Celebratio Mathematica

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

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

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

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.

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.

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

H. Whitney: “The self-intersections of a smooth $n$ -manifold in $2 n$ -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} $.

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.

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

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

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

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Hassler Whitney

Hassler Whitney: 1907–1989

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