Celebratio Mathematica — Whitney

[1]H. Whitney: “Abstract for ‘A peculiar function’,” Bull. Am. Math. Soc. 35 : 4 (1929), pp. 442–443. Abstract only; unpublished. JFM 55.0157.05 article

[2]H. Whitney: “Abstract for ‘Note on central motions’,” Bull. Am. Math. Soc. 36 : 5 (1930), pp. 352. Abstract only; unpublished. JFM 56.0685.18 article

[3]H. Whitney: “Abstract for ‘A theory of graphs and their coloring’,” Bull. Am. Math. Soc. 36 : 11 (1930), pp. 798. Abstract only; unpublished. JFM 56.0515.13 article

[4]H. Whitney: “Abstract for ‘A normal form for maps’,” Bull. Am. Math. Soc. 36 : 3 (1930), pp. 216–217. Abstract only; unpublished. JFM 56.0515.12 article

[5]H. Whitney: “Abstract for ‘A logical expansion in mathematics’,” Bull. Am. Math. Soc. 36 : 11 (1930), pp. 798. Abstract only; published in Bull. Am. Math. Soc. 38:8 (1932), pp. 572–579. JFM 56.0060.25 article

[6]H. Whitney: “A theorem on graphs,” Ann. Math. (2) 32 : 2 (April 1931), pp. 378–390. MR 1503003 JFM 57.0727.03 Zbl 0002.16101 article

[7]H. Whitney: “Non-separable and planar graphs,” Proc. Nat. Acad. Sci. U.S.A. 17 : 2 (February 1931), pp. 125–127. JFM 57.0727.05 article

[8]H. Whitney: “The coloring of graphs,” Proc. Natl. Acad. Sci. U.S.A. 17 : 2 (February 1931), pp. 122–125. JFM 57.0727.04 Zbl 0001.29301 article

[9]H. Whitney: “Abstract for ‘On homeomorphic graphs and the connectivity of graphs’,” Bull. Am. Math. Soc. 37 : 3 (1931), pp. 168. Abstract only; unpublished. JFM 57.0758.26 article

[10]H. Whitney: “Abstract for ‘A characterization of the closed 2-cell’,” Bull. Am. Math. Soc. 37 : 11 (1931), pp. 820–821. Abstract only; published in Trans. Am. Math. Soc. 35:1 (1933), pp. 261–273. JFM 57.0758.27 article

[11]H. Whitney: “Abstract for ‘Basic graphs’,” Bull. Am. Math. Soc. 38 : 1 (1932), pp. 37–38. Abstract only; unpublished. JFM 58.0647.17 article

[12]H. Whitney: “Abstract for ‘A set of topological invariants for graphs’,” Bull. Am. Math. Soc. 38 : 1 (1932), pp. 38. Abstract only; published in Am. J. Math. 55:1–4 (1933), pp. 231–235. JFM 58.0647.19 article

[13]H. Whitney: “Abstract for ‘Conditions that a graph have a dual’,” Bull. Am. Math. Soc. 38 : 1 (1932), pp. 37. Abstract only; unpublished. JFM 58.0647.18 article

[14]H. Whitney: “Regular families of curves, I,” Proc. Natl. Acad. Sci. U.S.A. 18 : 3 (March 1932), pp. 275–278. JFM 58.0633.02 Zbl 0004.07503 article

Given a family of curves, such as the paths of particles in a steady motion of a fluid, it is often convenient to introduce a continuous function \( x^{\prime} = g(x,t) \), with the interpretation that the particle at the point \( x \) has moved to the point \( x^{\prime} \) after a time \( t \). It is also useful to have cross-sections of the curves; if the curve through any given point has a direction and this direction varies continuously with the point, we can construct a cross-section in the neighborhood of a point by drawing rays at that point perpendicular to the direction of the curve.

In this note we associate with a general point set a function \( \mu \); with its help we can, given a family of curves satisfying very weak conditions, define a function and find cross-sections as above. Examples show the usefulness of the function in topology. Conditions under which a family of curves is “regular” will be given later; the question of orienting the curves and of introducing invariant points will also be discussed.

[15]H. Whitney: “Abstract for ‘On 2-congruent graphs’,” Bull. Am. Math. Soc. 38 : 5 (1932), pp. 354. Abstract only; unpublished. JFM 58.0647.20 article

[16]H. Whitney: “Non-separable and planar graphs,” Trans. Am. Math. Soc. 34 : 2 (1932), pp. 339–362. MR 1501641 JFM 58.0608.01 Zbl 0004.13103 article

In this paper the structure of graphs is studied by purely combinatorial methods. The concepts of rank and nullity are fundamental. The first part is devoted to a general study of non-separable graphs. Conditions that a graph be non-separable are given; the decomposition of a separable graph into its non-separable parts is studied; by means of theorems on circuits of graphs, a method for the construction of non-separable graphs is found, which is useful in proving theorems on such graphs by mathematical induction. In the second part, a dual of a graph is defined by combinatorial means, and the paper ends with the theorem that a necessary and sufficient condition that a graph be planar is that it have a dual.

[17]H. Whitney: “Note on Perron’s solution of the Dirichlet problem,” Proc. Natl. Acad. Sci. U.S.A. 18 : 1 (January 1932), pp. 68–70. JFM 58.0509.01 Zbl 0003.34901 article

[18]H. Whitney: “Abstract for ‘Regular families of curves, I’,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 183. Abstract only; published in Proc. Natl. Acad. Sci. U.S.A. 18:3 (1932), pp. 275–278. JFM 58.0648.10 article

[19]H. Whitney: “Regular families of curves, II,” Proc. Natl. Acad. Sci. U.S.A. 18 : 4 (April 1932), pp. 340–342. Zbl 0004.13201 article

[20]H. Whitney: “Congruent graphs and the connectivity of graphs,” Am. J. Math. 54 : 1 (January 1932), pp. 150–168. MR 1506881 JFM 58.0609.01 Zbl 0003.32804 article

[21]H. Whitney: “A logical expansion in mathematics,” Bull. Am. Math. Soc. 38 : 8 (1932), pp. 572–579. MR 1562461 JFM 58.0605.08 Zbl 0005.14602 article

Suppose we have a finite set of objects, (for instance, books on a table), each of which either has or has not a certain given property \( A \) (say of being red). Let \( n \), or \( n(1) \) be the total number of objects, \( n(A) \) the number with the property \( A \), and \( n(\overline{A}) \) the number without the property \( A \) (with the property not-\( A \) or \( \overline{A} \)). Then obviously \[ n(\overline{A}) = n - n(A). \] Similarly, if \( n(A\,B) \) denote the number with both properties \( A \) and \( B \), and \( n(\overline{A}\,\overline{B}) \) the number with neither property, that is, with both properties not-\( A \) and not-\( B \), then \[ n(\overline{A}\,\overline{B}) = n - n(A) - n(B) + n(A\,B), \] which is easily seen to be true.

