graph theory
Whitney has been interested in four-color problems all her life. His earliest and last math papers were all about four colors. He gave the equivalent proposition of the four-color problem and studied the reducibility problem. Starting with the four-color problem, he studied the general graph theory, especially the conditions of homeomorphism of two graphs: for example, G and G' are two connected graphs, and neither of them contains ab, ac and ad arcs. If there is any one-to-one correspondence between two arcs with common vertices and two arcs with common vertices of another graph, then the two graphs are homeomorphic. He defined the connectivity of graphs. The necessary and sufficient conditions for n-reconnection are given (the so-called n-reconnection refers to a graph with at least n+ 1 vertices, which cannot be broken by removing n- 1 or fewer vertices and connecting them with arcs. If the graph Gn is reconnected, but not n+ 1, then its connectivity is called n). He also defined the dual G' of graph G and proved that graph G can be disconnected.
His doctoral thesis is about the coloring of graphs, in which the formula of M(λ) is proved and calculated, where M(λ) is the number of different coloring methods of a graph with λ colors. He introduced a set of numbers mij, which can be used not only to calculate M(λ), but also to define the topological invariants of graph G;
Where r is the rank of graph G and n is the zero degree of G, he used these invariants to study the classification of graphs.
Whitney's greatest achievement in combinatorial theory is that he introduced matroid theory, which is an abstract linear correlation theory. It includes not only graph theory as a special case, but also network theory, comprehensive geometry and crosscutting theory. His starting point is simple. Consider the columns C 1, C2, …, Cn of the matrix m, and the subsets of these columns are either linearly independent or linearly related.
Any subset of the (1) independent set is also independent;
(2) If Np and Np+ 1 are independent sets of P column and p+ 1 column, Np plus one of Np+ 1 constitutes an independent set of p+ 1.
He called the system satisfying these two conditions matroid, and extended the properties of many graphs to matroid.
Differentiable mapping and singularity theory
Analytical extension of (1) differentiable function Whitney's main contribution to topology is the establishment of differential topology. Therefore, the continuous mapping considered by topology must be extended to differentiable cases. Whitney laid the foundation for this in his early work (1932- 1942).
1925 The Soviet mathematician улысон(uryson) proved that if A is a closed set (bounded or unbounded) in N-dimensional Euclidean space E and f(x) is a continuous function defined in A, then F can be extended to the whole E. If f(x) belongs to A.
(2) Singularity theory Singularity theory is one of Whitney's most important creations, which comes from the problem of differential embedding and immersion. Singularity is the generalization of critical point. In 20942, for the first time,
In this paper, the singularity of differEntial mapping F from N-dimensional Euclidean space en to E2n- 1 is studied. It is found that if you change f slightly, you can get f*. Its singularity is an arc singularity, which can be transformed into a standard form:
yi=xi(i=2，…，n)，
ym+i- 1=xixi(i=2，…，n)。
In 1955, he first classified the types of singularities in plane E2 to E; There are only two kinds of results, one is fold and the other is Cusp, and its standard is.
Through this paper, the singularity theory was founded. In 1956, he classified some cases of singular points of differential mapping of En→Em, and obtained standard forms, including n ≥ m)=(4 2,3 and (n, m) = (4,4), (5,5), (5,4), (n, 2n. Little was known at that time. This basic problem of singularity classification, together with other problems, has become a hot topic in singularity theory. In the same year, R. Thorm made a breakthrough by using his own transverse theory and universal folding theory, which became the basis of his catastrophe theory. Later, J. Mather established the stability theory in 1968- 197 1 year. Since 1967, the Soviet school headed by Soviet mathematician B. ирнолъв has made brilliant achievements in theory and application.
1948 also published On the Ideal of Two Differentiable Functions, which opened up another new direction of singularity theory. Later, B. Malgrange and others made great breakthroughs in this respect, including proving the "preparatory theorem".
(3) Hierarchy Theory Hierarchy theory is Whitney's last theory, and in a sense, it is also a natural continuation of singularity theory. Euclidean space and manifold usually studied have good homogeneous structure (local structure is the same), but this is not satisfied even for algebraic clusters. In particular, there are singularities in real algebraic clusters inherited from analytic geometry. From 1957 to 1965, Whitney studied the topology of real algebraic clusters and discussed the decomposition from clusters to manifolds. In 1957, Whitney introduced the concept of hierarchy, which decomposed algebraic clusters and analytic clusters into layers, and was later developed into hierarchical set theory by Tom. In 1965, S. Schaevitz proved that any semi-analytic set has Whitney stratification. In 1965, Whitney defined the concepts of tangent vector, tangent plane family and tangent cone for analytic clusters, and considered the harmony of tangent sets.
Differential popular topology
Although Poincare and even Riemann have studied the topology of differential manifolds, Whitney really founded the topology and differential topology of differential manifolds due to the lack of tools. In 1936, he laid the theoretical foundation of differential manifold. He gave the inherent definition of differential manifold. Define the Cr structure (1≤r≤∞) above. He proved that all Cr structures of Cr manifold contain C∞ coordinate system, and their C∞ structures are unique. This C∞ structure is called manifold microstructure or differential structure or smooth structure, and the corresponding manifold is called Hui manifold or differential manifold or smooth manifold. There are essential differences between differential manifolds and topological manifolds. That is, a topological manifold may not allow any differential structure or multiple differential structures, but any differential structure allows real analytic structure and Riemann metric, which is also proved by Whitney. In this paper, he proved some basic theorems, especially the embedding and immersion theorems: any N-dimensional differential manifold can be differentially embedded in R2n+ 1(2n+ 1 dimensional Euclidean space). In 1944, he improved that n-dimensional differential manifolds can be embedded in R2n and immersed in R2n- 1. For some manifolds, these results have reached perfection. This work has opened up an important field of differential manifolds, and many topologists such as Wu Wenjun have made contributions since then.
