sign in sign up
<<
Home , Hassler Whitney

Contemporary

Gian-Carlo Rota Editor Hassler Whitney in the 1970s (Photograph by Sally Whitney)

Hassler Whitney at age 14 in the Swiss Alps Hassler Whitney Collected Papers Volume I

James Eells Domingo Toledo Editors

Birkhiuser • Basel • Berlin 1992 Domingo Toledo Department of Department of Mathematics University of Warwick University of Utah Coventry CV 4 7AL Salt Lake City, Utah 84112 England U.S.A.

Library of Congress Cataloging-in-Publication Data

Whitney, Hassler. The collected papers of Hassler Whitney / edited by James Eells, Domingo Toledo p. cm. - (Contemporary mathematicians) Includes bibliographical references. ISBN-13: 978-1-4612-7740-8 e-ISBN-13: 978-1-4612-2972-8 DOl: 10.1007/978-1-4612-2972-8 1. . 2. Geometry, Differential. 3. Combinatorial analysis. 1. Eells, James, 1926- II. Toledo, Domingo. III. Title. IV. Series. QA611.W4854 1992 514-dc20 91-9885 CIP

Printed on acid-free paper.

© Birkhiiuser Boston, 1992 Softcover reprint of the hardcover 1st edition 1992 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any fonn or by any !p.eans, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner.

Permission to photocopy for internal or personal use, or the internal or personal use of specific clients is granted by Birkhiiuser Boston for libraries and other users registered with the Copyright Clearance Center (Ccq, provided that the base fee of $0.00per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. 3558-0/91 $0.00 + .20

Printed and bound by Quinn Woodbine, Woodbine, NJ

987 6 5 4 3 2 I The Collected Papers of Hassler Whitney

Contents - Volume 1 (Bracketed numbers are from the Bibliography)

Contents - Volume 1 · v Contents - Volume 2 · vii Preface · x Academic Appointments and Awards · xi Bibliography of Hassler Whitney . . · xii

Introduction ...... 1

[82] Moscow 1935: Topology Moving Toward America

Chapter 1 Graphs and ...... 23

[3] A theorem on graphs, Annals of Math. (2) v. 32, 1931, 378-390 ...... 24 [5] Non-separable and planar graphs, AMS Transac. v. 34, 1932, 339-362 . 37 [6] Congruent graphs and the connectivity of graphs, Am. Jour. Math. v. 54, 1932, 150-168 ...... 61 [10] The coloring of graphs, Annals of Math., (2) v. 33, 1932, 688-718 . 80 [12] A set of topological invariants for graphs, Am. Jour. Math., v. 55, 1933, 231-235 ...... 111 [13] On the classification of graphs, Am. Jour. Math., v. 55, 1933,236-244 116 [14] 2-Isomorphic graphs, Am. Jour. Math., v. 55, 1933, 245-254 . . 125 [17] Planar graphs, Fundamenta Math., V. 21, 1933, 73-84 . . . . . 135 [23] On the abstract properties of linear dependence, Am. Jour. Math., v. 57, 1935,509-533 ...... 147 [37] A numerical equivalent of the four color problem, Monatshefte fur Math. un Phys. 3, 1937-207-213 ...... 172 [77] On reducibility in the four color problem, unpublished manuscript, 1971 179 [78] (With W. T. Tutte) Kempe chains and the four colour problem, Utilitas Mathematica 2(1972), 241-281 ...... 185