The extension of these formulas to the general case where any number of properties are considered is quite simple, and is well known to logicians. It should be better known to mathematicians also; we give in this paper several applications which show its usefulness

[22]H. Whitney: “Abstract for ‘Cross sections of curves in 3-space’,” Bull. Am. Math. Soc. 38 : 11 (1932), pp. 808. Abstract only; published in Duke Math. J. 4:1 (1938), pp. 222–226. JFM 58.0648.11 article

[23]H. Whitney: “Abstract for ‘Characteristic functions and the algebra of logic’,” Bull. Am. Math. Soc. 38 : 1 (1932), pp. 38. Abstract only; published in Ann. of Math. (2) 34:3 (1933), pp. 405–414. JFM 58.0070.06 article

[24]H. Whitney: “The coloring of graphs,” Ann. Math. (2) 33 : 4 (October 1932), pp. 688–718. MR 1503085 JFM 58.0606.01 Zbl 0005.31301 article

[25]H. Whitney: “Abstract for ‘Note on Perron’s solution of the Dirichlet problem’,” Bull. Am. Math. Soc. 38 : 1 (1932), pp. 41. Abstract only; published in Proc. Natl. Acad. Sci. U.S.A. 18:1 (1932), pp. 68–70. JFM 58.0519.04 article

[26]H. Whitney: “Abstract for ‘Regular families of curves, II’,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 183–184. Abstract only; published in Proc. Natl. Acad. Sci. U.S.A. 18:4 (1932), pp. 340–342. article

[27]H. Whitney: The coloring of graphs. Ph.D. thesis, Harvard University, 1932. Advised by G. D. Birkhoff. MR 2936470 phdthesis

[28]H. Whitney: “2-Isomorphic graphs,” Am. J. Math. 55 : 1–4 (1933), pp. 245–254. MR 1506961 JFM 59.1235.01 Zbl 0006.37005 article

[29]H. Whitney: “On the classification of graphs,” Am. J. Math. 55 : 1–4 (1933), pp. 236–244. MR 1506960 JFM 59.1233.02 Zbl 0006.37004 article

R. M. Foster [1932] has given an enumeration of graphs, for use in electrical theory. He uses two distinct methods, classifying the graphs accoring to their nullity, and according to their rank. In either case, only a certain class of graphs is listed; the remaining graphs are easily constructed from these. In the present paper we give theorems sufficient to put the first method of classification on a firm foundation.

In this method (see §§8 and 9), only the elementary graphs (see §4), or graphs whose connected pieces are elementary, are listed. These graphs are most easly formed from the basic graphs (see §5), and these, from the basic graphs of nullity one less. This manner of constructing the graphs, and in particular, the important notion of basic graphs, is due to Foster. The definition of elementary graphs and the proofs are, in general, due to the author.

[30]H. Whitney: “Characteristic functions and the algebra of logic,” Ann. Math. (2) 34 : 3 (July 1933), pp. 405–414. MR 1503114 JFM 59.0051.05 Zbl 0007.19402 article

The algebra of logic differs from ordinary algebra in that there are two formulas \begin{align*} A \odot A &= A,\\ A \oplus A &= A \end{align*} which have no analogues in ordinary algebra. On this account the rules of the algebra of logic must be learned afresh; this usually involves studying a set of axioms and deducing formulas from them [Lewis 1918, Ch. II; Hilbert and Ackermann 1928, Ch. I–II]. We shall show in this paper how, by means of “characteristic functions” [de la Vallée Poussin 1916], the formulas of the algeba of logic can be expressed as equations of ordinary algebra; with this representation, the common operations on sets are easily understood.

In Part II “generalized sets” are taken up; these are useful in various mathematical theories.

[31]H. Whitney: “Abstract for ‘Differentiable functions defined on closed sets’,” Bull. Am. Math. Soc. 39 : 5 (1933), pp. 352–353. Abstract only; unpublished. JFM 59.0292.06 article

[32]H. Whitney: “Abstract for ‘Functions differentiable on the boundaries of regions’,” Bull. Am. Math. Soc. 39 : 11 (1933), pp. 874. Abstract only; published in Ann. Math. (2) 35:3 (1934), pp. 482–485. JFM 59.0292.09 article

[33]H. Whitney: “Abstract for ‘Differentiable functions defined in closed sets, I’,” Bull. Am. Math. Soc. 39 : 9 (1933), pp. 673. Abstract only; published in Trans. Am. Math. Soc. 36:2 (1934), pp. 369–387. JFM 59.0292.08 article

[34]H. Whitney: “Regular families of curves,” Ann. Math. (2) 34 : 2 (April 1933), pp. 244–270. MR 1503106 JFM 59.1256.04 Zbl 0006.37101 article

The topological properties of families of curves, occurring as solutions of differential equations, were studied extensively by Poincaré. Other authors have continued these investigations, building up a considerable theory of the behavior of such curves, as determined by the differential equations considered [Bierbach 1930, Part I, Ch. IV]. These equations usually are, or may be, taken in the form \[ \frac{dx_i}{dt} = X_i(x_1,\dots,x_n)\qquad (i=1,\dots,n); \] the solutions may then be considered as the paths of particles in \( n \)-space, \( t \) being the time.

In this paper we study families of curves, given abstractly. The curves are required to satisfy a certain condition, the “regularity condition” (§6); it is certainly satisfied by solutions of differential equations. The main object of the paper is first, to prove the existence of cross-sections of these curves (§17), and secondly, to show how a function \( q=f(p,t) \) can be defined (under slight further restrictions), which may be interpreted as defining a motion of particles along the curves of the family (§27).

[35]H. Whitney: “Planar graphs,” Fundam. Math. 21 (1933), pp. 73–84. JFM 59.1235.03 Zbl 0008.08501 article

[36]H. Whitney: “A set of topological invariants for graphs,” Am. J. Math. 55 : 1–4 (1933), pp. 231–235. MR 1506959 JFM 59.1235.02 Zbl 0006.37003 article

[37]H. Whitney: “Abstract for ‘Analytic extensions of differentiable functions defined on closed sets’,” Bull. Am. Math. Soc. 39 : 1 (1933), pp. 31. Abstract only; published in Trans. Am. Math. Soc. 36:1 (1934), pp. 63–89. JFM 59.0260.04 article

[38]H. Whitney: “A characterization of the closed 2-cell,” Trans. Am. Math. Soc. 35 : 1 (January 1933), pp. 261–273. MR 1501683 JFM 59.0566.02 Zbl 0006.08303 article

A number of characterizations have been given of the simple closed surface. The proofs involve considerable point set difficulties. We give here a characterization of the closed 2-cell, that is, a point set homeomorphic with a circle and its interior. The fundamental theorem is partly of a combinatorial and partly of a continuity nature. It reads

Let \( R \) be a continuous curve containing the simple closed curve \( J \) such that

\( J \) is irreducibly homologous to zero in \( R \), and
If \( \gamma \) is an arc with just its two end points \( a \) and \( b \) on \( J \), then \( R-\gamma \) is not connected.

Let \( R^{\prime} \) and \( J^{\prime} \) be defined similarly. Then \( R \) and \( R^{\prime} \) are homeomorphic, with \( J \) corresponding with \( J^{\prime} \).

[39]H. Whitney: “Abstract for ‘Derivatives, difference quotients, and Taylor’s formula, I’,” Bull. Am. Math. Soc. 39 : 7 (1933), pp. 508. Abstract only; published in Bull. Am. Math. Soc. 40:2 (1934), pp. 89–94. JFM 59.0292.07 article

[40]H. Whitney: “Abstract for ‘Derivatives, difference quotients, and Taylor’s formula, II’,” Bull. Am. Math. Soc. 39 : 11 (1933), pp. 875. Abstract only; published in Ann. Math. (2) 35:3 (1934), pp. 476–481. article

[41]H. Whitney: “Abstract for ‘On the abstract properties of linear dependence’,” Bull. Am. Math. Soc. 40 : 9 (1934), pp. 663. Abstract only; published in Am. J. Math. 57:3 (1935), pp. 509–533. JFM 60.0883.04 article

[42]H. Whitney: “Differentiable functions defined in closed sets, I,” Trans. Am. Math. Soc. 36 : 2 (1934), pp. 369–387. MR 1501749 JFM 60.0217.02 Zbl 0009.20803 article

In a recent paper [1934] the author has shown that if a function \( f(x) \) defined in a closed set \( A \) in \( n \)-space \( E \) satisfies certain conditions involving Taylor’s formula (in finite form), i.e. if it is “of class \( C^m \) in \( A \),” then its definition can be extended over \( E \) so that it will have continuous partial derivatives through the \( m \)-th order. In this paper we restrict ourselves to the one-dimensional case. (For the above theorem in this case, see §4.) Let \( x_0,\dots,x_m \) be distinct points of \( A \). If \[ P(x) = c_0 + \cdots + c_mx^m \] is the polynomial of degree at most \( m \) such that \( P(x_i) = f(x_i) \) (\( i=0,\dots,m \)), the \( m \)-th difference quotient of \( f(x) \) at these points is \[ \Delta_{0\cdots m}f = \Delta^mf(x) = m!\,c_m .\] The main object of this paper is to prove (see §§2 and 3 for definitions)

A necessary and sufficient condition that \( f(x) \) be of class \( C^m \) in \( A \) is that \( \Delta^m f(x) \) converge in \( A \).

