Table Of ContentContemporary Mathematicians
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
Boston • Basel • Berlin
1992
James Eells Domingo Toledo
Department of Mathematics 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. Topology. 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
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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 Combinatorics ....................... 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[77] On reducibility in the four color problem, unpublished manuscript, 1971 179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 Marston Morse),
Princeton, NJ, Princeton University 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
Manifolds 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 manifold, 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 Algebraic Topology . . . . . . . . . . . . . . . . . . . . . 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 cohomology 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
