Pinching, Pontrjagin classes, and negatively curved vector bundles

Belegradek, Igor

doi:10.1007/pl00005803

Cited by 8 publications

(8 citation statements)

References 37 publications

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“…[Bel98]), in every rank there are only finitely many vector bundles over F with nonnegatively curved total spaces. Also since the Euler and Pontrjagin classes determine a vector bundle up to finite ambiguity (see e.g.…”

Section: Proposition 11 Let N Be An Open Complete Nonnegatively Curmentioning

confidence: 99%

“…[Bel98]). the homotopy fiber F of c has finite homotopy groups) and the spaces F, BSO(n), X are simply-connected (see e.g.…”

Section: Producing Vector Bundlesmentioning

confidence: 99%

“…Also since the Euler and Pontrjagin classes determine a vector bundle up to finite ambiguity (see e.g. [Bel98]), in every rank there are only finitely many vector bundles over F with nonnegatively curved total spaces. Thus, 1.2 is a generalization of the main result of [ ÖW94].…”

Section: Introductionmentioning

confidence: 99%

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Topological obstructions to nonnegative curvature

Belegradek¹,

Kapovitch²

2001

Math Ann

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Section: Proposition 11 Let N Be An Open Complete Nonnegatively Curmentioning

confidence: 99%

“…[Bel98]). the homotopy fiber F of c has finite homotopy groups) and the spaces F, BSO(n), X are simply-connected (see e.g.…”

Section: Producing Vector Bundlesmentioning

confidence: 99%

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Topological obstructions to nonnegative curvature

Belegradek¹,

Kapovitch²

2001

Math Ann

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“…It is well-known to experts that a vector bundle over a finite cell complex is recovered up to finitely many possibilities by the total Pontrjagin class and the Euler class of its orientable (1 or 2-fold) cover (see [Bel98] for a proof). We are now going to reduce to this result.…”

Section: Local Convergence Resultsmentioning

confidence: 99%

Finiteness theorems for nonnegatively curved vector bundles

Belegradek¹,

Kapovitch²

2001

Duke Math. J.

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“…We refer to [Bel98a] or [Bel98b] for background. Here we only recall basic definitions and prove several new lemmas specific to the finite volume case.…”

Section: Convergence Of Finite Volume Manifoldsmentioning

confidence: 99%

On Mostow rigidity for variable negative curvature

Belegradek

2002

Topology

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We prove a finiteness theorem for the class of complete finite volume Riemannian manifolds with pinched negative sectional curvature, fixed fundamental group, and of dimension ≥ 3 . One of the key ingredients is that the fundamental group of such a manifold does not admit a small nontrivial action on an R -tree.Although this result does not appear in the literature, it is not new except in dimension 4 where only homeomorphism finiteness has been known. However, the proof we present is very different from the existing argument that runs as follows.M. Gromov and W. Thurston [Gro82] used straightening and bounded cohomology to deduce that volume is bounded by the simplicial volume:for any M ∈ M a,b,π,n . Since all the manifolds in M a,b,π,n belong to the same proper homotopy type, they must have equal simplicial volumes. Thus, volume is uniformly bounded from above on M a,b,π,n . Furthermore, a universal lower volume bound comes from the Heintze-Margulis theorem [BGS85]. For closed manifolds of dimension n ≥ 4 and sectional curvatures within [−1, 0), the diameter can be bounded in terms of volume [Gro78]. Hence the conclusion of 1.1 follows from the Cheeger-Gromov compactness theorem. Similarly, the conclusion of the theorem 1.1 for finite volume manifolds of dimension ≥ 5 can be deduced from the work of K. Fukaya [Fuk84]. The dimension restriction comes from treating the ends by the weak h-cobordism theorem. In fact, Fukaya proves a similar statement for n = 4 where diffeomorphism finiteness is replaced by homotopy finiteness. (The topological 4-dimensional weak h-cobordism theorem, unavailable at the time of [Fuk84], can be also applied here because we deal with virtually nilpotent fundamental groups [Gui92,FT95]. Yet this only gives homeomorphism rather than diffeomorphism finiteness.) By contrast, main ideas in our approach come from Kleinian groups and geometric group theory. Essentially, given a degenerating sequence of manifolds M k , one can use rescaling in the universal covers to produce a nontrivial action of π on an R-tree with virtually nilpotent arc stabilizers (cf. [Bes88, Pau88, Pau91]). Then results of Rips, Bestvina and Feighn [BF95] imply that π splits over a virtually nilpotent subgroup. We prove that this does not happen if M a,b,π,n is nonempty. Then the methods of the Cheeger-Gromov compactness theorem imply that M k subconverges in pointed C 1,α topology to a complete C 1,α Riemannian manifold M . We then prove that π 1 (M ) contains a subgroup isomorphic to π which implies that M has finite volume and, in fact, π 1 (M ) ∼ = π . Now the convergence M k → M is analogous to strong convergence of Kleinian groups. Studying this convergence yields the theorem 1.1. Instead of using the h-cobordism theorem to deal with ends, we find a direct geometric argument that works in all dimensions. Similarly to [Gro82], our methods provide a uniform upper bound on the volume of manifolds in M a,b,π,n .Note that the real Schwarz lemma of Besson-Courtois-Gallot [BCG98] gives yet

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Pinching, Pontrjagin classes, and negatively curved vector bundles

Cited by 8 publications

References 37 publications

Topological obstructions to nonnegative curvature

Topological obstructions to nonnegative curvature

Finiteness theorems for nonnegatively curved vector bundles

On Mostow rigidity for variable negative curvature

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