[43]H. Whitney: “Abstract for ‘Analytic approximations to manifolds’,” Bull. Am. Math. Soc. 40 : 9 (1934), pp. 663. Abstract only; unpublished. JFM 60.0536.01 article

[44]H. Whitney: “Analytic extensions of differentiable functions defined in closed sets,” Trans. Am. Math. Soc. 36 : 1 (1934), pp. 63–89. MR 1501735 JFM 60.0217.01 Zbl 0008.24902 article

Let \( A \) be a closed set, bounded or unbounded, in euclidean \( n \)-space \( E \), and let \( f(x) \) be a function, defined and continuous in \( A \). It is well known that this function can be extended so as to be continuous throughout \( E \). If \( A \) satisfies certain conditions, the solution of the Dirichlet problem is a function harmonic in \( E-A \) and taking on the given boundary values in \( A \). Two questions which arise are the following: Is there always a function differentiable, or perhaps analytic, in \( E-A \), and taking on the given values in \( A \)? If the given function \( f(x) \) is in some sense differentiable in \( A \), can the extension \( F(x) \) be made differentiable to the same order throughout \( E \)?

These questions are answered in the affirmative in Theorem I. We use a definition of the derivatives of a function in a general set which arises naturally from a consideration of Taylor’s formula. In Part II, a differentiable extension of \( f(x) \) is found, whether \( f(x) \) is differentiable to finite or infinite order. Part III is devoted to some general approximation theorems.

[45]H. Whitney: “Derivatives, difference quotients and Taylor’s formula, II,” Ann. Math. (2) 35 : 3 (July 1934), pp. 476–481. MR 1503173 JFM 60.0963.01 Zbl 0010.01504 article

[46]H. Whitney: “Functions differentiable on the boundaries of regions,” Ann. Math. (2) 35 : 3 (July 1934), pp. 482–485. MR 1503174 JFM 60.0217.03 Zbl 0009.30901 article

Let the function \( f(x_1,\dots,x_n) \) be defined in the bounded region \( R \) of \( n \)-space \( E \), and suppose \( f \) has continuous \( m \)-th partial derivatives in \( R \), i.e. \( f \) “is of class \( C^m \)” in \( R \). If \( B \) is the boundary of \( R \), how shall we decide whether \( f \) is of class \( C^m \) in \( R+B \)? If the dervatives of \( f \) take on boundary values on \( B \), it would be natural to define the derivatives on \( B \) as the limit of their values in \( R \). But it is easy to construct a region \( R \) and a function \( f \) such that the \( k \)-th partial dervatives of \( f \) (\( 0 < k\leq m \)) are continuous in \( R+B \), whereas at a certain boundary point \( P \) of \( B \), \( f \) is not continuous; it seems unreasonable in this case to say that \( f \) is of class \( C^m \) in \( R+B \).

If it is possible to extend the definition of \( f \) throughout a region containing \( R+B \) so that it has continuous \( m \)-th partial derivatives there, we may then surely say that \( f \) is of class \( C^m \) in \( R+B \); this is the definition we shall use. We show in this note that, for certain regions, for a function to be of class \( C^m \) in the closed region, it is sufficient that the \( m \)-th partial derivatives be continuous on the boundary.

[47]H. Whitney: “Erratum: ‘Derivatives, difference quotients, and Taylor’s formula’,” Bull. Am. Math. Soc. 40 : 12 (1934), pp. 894.

[48]H. Whitney: “Derivatives, difference quotients, and Taylor’s formula,” Bull. Am. Math. Soc. 40 : 2 (1934), pp. 89–94. MR 1562803 JFM 60.0962.02 Zbl 0009.05903 article

[49]H. Whitney: “Abstract for ‘A numerical equivalent of the four-color problem’,” Bull. Am. Math. Soc. 40 : 1 (1934), pp. 36. Abstract only; published in Monatsh. Math. Phys. 45:1 (1936), pp. 207–213. JFM 60.0535.04 article

[50]H. Whitney: “Abstract for ‘A function not constant on a connected set of critical points’,” Bull. Am. Math. Soc. 41 : 11 (1935), pp. 796. Abstract only; published in Duke Math. J. 1:4 (1935), pp. 514–517. JFM 61.0262.07 article

[51]H. Whitney: “Abstract for ‘Sphere-spaces, with applications’,” Bull. Am. Math. Soc. 41 : 5 (1935), pp. 337–338. Abstract only; unpublished. JFM 61.0641.13 article

[52]H. Whitney: “A function not constant on a connected set of critical points,” Duke Math. J. 1 : 4 (1935), pp. 514–517. MR 1545896 JFM 61.1117.01 Zbl 0013.05801 article

Let \( f(x_1,\dots,x_n) \) be a function of class \( C^m \) (i.e., with continuous partial derivatives through the \( m \)-th order) in a region \( R \). Any point at which all its first partial derivatives vanish is called a critical point of \( f \). Suppose every point of a connected set \( A \) of points in \( R \) is a critical point. It is natural to suspect then that \( f \) is a constant on \( A \). But this need not be so. We construct below an example with \( n=2 \), \( m=1 \), \( A = \) an arc. The example may be extended to the case \( n=n \), \( m=n-1 \), \( A = \) an arc. The arc and the function on the arc are easily defined. The extension of the function through the rest of the plane or space is given by a theorem of the author

[53]H. Whitney: “On the abstract properties of linear dependence,” Am. J. Math. 57 : 3 (July 1935), pp. 509–533. MR 1507091 JFM 61.0073.03 Zbl 0012.00404 article

Let \( C_1, C_2, \dots, C_n \) be the columns of a matrix \( \mathbf{M} \). Any subset of these columns is either linearly independent or linearly dependent; the subsets thus fall into two classes. These classes are not arbitrary; for instance, the two following theorems must hold:

Any subset of an independent set is independent.
If \( \mathbf{N}_p \) and \( \mathbf{N}_{p+1} \) are independent sets of \( p \) and \( p+1 \) columns respectively, then \( \mathbf{N}_p \) together with some column of \( \mathbf{N}_{p+1} \) forms an independent set of \( p + 1 \) columns.

There are other theorems not deducible from these; for in §16 we give an example of a system satisfying these two theorems but not representing any matrix. Further theorems seem, however, to be quite difficult to find. Let us call a system obeying (a) and (b) a “matroid.” The present paper is devoted to a study of the elementary properties of matroids. The fundamental question of completely characterizing systems which represent matrices is left unsolved. In place of the columns of a matrix we may equally well consider points or vectors in a Euclidean space, or polynomials, etc.

