F Then T(a/k) < T(b/k)mentioning

confidence: 99%

“…As above, we get [X,BU(n)] = 0 for n < m while [X,5i/(m)] 7t 0. By [4] or [11] there is a spectral sequence Hr(X,Ks(pt)) => F*(*). Now F*(pt) is 0 for s odd and Z for i even, while Hr(X,Z) = 0 for r even because i is odd.…”

Section: Ifk Is a Field And K C A C F Then T(a/k) < T(b/k)mentioning

confidence: 99%

Topological Examples of Projective Modules

Swan¹

1977

Transactions of the American Mathematical Society

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Abstract.A new and more elementary proof is given for Lonsted's theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of Lensted. In certain cases, methods associated with Hubert's 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If m = 2 mod 4, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of rank # m are free.In [29], Lonsted proved a remarkable theorem which shows that vector bundles over a finite CW complex can be represented by finitely generated projective modules over a noetherian ring. This means that by purely topological constructions one can produce examples of noetherian rings whose projective modules have certain specified properties. This method has the advantage that one can impose conditions on the totality of projective modules while the more elementary method of [42] only allows us to construct a finite number of such modules at a time.Lonsted's construction makes use of rather deep properties of analytic functions. In attempting to analyze his proof, I discovered a more elementary proof of the theorem which I will present here. This proof gives rings which are considerably smaller than the ones used by Lonsted. They are, in fact, localizations of algebras of finite type over R. In certain cases, one can even give a very simple and purely algebraic description of the ring. This will be done in §10 using methods associated with Hubert's 17th problem.The starting point for this work was a question of A. Geramita. He pointed out that the rings in [42] have nontrivial projective modules of low rank and asked whether, for all n, there are noetherian rings having nontrivial projective modules but such that all such modules of rank less than n are free. Three

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Some Remarks on Almost and Stable Almost Complex Manifolds

Dessai

1998

Mathematische Nachrichten

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In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H 1 ( M ; Z) = 0 and no 2-torsion in H i ( M ; Z) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the f-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.In Theorem 1.2 we give necessary and sufficient conditions for a 10-manifold M to be stably almost complex if H l ( M ; Z ) =: 0 and H i ( M ; Z ) , i = 2,3, has no 2torsion. No assumption on w4(M) is made. The theorem simplifies if M has "nice" cohomology, such as the one of complete intersections (cf. Corollary 1.3).In Theorem 1.4 we give, using the Classification Theorem of Donaldson, a reformulation of the classical conditions for a 4 -manifold M to be almost complex in terms of the Betti numbers and the dimension of the feigenspaces of the intersection form.In the second part we study the question whether an almost complex 2n -dimensional manifold admits infinitely many almost complex structures. In Theorem 2.2 and Theorem 2.3 we give a simple answer for a large class of manifolds. We apply these theorems to symplectic manifolds (cf. Corollary 2.4), homogeneous spaces (cf. Corollary 2.5) and complete intersections (cf. Example 2.6). Germany e -mail: dessai4mathpool. Uni -Augsburg.DE D -86135 Augsburg

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Vector bundles and homogeneous spaces

Cited by 318 publications

References 0 publications

On the K-theory of the loop space of a Lie group

On the K-theory of the loop space of a Lie group

Topological Examples of Projective Modules

Some Remarks on Almost and Stable Almost Complex Manifolds

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