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Immersions of manifolds
- Source :
- Hirsch, MW. (1959). Immersions of manifolds. Transactions of the American Mathematical Society, 93(2), 242-242. doi: 10.1090/S0002-9947-1959-0119214-4. UC Berkeley: Retrieved from: http://www.escholarship.org/uc/item/90q6s2cx, Transactions of the American Mathematical Society, vol 93, iss 2
- Publication Year :
- 1959
- Publisher :
- eScholarship, University of California, 1959.
-
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
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Hirsch, MW. (1959). Immersions of manifolds. Transactions of the American Mathematical Society, 93(2), 242-242. doi: 10.1090/S0002-9947-1959-0119214-4. UC Berkeley: Retrieved from: http://www.escholarship.org/uc/item/90q6s2cx, Transactions of the American Mathematical Society, vol 93, iss 2
- Accession number :
- edsair.doi.dedup.....030438cab467cc98fdef467c81900803