[54]H. Whitney: “Sphere-spaces,” Proc. Natl. Acad. Sci. U.S.A. 21 : 7 (1935), pp. 464–468. JFM 61.0624.01 Zbl 0012.12603 article

[55]H. Whitney: “Differentiable manifolds in Euclidean space,” Proc. Natl. Acad. Sci. U.S.A. 21 : 7 (July 1935), pp. 462–464. JFM 61.0624.02 Zbl 0012.12602 article

[56]H. Whitney: “Abstract for ‘Differentiable manifolds in Euclidean space’,” Bull. Am. Math. Soc. 41 : 9 (1935), pp. 625–626. Abstract only; published in Proc. Natl. Acad. Sci. U.S.A. 21:7 (1935), pp. 462–464. JFM 61.0641.14 article

[57]H. Whitney: “Abstract for ‘On a theorem of H. Hopf’,” Bull. Am. Math. Soc. 41 : 11 (1935), pp. 787. Abstract only; unpublished. JFM 61.0641.07 article

[58]H. Whitney: “Sphere-spaces,” Rec. Math. Moscou, n. Ser. 1 : 5 (1936), pp. 787–791. JFM 62.0662.03 Zbl 0016.13904 article

[59]H. Whitney: “A numerical equivalent of the four color map problem,” Monatsh. Math. Phys. 45 : 1 (1936), pp. 207–213. MR 1550643 JFM 63.0551.01 Zbl 0016.42002 article

[60]H. Whitney: “Abstract for ‘Differentiable functions defined in arbitrary subsets of Euclidean space’,” Bull. Am. Math. Soc. 42 : 1 (1936), pp. 23. Abstract only; published in Trans. Am. Math. Soc. 40:2 (1936), pp. 309–317. JFM 62.0272.04 article

[61]H. Whitney: “Abstract for ‘The maps of a 4-complex into a 2-sphere’,” Bull. Am. Math. Soc. 42 : 5 (1936), pp. 338–339. Abstract only; unpublished. JFM 62.0695.13 article

[62]H. Whitney: “Differentiable manifolds in Euclidean space,” Rec. Math. Moscou, n. Ser. 1 : 5 (1936), pp. 783–786. JFM 62.0806.03 Zbl 0016.08501 article

[63]H. Whitney: “Abstract for ‘Matrices of integers and combinatorial topology’,” Bull. Am. Math. Soc. 42 : 11 (1936), pp. 816. Abstract only; published in Duke Math. J. 3:1 (1937), pp. 35–45. JFM 62.0695.05 article

[64]H. Whitney: “Abstract for ‘On the maps of an \( n \)-sphere into another \( n \)-sphere’,” Bull. Am. Math. Soc. 42 : 11 (1936), pp. 810. Abstract only; published in Duke Math. J. 3:1 (1937), pp. 46–50. JFM 62.0695.12 article

[65]H. Whitney: “Differentiable functions defined in arbitrary subsets of Euclidean space,” Trans. Am. Math. Soc. 40 : 2 (1936), pp. 309–317. MR 1501875 JFM 62.0265.02 Zbl 0015.01001 article

In a former paper [1934] we studied the differentiability of a function defined in closed subsets of Euclidean \( n \)-space \( E \). We consider here the differentiability “about” an arbitrary point of a function defined in an arbitrary subset of \( E \). We show in Theorem 1 that any function defined in a subset \( A \) of \( E \) which is differentiable about a subset \( B \) of \( E \) may be extended over \( E \) so that it remains differentiable about \( B \). This theorem is a generalization of [1934, Lemma 2]. We show further that any function of class \( C^m \) about a set \( B \) is of class \( C^{m-1} \) about an open set \( B^{\prime} \) containing \( B \). In the second part of the paper we consider some elementary properties of differentiable functions, such as: the sum or product of two such functions is such a function. We end with the theorem that differentiability is a local property.

[66]H. Whitney: “The imbedding of manifolds in families of analytic manifolds,” Ann. Math. (2) 37 : 4 (October 1936), pp. 865–878. MR 1503315 JFM 62.0806.02 Zbl 0015.18002 article

[67]H. Whitney: “Abstract for ‘On regular closed curves in the plane’,” Bull. Am. Math. Soc. 42 : 9 (1936), pp. 639. Abstract only; published in Compositio Math. 4 (1937), pp. 276–284. JFM 62.0809.05 article

[68]H. Whitney: “Differentiable manifolds,” Ann. Math. (2) 37 : 3 (July 1936), pp. 645–680. MR 1503303 JFM 62.1454.01 Zbl 0015.32001 article

The main purpose of this paper is to provide tools of a purely analytic character for a general study of the topology of differentiable manifolds, and maps of them into other manifolds. A differentiable manifold is generally defined in one of two ways; as a point set with neighborhoods homeomorphic with Euclidean space \( E_n \), coördinates in overlapping neighborhoods being related by a differentiable transformation, or as a subset of \( E_n \), defined near each point by expressing some of the coördinates in terms of the others by differentiable functions.

The first fundamental theorem is that the first definition is no more general than the second; any differentiable manifold may be imbedded in Euclidean space. In fact, it may be made into an analytic manifold in some \( E_n \). As a corollary, it may be given an analytic Riemannian metric. The second fundamental theorem (when combined with the first) deals with the smoothing out of a manifold. Let \( f \) be a map of any character (continuous or differentiable, without an inverse) of a differentiable manifold \( M \) of dimension \( m \) into another, \( N \), of dimension \( n \). (Either manifold might be an open subset of Euclidean space.) Then if \( n\geq 2m \), we may alter \( f \) as little as we please, forming a regular map \( F \). (A map is regular if, near each point, it is differentiable and has a differentiable inverse.). Moreover, if \( n\geq 2m+1 \), \( F \) may be made (1-1). We show in Theorem 6 that if \( n\geq 2m + 2 \), then any two regular maps \( f_0 \), \( f_1 \) of \( M \) into \( E_m \) are equivalent, in the following sense. \( f_0(M) \) may be deformed into \( f_1(M) \) by maps \( f_t \) (\( 0\leq t\leq 1 \)) so that the path crossed by the manifold is the regular map of an \( (m+1) \)-dimensional manfold. Moreover, if \( n\geq 2m + 3 \), and \( f_0(M) \) and \( f_1(M) \) are non-singular, so is the \( (m+1) \)-manifold.

[69]H. Whitney: “On regular closed curves in the plane,” Compositio Math. 4 (1937), pp. 276–284. MR 1556973 JFM 63.0647.01 Zbl 0016.13804 article

We consider in this note closed curves with continuously turning tangent, with any singularities. To each such curve may be assigned a “rotation number” \( \gamma \), the total angle through which the tangent turns while traversing the curve. (For a simple closed curve, \( \gamma = \pm 2\pi \).) Our object is two-fold; to show that two curves with the same rotation number may be deformed into each other, and to give a method of determining the rotation number by counting the algebraic number of times that the curve cuts itself (if the curve has only simple singularities,–see Lemma 2).

This paper may be considerered as a continuation of a paper of H. Hopf [1935]; we assume a knowledge of the first part of his paper.

[70]H. Whitney: “Topological properties of differentiable manifolds,” Bull. Am. Math. Soc. 43 : 12 (1937), pp. 785–805. MR 1563640 JFM 63.0556.03 Zbl 0018.23902 article

Suppose we wish to study differential geometry in the \( n \)-dimensional manifold \( M^n \). At each point \( p \) of \( M^n \), the possible differentials (or “tangent” vectors) form an \( n \)-dimensional vector space \( V(p) \), the so-called tangent space at \( p \). For topological considerations, it is sufficient to consider vectors of unit length, or directions, which form a sphere \( S(p) \) of dimension \( n-1 \). This set of spheres forms the tangent sphere-space of \( M^n \). Most of our work will be on the problem, how do the spheres fit together over the whole manifold? Suppose \( M^n \) is imbedded in a higher dimensional manifold \( M^m \) (for instance, euclidean \( E^m \)). Then we may consider the normal unit vectors at each point, forming an \( (m-n-1) \)-sphere, and thus the normal sphere-space.