v Chapter 2 Differentiable Functions and Singularities .... 227

[18] Analytic extensions of differentiable functions defined in closed sets, AMS Transac., v. 36, 1934, 63-89 ...... 228 [19] Derivatives, difference quotients and Taylor's formula, AMS Bull., v. 40, 1934, 89-94 ...... 255 [20] Differentiable functions defined in closed sets I, AMS Transac., v. 36, 1934, 369-387 ...... 261 [21] Derivatives, difference quotients and Taylor's formula II, Annals of Math. (2) v. 35, 1934,476--481 ...... 280 [22] Functions differentiable on the boundaries of regions, Annals of Math. (2) v. 35, 1934,482-485 ...... 286 [26] A function not constant on a connected set of critical points, Duke Math. J., v. 1, 1935, 514-517 ...... 290 [27] Differentiable functions defined in arbitrary subsets of Euclidean space, AMS Transac., v. 40, 1936,309-317 ...... 294 [45] Differentiability of the remainder term in Taylor's formula, Duke Math. J., 10, 1943, 153-158 ...... 303 [46] Differentiable even functions, Duke Math. J., 10, 1943, 159-160 309 [47] The general type of singularity of a set of 2n - 1 smooth functions of n variables, Duke Math. J., 10, 1943, 161-172 ...... 311 [49] On the extension of differentiable functions, AMS Bull., 50, 1944, 76-81 323 [55] On ideals of differentiable functions, Am. Jour. Math. 70, 1948, 635-658 329 [61] On totally differentiable and smooth functions, Pacific J. Math. 1, 1951, 143-159 ...... 353 [63] On singularities of mappings of Euclidean spaces, I. Mappings of the plane into the plane, Annals of Math. (2) 62, 1955, 374-410 370 [64] On functions with bounded n-th differences, J. de Maths. Pures et Appl. 36, 1957, 67-95 ...... 407 [67] Singularities of mappings of Euclidean spaces, Symposium Intemacional de Topologia Algebraica, Mexico, 1956,285-301, Mexico, La Universidad Nacional Autonoma, 1958 ..... 436 [70] On bounded functions with bounded n-th differences, AMS Proc. 10, 1959,480-481 ...... 453

Chapter 3 Analytic Spaces ... 455

[66] Elementary structure of real algebraic varieties, Annals of Math. (2) 66, 1957, 545-556 ...... 456 [68] (With F. Bruhat) Quelques proprietes fondamentales des ensembles analytiques-reels, Comm. Math. Helv. 33, 1959, 132-160 468 [73] Local properties of analytic varieties, in: differential and combinatorial topology (Symposium in Honor of ), Princeton, NJ, Press, 1965, 205-244 . . . . 497 [74] Tangents to an analytic variety, Annals of Math (2) 81, 1965, 496-549 537

Permissions

vi The Collected Papers of Hassler Whitney

Contents - Volume 2 (Bracketed numbers are from the Bibliography)

Contents - Volume 1 . v Contents - Volume 2 vii Preface . x Academic Appointments and Awards .xi Bibliography of Hassler Whitney . . xii

Chapter 1 xiii

[28] Differentiable Manifolds, Annals of Math.(2) v.37, 1936,645-680 . . 1 [29] The imbedding of Manifolds in families of analytic manifolds, Annals of Math.(2) v. 37, 1926, 865-878 ...... 37 [30] On regular closed curves in the plane, Compositio Math. 4, 1937,276-284 51 [35] Analytic coordinate systems and arcs in a , Annals of Math.(2) 38, 1937,809-818 ...... 60 [50] The self-intersections of a smooth n-manifold in 2n-space, Annals of Math. (2), 45, 1944, 220-246 ...... 70 [51] The singularities of a smooth n-manifold in (2n - I)-space, Annals of Math.(2), 45, 1944,247-293...... 97 [72] The work of John W. Milnor, Proceedings ICM 1962, Institut Mittag-Leffler, Djursholm, Sweden, xlviii-I...... 144

Chapter 2 Bundles and Characteristic Classes ...... 147

[36] Topological properties of differentiable manifolds, AMS Bull. 43, 1937, 785-805 ...... 148 [42] On the theory of sphere-bundles, NAS Proc., 26, 1940, 148-153 . 169 [44] On the topology of differentiable manifolds, Lectures in Topology, U. of Michigan Press, 1941, 101-141 ...... 175 [69] (With A. Dold) Classification of oriented sphere bundles over a 4-complex, Annals of Math. (2) 69, 1959,667-677 ...... 216

