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J. H. C. Whitehead

Bio: J. H. C. Whitehead is an academic researcher from University of Oxford. The author has contributed to research in topics: Homotopy group & Homotopy. The author has an hindex of 33, co-authored 77 publications receiving 5097 citations. Previous affiliations of J. H. C. Whitehead include Princeton University & Balliol College.


Papers
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Journal ArticleDOI
TL;DR: The sequence 2(X) is followed by the main theorem, which states that the sequence r(A) is equivalent to 2(C, A), and the sequence K is a combination of 2 (C and A) and 2 (A), which are equivalent to 1 (A and C).
Abstract: Page INTRODUCTION 51 CHAPTER I. THE SEQUENCE 2(C, A) 53 1. Definition of 2(C, A) 53 2. The secondary modular boundary operator 55 3. Induced homomorphisms of 2 55 4. Combinatorial realizability 57 CHAPTER II. THE GROUP r(A). . 60 5. Definition of r(A) 60 6. Induced homomorphisms of r(A) 64 7. Relation to tensor products 67 8. "A" finitely generated 67 9. Direct systems 70 CHAPTER III. THE SEQUENCE 2(K) 71 10. Definition of 2(K) 71 11. The invariance of 2(K) 73 12. The sufficiency of 2(K) 75 13. Expression for r3(K) 75 14. Geometrical realizability 77 15. q-types 81 CHAPTER IV. THE PONTRJAGIN SQUARES 83 16. The main theorem 83 17. Secondary boundary operators 92 18. The calculation of 24(K) 97 CHAPTER V. THE SEQUENCE OF A GENERAL SPACE . . 98 19. The complex K(X) 98 20. The maps K and X 101 21. The sequence 2(X) 106 APPENDIX A. ON SPACES DOMINATED BY COMPLEXES...... . 107 APPENDIX B. ON SEPARATION COCHAINS 108

321 citations

Journal ArticleDOI
TL;DR: The relation between 7rn(X) and 7r(X*) when the maps fi(S'-') are arbitrary is studied in this paper, where the relation is expressed in terms of a product A e 7rm+n-1(X).
Abstract: In a recent paper2 I described in algebraical terms the relation between 7rn-i(X) and 7rn1-(X*), and also the relation between 7rn(X) and 7rn(X*) in case each of fi(St'') is homotopic to a point. Here we study the relation between 7rn(X) and 7r,,(X*) when the maps fi(S'-') are arbitrary. There is a considerable difference between the cases n = 2 and n > 2. In case n > 2 the relation between 7rn(X) and 7rn(X*) is expressed in terms of a product a A e 7rm+n-1(X), where a e rm(X), 0 E 7rn(X). The case n = 2 is, in many ways, the more interesting of the two. Among other things a method is found for calculating3 7r2(K) algebraically, where K is any simplicial complex. Of course K. Reidemeister's4 theory of homology in K, the universal covering complex of K, together with a theorem due to, W. Hurewicz5 lead to a theoretical definition of 7r2(K), which may be stated in purely algebraic terms. But since there is no general algorithm for deciding whether or no given elements Pi X n * p*. in the group ring, 9?, of K, satisfy given equations

284 citations

Journal ArticleDOI
TL;DR: In this article, the authors give a mechanical process for deciding whether or no two sets are equivalent and also a process for reducing (a) to one of a finite number of normal forms.
Abstract: together with the 'simple automorphism' which replaces a, by its inverse. Relative to this kind of equivalence we have very little to add to a paper by J. Nielsen,2 in which he gives a mechanical process for deciding whether or no two sets are equivalent and also a process for reducing (a) to one of a finite number of normal forms. When reduced in this way we shall describe a set of elements as reduced (N), and we recall that (a) is reduced (N) if it contains no two words of the form (AB)" and (AC)" respectively,3 where 1(A) > 1(B) or > I(C), and if the last half of every word with an even number of letters is an 'isolated ending.' That is to say, if a AB and 1(A) = 1(B) no other word in (a) ends with B or begins with B'. In ?2 it is assumed that any empty words which appear' during the process of reduction are discarded, as in J. N., while in ?4 they are retained. Theorem 1 below is essentially a restatement of various arguments used by Nielsen, while Theorem 2 adds a detail to J. N. The second kind of equivalence refers to the effect on (a) of automorphisms5 of G, two ordered sets of elements (a) and (I), both of which contain the same number of words, being equivalent if ax, corresponds to #X(X = 1, 2, *. * ) in some automorphism of G. That is to say they are equivalent if there is an automorphism

273 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Abstract: © Publications mathématiques de l’I.H.É.S., 1977, 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.

1,667 citations

Journal ArticleDOI
TL;DR: In this article, a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary was proposed.
Abstract: 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained. The discovery of this principle was made in three stages. From 1973 to 1975 Andrew Casson developed his theory of "flexible handles". These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D X D, dD X D) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion. It was clear to Casson that: (1) no known invariant—link theoretic

1,566 citations

Book
30 Sep 1997
TL;DR: Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly real compact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifold, infinite dimensional differential geometry Manifolds of Mappings Further applications References as mentioned in this paper.
Abstract: Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

1,291 citations

Book
01 Jan 1993
TL;DR: In this article, the authors present a general theory of Lie Derivatives and their application in a variety of fields and functions, including bundles and bundles of bundles on manifolds.
Abstract: I. Manifolds and Lie Groups.- II. Differential Forms.- III. Bundles and Connections.- IV. Jets and Natural Bundles.- V. Finite Order Theorems.- VI. Methods for Finding Natural Operators.- VII. Further Applications.- VIII. Product Preserving Functors.- IX. Bundle Functors on Manifolds.- X. Prolongation of Vector Fields and Connections.- XI. General Theory of Lie Derivatives.- XII. Gauge Natural Bundles and Operators.- References.- List of symbols.- Author index.

1,251 citations