In the last part we give some fundamental results, due to de Rham, on the theory of multiple integration in a manifold. If we wish to integrate over an \( r \)-dimensional subset, and no measure function is given, we integrate “differential forms,” in other words, “alternating covariant tensors” of order \( r \); we shall call these simply \( r \)-functions. It is necessary for various purposes to find what “exact” \( r \)-functions exist.

[71]H. Whitney: “The maps of an \( n \)-complex into an \( n \)-sphere,” Duke Math. J. 3 : 1 (1937), pp. 51–55. MR 1545972 JFM 63.1162.02 Zbl 0016.22901 article

The classes of maps of an \( n \)-complex into an \( n \)-sphere were classified by H. Hopf in [1933]. Recently, W. Hurewicz [1935–1936] has extended the theorem by replacing the \( n \)-sphere by much more general spaces. Freudenthal [1935, footnote 8] and Steenrod have noted that the theorem and proof are simplified by using real numbers reduced mod 1 in place of integers as coefficients in the chains considered. We shall give here a statement of the theorem which seems the most natural; the proof is quite simple. As in the original proof by Hopf, we shall base it on a more general extension theorem.

The fundamental tool of the paper is the relation of “coboundary” [Whitney 1937]; it has come into prominence in the last few years.

In later papers we shall classify the maps of a 3-complex into a 2-sphere and of an \( n \)-complex into projective \( n \)-space.

[72]H. Whitney: “On the maps of an \( n \)-sphere into another \( n \)-sphere,” Duke Math. J. 3 : 1 (1937), pp. 46–50. MR 1545971 JFM 63.1162.01 Zbl 0016.22808 article

[73]H. Whitney: “On matrices of integers and combinatorial topology,” Duke Math. J. 3 : 1 (1937), pp. 35–45. MR 1545970 JFM 64.1265.03 Zbl 0016.27805 article

Our object is to give an elementary account of some algebraic theorems, with some immediate applications in combinatorial topology, in particular, in the theory of homology and cohomology groups. The theorems are to a certain extent known, if in somewhat different forms.

The main tool in the algebraic part is the theory of group pairs, and in particular the question of when one group “resolves” or “completely resolves” another. The main theorems are on the existence of extensions of a homomorphism, and the existence of solutions of linear equations, with a matrix of integers and elements of an abelian group as unknowns. In each theorem, two types of conditions are employed, one using mod \( m \) properties, the other using group pairs. We shall use only discrete groups (except in Theorem 1).

[74]H. Whitney: “Analytic coordinate systems and arcs in a manifold,” Ann. Math. (2) 38 : 4 (October 1937), pp. 809–818. MR 1503372 JFM 63.0651.01 Zbl 0017.42802 article

[75]H. Whitney: “On products in a complex,” Proc. Natl. Acad. Sci. U.S.A. 23 : 5 (May 1937), pp. 285–291. JFM 63.1160.02 Zbl 0016.42001 article

[76]H. Whitney: “Tensor products of Abelian groups,” Duke Math. J. 4 : 3 (1938), pp. 495–528. MR 1546071 JFM 64.0065.01 Zbl 0019.39802 article

Let \( G \) and \( H \) be Abelian groups. Their direct sum \( G \oplus H \) consists of all pairs \( (g,h) \), with \[ (g,h)+(g^{\prime},h^{\prime}) = (g+g^{\prime},h+h^{\prime}) .\] If we consider \( G \) and \( H \) as subgroups of \( G\oplus H \), with elements \( g=(g,0) \) and \( h=(0,h) \), then we may form \( g+h \), and the ordinary laws of addition hold. Our object in this paper is to construct a group \( G\circ H \) from \( G \) and \( H \), in which we can form \( g\cdot h \), with the properties of multiplication; that is, the distributive laws \begin{equation*}\tag{1} \begin{aligned} (g+g^{\prime})\cdot h &= g\cdot h + g^{\prime}\cdot h,\\ g\cdot (h+h^{\prime}) &= g\cdot h + g\cdot h^{\prime} \end{aligned} \end{equation*} hold. Clearly \( G\circ H \) must contain elements of the form \( \sum g_i\cdot h_i \); it turns out (Theorem 1) that with these elements, assuming only (1), we obtain an Abelian group, which we shall call the tensor product of \( G \) and \( H \).

[77]H. Whitney: “Book review: Leçons sur les principes topologiques de la théorie des fonctions analytiques, by S. Stöilow,” Bull. Am. Math. Soc. 44 : 11 (1938), pp. 758–759. MR 1563864 article

[78]H. Whitney: “On products in a complex,” Ann. Math. (2) 39 : 2 (April 1938), pp. 397–432. MR 1503416 JFM 64.1265.04 Zbl 0019.14204 article

In classical homology theory, founded by Poincaré, the fundamental operation is that of forming the boundary \( \partial A^p \) of a chain \( A^p \). This is found by multiplying the coefficients of \( A^p \) into a matrix of incidence. Algebraically, an equally obvious operation, using the same matrix of incidence, forms the “coboundary” \( \delta A^{p-1} \) from a given \( A^{p-1} \). It has recently been discovered that the algebraic part of the theory of intersections of chains in a manifold, when interpreted with the other operation, could be generalized to arbitrary complexes. It is the object of this paper to give a complete treatment of the fundamentals of this theory. We use a general type of complex and general coefficient groups, and prove the required invariance theorems. Parts of the paper are new only in form. Various notions used here appear first in Tucker’s thesis [1933].

[79]H. Whitney: “Cross-sections of curves in 3-space,” Duke Math. J. 4 : 1 (1938), pp. 222–226. MR 1546045 JFM 64.0626.03 Zbl 0018.42603 article

[80]H. Whitney: “Some combinatorial properties of complexes,” Proc. Nat. Acad. Sci. U.S.A. 26 : 2 (February 1940), pp. 143–148. MR 0001337 JFM 66.0947.04 Zbl 0023.38303 article

[81]H. Whitney: “On the theory of sphere-bundles,” Proc. Nat. Acad. Sci. U.S.A. 26 : 2 (February 1940), pp. 148–153. MR 0001338 JFM 66.0952.01 Zbl 0023.17603 article

[82]H. Whitney: “On the topology of differentiable manifolds,” pp. 101–141 in Lectures in topology (Ann Arbor, MI, 24 June–6 July 1940). Edited by R. L. Wilder and W. L. Ayres. University of Michigan Press (Ann Arbor, MI), 1941. MR 0005300 Zbl 0063.08233 incollection

Raymond Louis Wilder	Related
William Leake Ayres	Related

[83]H. Whitney: “On regular families of curves,” Bull. Am. Math. Soc. 47 (1941), pp. 145–147. MR 0004115 JFM 67.0744.03 Zbl 0025.23602 article

[84]H. Whitney: “Differentiability of the remainder term in Taylor’s formula,” Duke Math. J. 10 : 1 (1943), pp. 153–158. MR 0007782 Zbl 0063.08234 article

Taylor’s formula with exact remainder may be written \begin{align*} f(x) = f(0) + \frac{f^{\prime}(0)}{1!}x &+ \cdots\\ &+ \frac{f^{(n-1)}(0)}{(n-1)!}x^{n-1} + \frac{1}{n!}x^nf_n(x), \end{align*} where \[ f_n(x) = \frac{n}{x^n}\int_0^x (x-t)^{n-1}f^{(n)}(t)\,dt \qquad (x\neq 0). \] The main object of the paper (Theorem 1) is to show that if \( f \) is of class \( C^{n+p} \) (i.e., it has continuous derivatives through the order \( n+p \)), then \( f_n \) is of class \( C^p \), but not necessarily of higher class. The condition that \( f \) be of class \( C^{n+p} \) may be expressed purely in terms of \( f_n \) (Theorem 2). A generalization is given to several dimensions (Theorem 3). The functions are always assumed defined in a neighborhood of the origin. The second theorem will be used in the following note; the results of both papers are essential in studying the singularities of certain mappings (see the next following paper).