vii Chapter 3 Topology and ...... 227

[11] A characterization of the closed 2-cell, AMS Transac., v. 35, 1933,261-273 228 [15] Regular families of curves, Annals. of Math.(2) v. 34, 1933,244-270 241 [31] On matrices of integers and combinatorial topology, Duke Math. J., 3, 1937,35-45...... 268 [32] On the maps of an n-sphere into another n-sphere, Duke Math. J., 3, 1937,46-50...... 279 [33] The maps of an n-complex into an n-sphere, Duke Math. J., 3, 1937,51-55...... 284 [38] Cross sections of curves in 3-space, Duke Math. J.,4, 1938,222-226 289 [39] On products in a complex, Annals of Math.(2) 39, 1938,397-432. . 294 [40] Tensor products of abelian groups, Duke Math. J., 4, 1938, 495-528 330 [41] Some combinatorial properties of complexes, NAS Proc., 26, 143-148 364 [43] On regular families of curves, AMS Bull., 47, 1941, 145-147 370 [48] Topics in the theory of A~lian groups, I. Divisibility of Homomorphisms, AMS Bull., 50, 1944, 129-134 ...... 373 [54] Complexes of manifolds, NAS Proc., v. 33, 1947, 10-11 379 [56] Relations between the second and third homotopy groups of a simply-connected space, Annals of Math.(2) 50, 1949, 180-202 . 381 [57] Classification of the mappings of a 3-complex into a simply-connected space, Annals of Math.(2) 50, 1949, 270-284 ...... 404 [58] An extension theorem for mappings into simply-connected spaces, Annals of Math.(2) 50, 1949, 285-296 ...... 419

Chapter 4 Geometric Integration Theory ...... 431

[52] Algebraic topology and integration theory, NAS Proc., v. 33, 1947, 1-6 . .. 432 [53] Geometric methods in theory, NAS Proc., v. 33, 1947,7-9 . .. 438 [59] La topologie algebrique et la theorie de l'integration, Colloques Internationaux du CNRS XII, Topologie Algebrique, 1947, 107-113, published by CNRS, Paris, 1949 ...... 441 [62] r-dimensional integration in n-space, Proc. Int Congo Math., 1950, vol. 1,245-256, Amer. Math. Soc. 1952 ...... 448 [65] Introduction to "Geometric Integration Theory", Princeton, NJ, Princeton University Press, 1957, pp. 3-31 ...... 460

Chapter 5 Other Subjects ...... 489

[9] A logical expansion in mathematics, AMS Bull., v. 38, 1932,572-579. .. 490 [16] Characteristic functions and the algebra of logic, Annals of Math.(2) v. 34, 1933,405-414 ...... 498

viii [60] (With L.H. Loomis) An inequality related to the isoperimetric inequality related to the isoperimetric inequality, AMS Bull., 55, 1949,961-962 508 [71] (With A.M. Gleason) The extension of linear functionals defined on H-infinity, Pacific J. Math. 12, 1962, 163-182 ...... 510 [75] The mathematics of physical quantities. Part I, Mathematical models for measurement, Am. Math. Monthly 75(1968), 115-138, Part II, Quantity structures and dimensional analysis, ibid. 237-256 . 530 [76] Logic fad or tool? nico 4, 1969, Revue per . du centre Beige de Pedagogie de la Mathematique, 2-14 . . . 584 [81] Comment on the division of the plane by lines, Am. Math. Monthly 86(1979), p. 700 . . . . 597