Fiber bundle and indicator class
Whitney first defined the real "fiber space" in 1935, then he called it "ball space", and in 1940 he changed it to "ball cluster". In 1937 and 194 1, he made two reports on this, including many fundamental results, and he intends to do the same. It was never finished. His interest has been focused on "special classes". He independently defined this kind of feature class in 1936 and Swiss mathematician E. Stiefel in 1935, which was later called Stiefel-Whitney feature class. His purpose is to study the topology of differential manifolds with characteristic classes. In this respect, fiber bundle is just a tool, so his definition is not clear in every detail, but very general. 1940- 1950, fiber bundle became the main tool to study many topological problems (especially homotopy, homotopy and differential geometry problems). 195 1 year, the publication of the monograph "Fiber Bundle Topology" by N.E. Steenrod marked fiber bundle topology.
(1) classification from the beginning, Whitney mainly studied the classification of fiber bundles. In 1937, he obtained the classification space of the ball bundle, that is, the Glassman manifold Gn, r, and asserted that the isomorphic class of the ball bundle with base space b and rank r is [B, Gn, R], that is, the homotopy class (nr) of the mapping from b to Gn, r.
Whitney also knows that the cluster space of ball clusters based on B only depends on the homotopy type of B, which was proved by J. Feldbau in 1939. On the other hand, as early as 1935, Whitney constructed a new fiber bundle g *(ξ) for fiber bundle ξ and continuous mapping G: b' → B.
(2) The characteristic class Stiefel only considers the characteristic class of tangent bundle of differential manifold, while Whitney considers it much wider. He thinks that the bottom space b of any ball cluster (e, b, p) can also be an arbitrarily locally finite simple complex. He defined the characteristic class as the integer coefficient homology class of Stiefel manifold Sn, m. He pointed out the homology group of Sn and m.
In 1937, he defined the sex class with homology. In 1940, he pointed out that for continuous mapping,
g:B'0→B，
If E'=g*(E) is the callback of e, then
Wr(E')=g*(Wr(E))。
At the same time, he gave Whitney's summation formula: define the income of two ball clusters e' and e "on the same bottom space.
Among them, the area of ∪ curve, he pointed out that when r≥4, it is "extremely difficult" to prove. In 194 1, he only gave the proof that E and E' are both line bundles. The first published proof was given by Wu Wenjun in 1948. He also replaced the ball bundle with a vector bundle. In the same year, Chen Shengshen also published another proof.
Whitney also gave the concept of demonstrative class or even formal power series of demonstrative class. At this point, the theoretical basis of Steefel-Whitney demonstration class was formally established. Later, based on the four theorems put forward by Whitney, Milnor opened the theory of deixis and other deixis, especially Pontryagin (понт).
(3) The application of demonstrative classes plays an extremely important role in topology and geometry. Whitney herself mainly uses indicative courses to learn immersion. For example, he proved that the 8-dimensional real projective space P8(R) can not be immersed in R 14, but can be immersed in R 15. His theory was later developed by Wu Wenjun and others.
algebraic topology
1935 is a turning point in algebraic topology, which is mainly marked by the establishment of homology theory and homology ethics. Forty years after Poincare introduced the concept of homology, four mathematicians introduced the concept of homology almost simultaneously and independently. They are J.W. Alexander, Whitney, E. Cech (Cech) and A.H. André Andrey Kolmogorov (колмогоров). The other three were at 660.
In the same ethics theory, in 1937, Whitney expressed hopf-Hurewicz criterion in the same tone. If x is an n-dimensional locally finite cell complex and y is an n-dimensional (n- 1) connected space, then f, G: X→ Y is homotopy if and only if.
HN(Y； z)→Hn(X； z)。
Infer from this
〔X，x0； y，y0】úHn(X； πn(Y))
There is a one-to-one correspondence. These conditions are not necessarily true for the mapping of different dimensions. Whitney gave the homotopy algebraic condition of two-dimensional complex mapping to two-dimensional or three-dimensional projective space in 1936, but it was not published. In 194 1, H.E. Robbins extended the homotopy classification of mappings from two-dimensional complex to arbitrary space. Later, Olum simplified and popularized it on a large scale. For the three-dimensional complex, Pontryagin considered the homotopy classification mapped to S2 in 194 1, in which the newly emerging upper product was applied for the first time. In fact, Whitney got the corresponding results as early as 1936. He studies simple connectivity in 1948. On this basis, the necessary and sufficient conditions for homotopy of two continuous mappings in three-dimensional complex K to R and the obstacle classes of mapping expansion are given. It should also be pointed out that Whitney introduced the concept of tensor product of Abel group in 1938, which is an essential tool for algebraic topology and homology algebra.
Geometric integral theory
During the period of 1946- 1957, Whitney established geometric integration theory, which is a more general integration theory, such as R-dimensional integration in N-dimensional space. From this, he gave an analytical explanation of winding and upper closed chain, such as geometric chain is a function on the singular chain of "general position" He used Lipschitz condition to replace the differentiable condition in the external differential form theory of E Cartan and G de Rham, and the obtained integral theory is equivalent to algebraic homology theory, which is also true for more general Lipschitz spaces, including polyhedron and absolute neighborhood contraction kernel as its special cases, especially extending Stokes theorem to Lipschitz spaces. His theory is summarized in ""