[85]H. Whitney: “Differentiable even functions,” Duke Math. J. 10 : 1 (1943), pp. 159–160. MR 0007783 Zbl 0063.08235 article

[86]H. Whitney: “The general type of singularity of a set of \( 2n-1 \) smooth functions of \( n \) variables,” Duke Math. J. 10 : 1 (1943), pp. 161–172. MR 0007784 Zbl 0061.37207 article

Let a region \( R \) of \( n \)-space \( E^n \), or more generally, of a differentiable \( n \)-manifold, be mapped differentiably into \( m \)-space \( E^m \). If \( m\geq 2n \), it is always possible [1937, p. 818; 1936], by a slight alteration of the mapping function \( f \) (letting also any finite number of derivatives change arbitrarily slightly), to obtain a mapping \( f^* \) which is everywhere regular. That is, for any \( p \) in \( R \), and any set of independent vectors \( u_1,\dots,u_n \), in \( R \) at \( p \), \( f^* \) carries these vectors into independent vectors. Here, vector equals the vector in “tangent space” equals the differential. As a consequence, some neighborhood \( U \) of \( p \) is mapped by \( f \) in a one-one way. The object of this paper is to determine what can be obtained by slight alterations of \( f \) in case \( m=2n-1 \). It turns out that any singularities may be made into a fixed kind. (It will be shown in other papers that any smooth \( n \)-manifold may be imbedded in \( (2n) \)-space, and may be immersed (self-intersections allowed) in \( (2n-1) \)-space.)

[87]H. Whitney: “The self-intersections of a smooth \( n \)-manifold in \( 2n \)-space,” Ann. Math. (2) 45 : 2 (April 1944), pp. 220–246. MR 0010274 Zbl 0063.08237 article

Let \( f \) be a regular mapping (see the definitions below) of the simple closed curve \( M^1 \) into the plane \( E^2 \). The resulting curve \( C=f(M) \) may cut itself a number of times; if this number is finite, and the “positive” and “negative” self-intersections are distinguished, the algebraic number \( I_f \) of them is invariant under “regular deformations,” which keep the mapping always regular. We may determine \( I_f \) by considering the space \( \mathfrak{T} \) of ordered pairs of points of \( M^1 \), mapping it into \( E^2 \) in a manner determined by \( f \) (see below), and counting the coverings of the origin. The object of this paper is to study the situation in \( n \) dimensions, for regular mappings of a manifold \( M^n \) into \( E^{2n} \). The main theorem (which is trivial for \( n=1 \) and well known for \( n=2 \)) is that \( M^n \) may be imbedded in \( E^{2n} \).

[88]H. Whitney: “Topics in the theory of Abelian groups, I: Divisibility of homomorphisms,” Bull. Am. Math. Soc. 50 (1944), pp. 129–134. MR 0009404 Zbl 0061.03504 article

The theory of character groups of Abelian groups has in recent years become of great importance, especially through applications to topology and algebra. The character group of \( G \) is the group \( H \) of homomorphisms of \( G \) into the real numbers mod 1. Extending this, we may consider the group \( H=\operatorname{Hom}(G,Z) \) of homomorphisms of \( G \) into a third group \( Z \), or more generally, a “pairing” of groups \( H \) and \( G \) into \( Z \): a multiplication \( h\cdot g=z \), satisfying both distributive laws.

Of course the duality theorems for character groups will not hold in the more general cases; but, under certain conditions, substitutes may hold. We expect in later notes to give various facts about pairings, and the closely associated problem of divisibility by integers. In the present note, we answer the question of when a homomorphism of \( G \) into \( Z \) is divisible by an integer \( m \); this has an immediate application to a theorem in combinatorial topology.

[89]H. Whitney: “The singularities of a smooth \( n \)-manifold in \( (2n-1) \)-space,” Ann. Math. (2) 45 : 2 (April 1944), pp. 247–293. MR 0010275 Zbl 0063.08238 article

[90]H. Whitney: “On the extension of differentiable functions,” Bull. Am. Math. Soc. 50 (1944), pp. 76–81. MR 0009785 Zbl 0063.08236 article

[91]H. Whitney: “Complexes of manifolds,” Proc. Nat. Acad. Sci. U.S.A. 33 : 1 (January 1947), pp. 10–11. MR 0019306 Zbl 0029.41903 article

[92]H. Whitney: “Geometric methods in cohomology theory,” Proc. Nat. Acad. Sci. U.S.A. 33 : 1 (January 1947), pp. 7–9. MR 0019305 Zbl 0030.37402 article

[93]H. Whitney: “Algebraic topology and integration theory,” Proc. Nat. Acad. Sci. U.S.A. 33 : 1 (January 1947), pp. 1–6. MR 0019304 Zbl 0029.42002 article

A smooth manifold \( M \) (smooth means continuously differentiable) is a well-defined concept. A bounded manifold may have, for boundary, anything from a smooth manifold to quite a general point set. For analytical purposes, the boundary should at least be made up of pieces of manifolds, joining together smoothly; that is, it should be formed of a “complex of manifolds,” or “complifold” for short. Then \( M = \sigma^n \), together with the boundary cells \( \sigma_i^r \), forms a complifold \( K \). If one tries to define \( K \) abstractly, one finds that all sorts of curious situations may occur locally. In this preliminary note, we wish to point out some of the properties that will bring \( K \) back to reasonableness. If \( K \) is imbeddable in a Euclidean space \( E^m \), each cell being imbedded smoothly, this should be satisfactory.

[94]H. Whitney: “On ideals of differentiable functions,” Am. J. Math. 70 : 3 (July 1948), pp. 635–658. MR 0026238 Zbl 0037.35502 article

[95]L. H. Loomis and H. Whitney: “An inequality related to the isoperimetric inequality,” Bull. Am. Math. Soc. 55 (1949), pp. 961–962. MR 0031538 Zbl 0035.38302 article

[96]H. Whitney: “Relations between the second and third homotopy groups of a simply connected space,” Ann. Math. (2) 50 : 1 (January 1949), pp. 180–202. MR 0034019 Zbl 0031.28403 article

In the study of the classification and extension of mappings into simply connected spaces \( R \) (see two forthoming papers), certain special mappings \( \varphi \) of a 3-sphere \( S_0^3 \) into \( R \) turn out to be of importance. There is a set of “tubes” \( T_i \) in \( S_0^3 \); \( \varphi \) maps each cross section of a single tube in the same manner into \( R \), and maps all of \( S_0^3 \) outside these tubes into a fixed point of \( R \).

The mapping \( \varphi \), in each cross section of a tube, determines an element \( \alpha \) of the second homotopy group \( \pi_2 = \pi_2(R) \) of \( R \); in \( S_0^3 \), it determines an element \( \xi \) of the third homotopy group \( \pi_3 = \pi_3(R) \). It is the purpose in Part II of this paper to study the relation between these elements. The formulas are based on two fundamental operations on elements of \( \pi_2 \), giving elements of \( \pi_3 \); these operations have been defined by J. H. C. Whitehead [1941]. (His operations are defined for more general dimensions.)