Permissions

ix Preface

We present here the mathematical papers of Hassler Whitney. This collection contains all the published papers, with the exception of some short announcements that Whitney did not wish to be included. We also include the introduction to his book Geometric Integration Theory, and one previously unpublished manuscript on the four-color problem. The papers are presented under some broad categories: graphs· and combinatorics, differentiable functions and singularities, analytic spaces, manifolds, bundles and characteristic classes, topology and algebraic topology, geometric integration theory. Whitney intended to write an introduction to this collection. Unfortunately he left us no manuscript at the time of his death, May 10, 1989. We had discussed the possibility of using his paper "Moscow 1935 - Topology moving toward America," written for the Centennial of the American Mathematical Society, as part of his introduction to this collection, an idea which he much liked. We therefore include this paper, which contains personal information as well as mathematical reflections, as Whitney's own introduction to these volumes. Whitney's mathematical style, like his personal style, was that of an explorer and pioneer. One of the pictures included in these volumes shows him as a mountain climber. In mathematics, he preferred to work on undeveloped areas: break new ground and build foundations. During the last twenty years of his life he concentrated his efforts on developing an educational system that builds on the natural tendency in children to be explorers. We find that his papers are as fresh today as when they were written, and feel that any comments from us on their contents could only detract from their clarity. The influence of his work is too vast to review meaningfully in a few pages, event if we felt competent to do so. We do find it appropriate to comment on his unpublished manuscript (dating from around 1940) of a book that contained the details of all the results announced in "On the theory of sphere bundles," [42]. It contains a lengthy proof of his duality theorem (the formula for the Stiefel-Whitney classes of the direct sum of two vector bundles) obtained by developing an for direct sums, and computations with cochains. Whitney has referred to that proof as the hardest thing he ever did. In an unguarded moment he at once remarked that he felt that the duality theorem was his most original contribution to mathematics.

J. Eells D. Toledo

x Academic Appointments and Awards

Born March 23, 1907, in , Ph. B., 1928; Mus. B., 1929; honorary Sc. D., 1947 , Ph. D. 1932 National Research Council Fellow in Mathematics, 1931-33 Instructor to Professor of Mathematics, Harvard University, 1933-52 Professor of Mathematics, Institute for Advanced Study, 1952-77; Professor Emeritus, 1977-89

Member: American Mathematical Society, Mathematical Association of America, National Council of Teachers of Mathematics, National Academy of Sciences, American Philo• sophical Society, Swiss Mathematical Society (honorary)

American Mathematical Society: Colloquium Lecturer, 1946 Vice President, 1948-50 Editor, American Journal of Mathematics, 1944-49 Editor, Mathematical Reviews, 1949-54 Committee on Visiting Lectureship (chairman), 1946-51 Committee for Summer Institutes (chairman), 1953-54

National Science Foundation: Chairman of Panel on Mathematics, 1953-56 Researcher for Applied Mathematics Panel of NDRC of OSRD, 1943-45 Exchange Professor to France from Harvard, 1951-52 Exchange Professor to College de France (Fulbright), 1957 Committee on Support of Research in the Mathematical Sciences of the National Research Council, 1966--67 L. R. Ford Award for paper, "The Mathematics of Physical Quantities," 1969 National Medal of Science, 1967 Wolf Prize in Mathematics (shared) for 1982 Steele Prize, American Mathematical Society, 1985 Consultant to School Mathematics Study Group, Cambridge Conference in School Mathematics, Education Development Center, and other groups International Commission on Mathematical Instruction, Pres• ident, 1979-82

