Rank (differential topology)

About: Rank (differential topology) is a research topic. Over the lifetime, 1934 publications have been published within this topic receiving 21967 citations.

On hyperbolic groups

Abstract: We prove that a δ-hyperbolic group for δ < 1 2 is a free product F ∗ G1 ∗ . . . ∗Gn where F is a free group of finite rank and each Gi is a finite group.

Immersions of manifolds

Abstract: IMMERSIONS OF MANIFOLDSC) BY MORRIS W. HIRSCH Introduction Let M and N be differentiable manifolds of dimensions k and n respec- tively, k N is called an immersion if / is of class C1 and the Jacobian matrix of/ has rank k at each point of M. Such a map is also called regular. Until recently, very little was known about the ex- istence and classification of immersions of one manifold in another. The present work addresses itself to this problem and reduces it to the problem of constructing and classifying cross-sections of fibre bundles. In 1944, Whitney [15] proved that every ^-dimensional manifold can be immersed in Euclidean space of 2k — 1 dimensions, P2*-1. The Whitney- Graustein theorem [13] classifies immersions of the circle S1 in- the plane E2 up to regular homotopy, which is a homotopy / En, k /') and (g, g') are regularly homotopic (in a sense to be defined later). Given two immersions/, g: £>*—»£ that agree on S*_1 and have the same first derivatives at points of S*_1, Q(f, g) is an element of a certain homotopy group, and has the following properties: (1) Q(f, g) =0 if and only if/and g are regularly homotopic rel S*-1, i.e., the homotopy agrees with / and g on Si_1 at each stage, up to the first derivative; (2) fl(/, g) enjoys the usual algebraic properties of a difference cochain. At this point we should like to be able to make the following statement: If / is an immersion of the Received by the editors September 29, 1958. (*) The material in this paper is essentially a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Chicago, 1958. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Harmonic maps into singular spaces and $p$ -adic superrigidity for lattices in groups of rank one

Abstract: © Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Oka’s principle for holomorphic sections of elliptic bundles

Abstract: 2. The h-principle for locally trivial fibrations 2.1. The h-principle over small subsets 2.2. The h-principle over totally real submanifolds 2.3. Local extension of the h-principle 2.4. Localizable extensions 2.5. The h-principle for Cm-bundles 2.6. Manifolds with totally real souls 2.7. Totally real extensions 2.8. Nicely localizable extensions 2.9. The basic h-principle for Z -* X 2.10. Homomorphisms and holomorphic maps of rank > r

Rank-one convexity does not imply quasiconvexity

Abstract: We consider variational integralsdefined for (sufficiently regular) functions u: Ω→Rm. Here Ω is a bounded open subset of Rn, Du(x) denotes the gradient matrix of u at x and f is a continuous function on the space of all real m × n matrices Mm × n. One of the important problems in the calculus of variations is to characterise the functions f for which the integral I is lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).