If \( \pi_2 \) has elements of finite order, a somewhat more complicated system is necessary. Suppose \( \alpha \) is of order \( n \). Then we may have cylinders, which we call “junctions”, into each of which \( n \) tubes enter; the cylinders are mapped into \( R \) in some fixed fashion, which need not be specified. The resulting element of \( \pi_3 \) is studied in Part III.

[97]H. Whitney: “La topologie algébrique et la théorie de l’intégration” [Algebraic topology and the theory of integration], pp. 107–113 in Topologie algébrique (Paris, 26 June–2 July 1947). Edited by A. Denjoy. Colloques Internationaux du Centre National de la Recherche Scientifique 12. CNRS (Paris), 1949. MR 0034022 Zbl 0041.52101 incollection

[98]H. Whitney: “An extension theorem for mappings into simply connected spaces,” Ann. Math. (2) 50 : 2 (April 1949), pp. 285–296. MR 0034021 Zbl 0037.10002 article

[99]H. Whitney: “Classification of the mappings of a 3-complex into a simply connected space,” Ann. Math. (2) 50 : 2 (April 1949), pp. 270–284. MR 0034020 Zbl 0040.25803 article

[100]H. Whitney: “On totally differentiable and smooth functions,” Pacific J. Math. 1 : 1 (1951), pp. 143–159. MR 0043878 Zbl 0043.05803 article

H. Rademacher has proved that a function of \( n \) variables satisfying a Lipschitz condition is totally differentiable a.e. (almost everywhere) (see, for instance [Saks 1937, pp. 310–311]). It was discovered by H. Federer (though not stated as a theorem; see [1944, p. 442]) that if \( f \) is totally differentiable a.e. in the bounded set \( P \), then there is a closed set \( Q\subset P \) with the measure \( |P-Q| \) as small as desired, such that \( f \) is smooth (continuously differentiable) in \( Q \); that is, the values of \( f \) in \( Q \) may be extended through space so that the resulting function \( g \) is smooth there.

Theorem 1 of the present paper strengthens the latter theorem by showing that \( f \) is approximately totally differentiable a.e. in \( P \) if and only if \( Q \) exists with the above property. The rest of the paper gives further theorems in the direction of Federer’s Theorem.

[101]H. Whitney: “\( r \)-dimensional integration in \( n \)-space,” pp. 245–256 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 3 August–6 September 1950), vol. I. American Mathematical Society (Providence, RI), 1952. MR 0043879 Zbl 0049.04102 inproceedings

[102]H. Whitney: “On singularities of mappings of Euclidean spaces, I: Mappings of the plane into the plane,” Ann. Math. (2) 62 : 3 (November 1955), pp. 374–410. MR 0073980 Zbl 0068.37101 article

Let \( f_0 \) be a mapping of an open set \( R \) in \( n \)-space \( E^n \) into \( m \)-space \( E^m \). Let us consider, along with \( f_0 \), all mappings \( f \) which are sufficiently good approximations to \( f_0 \). By the Weierstrass Approximation Theorem, there are such mappings \( f \) which are analytic; in fact, (see [1934, Lemma 6]) we may make \( f \) approximate to \( f_0 \) throughout \( R \) arbitrarily well, and if \( f_0 \) is \( r \)-smooth (i.e., has continuous partial derivatives of orders \( \leq r \)), \( r \) finite, we may make corresponding derivatives of \( f \) approximate to those of \( f_0 \).

Supposing \( f \) is smooth (i.e., 1-smooth), the Jacobian matrix \( \mathbf{J} \) of \( f \) is defined (using fixed coordinate systems); we say the point \( p\in R \) is a regular point or a singular point of \( f \) according as \( \mathbf{J} \) is of maximum rank (i.e. of rank \( \min(n,m) \)) or of lesser rank. In general we cannot expect \( f \) to be free of singular points. A fundamental problem is to determine what sort of singularities any good approximation \( f \) to \( f_0 \) must have; what sort of sets they occupy, what \( f \) is like near such points, what topological properties hold with reference to them, etc.

[103]H. Whitney: “Elementary structure of real algebraic varieties,” Ann. Math. (2) 66 : 3 (November 1957), pp. 545–556. MR 0095844 Zbl 0078.13403 article

A real (or complex) algebraic variety \( V \) is a point set in real \( n \)-space \( R^n \) (or complex \( n \)-space \( C^n \)) which is the set of common zeros of a set of polynomials. The general properties of a real \( V \) as a point set have not been the subject of much study recently (but see for instance [Lefschetz 1924; Nash 1952; Oleĭnkik 1951]); attention has turned much more to the complex case, the complex projective case, and especially the abstract algebraic theory. Facts about the real case are sometimes needed in the applications; proofs are commonly very difficult to locate.

Our first object is to give a proof involving no algebraic theory that through a certain splitting process, \( V \) may be expressed as a union of “partial algebraic manifolds”.

[104]H. Whitney: “On functions with bounded \( n \)-th differences,” J. Math. Pures Appl. (9) 36 (1957), pp. 67–95. MR 0084611 Zbl 0077.06901 article

[105]H. Whitney: Geometric integration theory. Princeton Mathematical Series 21. Princeton University Press, 1957. MR 0087148 Zbl 0083.28204 book

[106]H. Whitney: “Singularities of mappings of Euclidean spaces,” pp. 285–301 in Symposium internacional de topología algebraica (Mexico City, August 1956). Edited by J. Adem. Universidad Nacional Autónoma de México (Mexico City), 1958. MR 0098418 Zbl 0092.28401 incollection

[107]A. Dold and H. Whitney: “Classification of oriented sphere bundles over a 4-complex,” Ann. Math. (2) 69 : 3 (May 1959), pp. 667–677. MR 0123331 Zbl 0124.38103 article

The classification of \( (n-1) \)-sphere bundles with structure group \( SO(n) \) (special orthogonal group) over a complex \( K \) of dimension at most 4 has been carried out in several special cases. If \( n=2 \) or if the dimension of \( K \) does not exceed 3 then the characteristic class \( W_2 \) is a complete invariant (see [1937]; note that \( W_2 \) is an integer class for \( n=2 \), and is a class mod 2 if \( n\geq 3 \)). 2-sphere bundles with vanishing class \( W_2 \) were classified by the second author in an unpublishedanuscript (1938; announced in [1940]); these bundles are not determined by their characteristic classes (see our example in Section 3). Pontrjagin in [1945] gave a solution for arbitrary \( n \) provided that \( H^4(K;Z) \) has no 2-torsion; in this case \( (n-1) \)-sphere bundles are characterized by \( W_2 \), \( W_4 \) (for \( n\geq 4 \)) and \( P_4 \) (see [Pontrjagin 1945] or the Corollary in Section 3). In this paper we give the classification for the general case. Throughout the paper we assume \( n\geq 3 \).

[108]H. Whitney and F. Bruhat: “Quelques propriétés fondamentales des ensembles analytiques-réels” [Some fundamental properties of real-analytic sets], Comment. Math. Helv. 33 (1959), pp. 132–160. MR 0102094 Zbl 0100.08101 article

[109]H. Whitney: “On bounded functions with bounded \( n \)-th differences,” Proc. Am. Math. Soc. 10 : 3 (June 1959), pp. 480–481. MR 0106969 Zbl 0089.27601 article

[110]H. Whitney: Geometric integration theory. Edited by V. G. Boltyanski. 1960. Zbl 0106.26703 book

[111]A. M. Gleason and H. Whitney: “The extension of linear functionals defined on \( H^{\infty} \),” Pacific J. Math. 12 : 1 (1962), pp. 163–182. Dedicated to Marston Morse. MR 0142013 Zbl 0191.15202 article

We consider the Banach space \( L^{\infty} \) of (classes of) bounded measurable complex functions on the unit circle \( \Gamma_1 \). It has a subspace \( H = H^{\infty} \) consisting of those functions \( h \) which are the boundary value functions (existing almost everywhere by Fatou’s theorem) of bounded analytic functions \( \hat{h} \) in the interior \( S_1 \) of \( \Gamma_1 \). By the Hahn–Banach theorem, any bounded linear functional \( \varphi \) defined on \( H^{\infty} \) can be extended over \( L^{\infty} \) with no increase in norm. It is the primary purpose of this paper to prove that this extension is unique, provided that \( \varphi \) is defined (over \( H \)) by an integral with kernel in \( L^1 \). Without this hypothesis, uniqueness may fail.