xi Bibliography of Hassler Whitney

[1] The coloring of graphs, NAS Proc. v. 17, 1931, 122-125. [2] Non-separable and planar graphs, NAS Proc. v. 17, 1931, 125-127 • [3] A theorem on graphs, Annals of Math. (2) v. 32, 1931, 378-390 . [4] Note on Perron's solution of the Dirichlet problem, NAS Proc., v. 18, 1932, 68-70. [5] Non-separable and planar graphs, AMS Transac. v. 34, 1932, 339-362. [6] Congruent graphs and the connectivity of graphs, Am. Jour. Math. v. 54, 1932, 150-168. [7] Regular families of curves I, NAS Proc., v. 18, 1932, 275-278. [8] Regular families of curves TI, NAS Proc., v. 18, 1932, 340-342. [9] A logical expansion in mathematics, AMS Bull., v. 38, 1932,572-579. [10] The coloring of graphs, Annals of Math., (2) v. 33, 1932, 688-718. [11] A characterization of the closed 2-cell, AMS Transac., v. 35, 1933, 261-273. [12] A set of topological invariants for graphs, Am. Jour. Math., v. 55, 1933, 231-235. [13] On the classification of graphs, Am. Jour. Math., v. 55, 1933, 236-244. [14] 2-Isomorphic graphs, Am. Jour. Math., v. 55, 1933, 245-254. [15] Regular families of curves, Annals of Math. (2) v. 34, 1933, 244-270. [16] Characteristic functions and the algebra of logic, Annals of Math. (2) v. 34, 1933, 405-414. [17] Planar graphs, Fundamenta Math., v. 21, 1933, 73-84. [18] Analytic extensions of differentiable functions defined in closed sets, AMS Transac., v. 36,1934,63-89. [19] Derivatives, difference quotients and Thylor's formula, AMS Bull., v. 40,1934,89-94. [20] Differentiable functions defined in closed sets I, AMS Transac., v. 36, 1934,369-387. [21] Derivatives, difference quotients and Thylor's formula TI, Annals of Math. (2) v. 35, 1934, 476-481. [22] Functions differentiable on the boundaries of regions, Annals of Math. (2) v. 35, 1934, 482-485. [23] On the abstract properties of linear dependence, Am. Jour. Math., v. 57, 1935, 509-533. [24] Differentiable manifolds in Euclidean space, NAS Proc., v. 21, 1935, 462-464, reprinted in Receuil Mathematique. T. 1(43) N. 5 (1936) 783-786. [25] Sphere-spaces, NAS Proc., v. 21, 1935, 464-468, reprinted in Receuil Mathematique. T. 1(43) N. 5 (1936) 787-791. [26] A function not constant on a connected set of critical points, Duke Math. J., v. 1, 1935, 514-517. [27] Differentiable functions defined in arbitrary subsets of Euclidean space, AMS Transac., ~40, 1936,309-317. [28] Differentiable manifolds, Annals of Math. (2) v. 37, 1936,645-680. [29] The imbedding of manifolds in families of analytic manifolds, Annals of Math. (2) ~ 37,1936,865-878. [30] On regular closed curves in the plane, Compositio Math. 4, 1937,276-284. [31] On matrices of integers and combinatorial topology, Duke Math. J., 3, 1937, 35-45. [32] On the maps of an n-sphere into another n-sphere, Duke Math. J., 3, 1937,46-50. [33] The maps of an n-complex into an n-sphere, Duke Math. J., 3, 1937, 51-55. [34] On products in a complex, NAS Proc., v. 23, 1937, 285-291.