Andrew Mattei Gleason	Related	Volume
Harold Calvin Marston Morse	Related

[112]H. Whitney: “The work of John W. Milnor,” pp. XLVIII–L in Proceedings of the International Congress of Mathematicians (Stockholm, 15–22 August 1962). Institut Mittag-Leffler (Djursholm, Sweden), 1963. Zbl 0112.24401 incollection

[113]H. Whitney: “Tangents to an analytic variety,” Ann. Math. (2) 81 : 3 (May 1965), pp. 496–549. MR 0192520 Zbl 0152.27701 article

[114]H. Whitney: “Local properties of analytic varieties,” pp. 205–244 in Differential and combinatorial topology: A symposium in honor of Marston Morse (Institute for Advanced Study, Princeton, NJ, 1964). Edited by S. S. Cairns. Princeton Mathematical Series 27. Princeton University Press, 1965. MR 0188486 Zbl 0129.39402 incollection

Harold Calvin Marston Morse	Related
Stewart Scott Cairns	Related

[115]H. Whitney: “The mathematics of physical quantities, II: Quantity structures and dimensional analysis,” Am. Math. Mon. 75 : 3 (March 1968), pp. 227–256. MR 0228220 article

[116]H. Whitney: “The mathematics of physical quantities, I: Mathematical models for measurement,” Am. Math. Mon. 75 : 2 (February 1968), pp. 115–138. MR 0228219 Zbl 0186.57901 article

[117]H. Whitney: Complex analytic varieties. Addison-Wesley (Reading, MA), 1972. MR 0387634 Zbl 0265.32008 book

[118]H. Whitney and W. T. Tutte: “Kempe chains and the four colour problem,” Util. Math. 2 (1972), pp. 241–281. MR 0309782 Zbl 0253.05120 article

[119]H. Whitney: “Are we off the track in teaching mathematical concepts?,” pp. 283–296 in Developments in mathematical education: Proceedings of the Second International Congress on Mathematical Education (Cambridge, UK). Edited by A. G. Howson. Cambridge University Press, 1973. incollection

[120]H. Whitney and W. T. Tutte: “Kempe chains and the four colour problem,” pp. 378–413 in Studies in graph theory, part II. Edited by C. Berge and D. R. Fulkerson. Studies in Mathematics 12. Mathematical Association of America (Washington, DC), 1975. MR 0392651 Zbl 0328.05107 incollection

William Thomas Tutte	Related
Claude Berge	Related
Delbert Ray Fulkerson	Related

[121]H. Whitney: “Comment on the division of the plane by lines,” Am. Math. Mon. 86 : 8 (October 1979), pp. 700. MR 1539143 article

[122]H. Whitney: “Taking responsibility in school mathematics education,” J. Math. Behav. 4 : 3 (December 1985), pp. 219–235. article

[123]H. Whitney: “Letting research come naturally,” Math. Chronicle 14 (1985), pp. 1–19. MR 826955 Zbl 0617.00022 article

[124]A. Tucker and W. Aspray: Hassler Whitney, 1985. Online interview transcript, 10 April 1984. misc

William Aspray	Related
Albert William Tucker	Related

[125]F. M. Hechinger: “Learning math by thinking,” New York Times (10 June 1986), pp. C1. Article is based around an interview with Whitney. article

[126]H. Whitney: “Coming alive in school math and beyond,” J. Math. Behav. 5 : 2 (1986), pp. 129–140. article

In spite of ever-increasing efforts, the failure of schools to help children learn mathematics in a relevant and useful manner continues. We know that the enforced rote learning is a direct cause of this, so we try to promote better teaching methods. But having given up on the children, believing most of them incapable, we continue teaching everything with drill and testing, thus ensuring the continued rote learning and confirming our beliefs. Thus the children’s power of thought, vitality and responsibility remain wiped out.

A strong shift in attitudes of students to “I can explore, I can control my study and learning” is not difficult to obtain, and we have seen many examples of this, with excellent results. In any community where there is sufficient concern, cooperation and communication, this can be achieved. The need is to let better ways come in in little bits, not by trying to stop standard methods. We describe some ways in which such a process may be carried out.

[127]H. Whitney: “Coming alive in school math and beyond,” Educ. Stud. Math. 18 : 3 (1987), pp. 229–242. article

In spite of ever-increasing efforts, the failure of schools to help children learn mathematics in a relevant and useful manner continues. We know that the enforced rote learning is a direct cause of this, so we try to promote better teaching methods. But having given up on the children, believing most of them incapable, we continue teaching everything with drill and testing, thus ensuring the continued rote learning and confirming our beliefs. Thus the children’s power of thought, vitality and responsibility remain wiped out.

A strong shift in attitudes of students to “I can explore, I can control my study and learning” is not difficult to obtain, and we have seen many examples of this, with excellent results. In any community where there is sufficient concern, cooperation and communication, this can be achieved. The need is to let better ways come in in little bits, not by trying to stop standard methods. We describe some ways in which such a process may be carried out.

[128]H. Whitney: “Moscow 1935: Topology moving toward America,” pp. 97–117 in A century of mathematics in America, part I. Edited by P. Duren. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. Edited with the assistance of Richard A. Askey and Uta C. Merzbach. MR 1003166 Zbl 0666.01008 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[129]A. Shields: “Differentiable manifolds: Weyl and Whitney,” Math. Intell. 10 : 2 (June 1988), pp. 5–9. article

Hermann Weyl	Related
Allen Shields	Related

[130]U. D’Ambrosio: “The visits of Hassler Whitney to Brazil,” Humanistic Mathematics Network Newsletter 4 (1989), pp. 8. article

[131]A. Lax: “Hassler Whitney 1907–1989: Some recollections 1979–1989,” Humanistic Mathematics Network Newsletter 4 (1989), pp. 2–7. article

[132]H. Whitney: “Education is for the students’ future (draft),” Humanistic Mathematics Network Newsletter 4 (1989), pp. 9–12. article

[133]G. Fowler: “Obituary: Hassler Whitney, geometrician: He eased ‘mathematics anxiety’,” New York Times (12 May 1989), pp. B10. article

[134]R. Thom: “La vie et l’œuvre de Hassler Whitney” [The life and works of Hassler Whitney], C. R. Acad. Sci., Paris, Sér. Gén., Vie Sci. 7 : 6 (1990), pp. 473–476. MR 1105198 article

[135]H. Whitney: Collected papers, vol. I. Edited by J. Eells and D. Toledo. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1992. With a foreword by the editors. MR 1138982 Zbl 0746.01016 book

Domingo Toledo	Related
James Eells	Related

[136]H. Whitney: Collected papers, vol. II. Edited by J. Eells and D. Toledo. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1992. With a foreword by the editors. MR 1138983 book

Domingo Toledo	Related
James Eells	Related

[137]J. D. Zund: “Whitney, Hassler,” pp. 303–304 in American National Biography, vol. 23. Edited by J. A. Garraty and M. C. Carnes. Oxford University Press (New York), 1999. incollection

Mark Christopher Carnes	Related
Joseph D. Zund	Related
John Arthur Garraty	Related

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