xii [35] Analytic coordinate systems and arcs in a manifold. Annals of Math. (2) 38, 1937, 809-818. [36] Topological properties of differentiable manifolds, AMS Bull. 43, 1937, 785-805. [37] A numerical equivalent of the four color problem, Monatshefte fur Math. und Phys. 3, 1937, 207-213. [38] Cross sections of curves in 3-space, Duke Math. J.,4, 1938,222-226. [39] On products in a complex, Annals of Math. (2) 39, 1938,397-432. [40] Tensor products of abelian groups, Duke Math. J.,4, 1938,495-528. [41] Some combinatorial properties of complexes, NAS Proc., 26, 43-148. [42] On the theory of sphere-bundles, NAS Proc., 26, 1940, 148-153. [43] On regular families of curves, AMS Bull., 47, 1941, 145-147. [44] On the topology of differentiable manifolds, Lectures in Topology, U. of Michigan Press, 1941, 101-141. [45] Differentiability of the remainder term in Taylor's formula, Duke Math. J.,10, 1943, 153-158. [46] Differentiable even functions, Duke Math. J., 10, 1943, 159-160. [47] The general type of singularity of a set of 2n - 1 smooth functions of n-variables, Duke Math. J., 10, 1943, 161-172. [48] Topics in the theory of Abelian groups, I. Divisibility of Homomorphisms, AMS BulL, 50, 1944, 129-134. [49] On the extension of differentiable functions, AMS Bull., 50, 1944, 76-81. [50] The self-intersections of a smooth n-manifold in 2n-space, Annals of Math. (2), 45, 1944, 22~246. [51] The singularities of a smooth n-manifold in (2n-l)-space, Annals of Math. (2), 45, 1944, 247-293. [52] Algebraic topology and integration theory, NAS Proc., v. 33, 1947, 1-6. [53] Geometric methods in cohomology theory, NAS Proc., v. 33, 1947, 7-9. [54] Complexes of manifolds, NAS Proc., v. 33, 1947, 1~11. [55] On ideals of differentiable functions, Am. Jour. Math. 70, 1948,635-658. [56] Relations between the second and third homotopy groups of a simply-connected space, Annals of Math. (2) 50, 1949, 18~202. [57] Classification of the mappings of a 3-complex into simply-connected spaces, Annals of Math. (2) 50, 1949, 27~284. [58] An extension theorem for mappings into simply-connected spaces, Annals of Math. (2) 50, 1949, 285-296. [59] La topologie algebrique et la theorie de l'integration, Colloques Intemationaux du CNRS XII, Topologie Algebrique, 1947, 107-113, published by CNRS, Paris, 1949. [60] (With L. H. Loomis) An inequality related to the isoperimetric inequality, AMS BulL, 55, 1949, 961-962. [61] On totally differentiable and smooth functions, Pacific J. Math. 1, 1951, 143-159. [62] r-dimensional integration in n-space, Proc. Int Congo Math., 1950, vol. 1, 245-256, Amer. Math. Soc. 1952. [63] On singularities of mappings of Euclidean spaces, I. Mappings of the plabe into the plane, Annals of Math. (2) 62, 1955, 374-410. [64] On functions with bounded n-th differences, J. de Maths. Pres et Appl. 36, 1957, 67-95. [65] Geometric Integration Theory, Princeton University Press, 1957, Princeton, NJ, xv + 397 pages. (Princeton Math. Series 21) Translated into Russian [book]. [66] Elementary structure of real algebraic varieties, Annals of Math. (2) 66, 1957, 545-556.

xiii [67] Singualrities of mappings of Euclidean spaces, Symposium Intemacional de Topologia Algebraica, Mexico, 1956,285-301, Mexico, La Universidad Nacional Autonoma. [68] (With F. Bruhat) Quelques proprietes fondamentales des ensembles analytiques-reels, Comm. Math. Helv. 33, 1959, 132-160. [69] (With A.Dold) Classification of oriented sphere bundles over a 4-complex, Annals of Math. (2) 69, 1959, 667-677. [70] On bounded functions with bounded n-th differences, AMS Proc. 10, 1959,480-481. [71] (With A. M. Gleason) The extension of linear functionals defined on H -infinity, Pacific J. Math. 12, 1962, 163-182. [72] The work of John W. Milnor, Proceedings ICM 1962m Institut Mittag-Leffler Djursholm, Sweden, x 1viii-I. [73] Local properties of analytic varieties, in: differential and combinatorial topology (Sym• posium in Honor of Marston Morse), Princeton, NJ., Princeton University Press. [74] Tangents to an analytic variety, Annals of Math (2) 81, 1965,496-549. [75] The mathematics of physical quantities. Part I, Mathematical Models for measurement, Am. Math. Monthly 75 (1968), 115-138, Part II, Quantity structures and dimensional analysis, ibid., 237-256. [76] Logic Fad or tool? Nico 4, 1969, Revue per du Centre BeIge de Pedagogie de la Mathematique, 2-14. [77] On reducibility in the four color problem, unpublished manuscript, 1971. [78] (With W. T. Tutte) Kempe chains and the four color problem, Utilitas Mathematica 2(1972), 241-281. [79] Complex Analytic Varieties, Addison-Wesley Pub. Co., Reading, MA 1972, xii +399 pp. [book]. [80] Math Activities, multilithed, Institute for Adv. Study, 1974 [book]. [81] Comment on the division of the plane by lines, Am. Math. Monthly 86(1979), p. 700. [82] Moscow 1935: Topology moving toward America, in A Century of Mathematics in America, Part I, P. L. Duren Ed., 96-117, Amer. Math.

xiv