Borsuk–Ulam theorem

Abstract: The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singulari- ties of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem. In 1963, Hirsch (13) gave a short-and by now classic-proof of the Brouwer Fixed Point Theorem by a generic argument. This theorem is usually proved by some kind of degree argument and degree genetically counts inverse images of points. Hirsch directly looked at the inverse image of a generic point. The idea that one could replace degree arguments by looking at the inverse images of points of maps was made the theme of a book by J. Milnor (24) and later by V. Guillemin and A. Pollack (12, especially Chapters 2, 3) and Hirsch (14, especially Chapter 5). We offer these books as general references for transversality and Sard's theorem. Without knowledge of Hirsch's paper, H. Scarf (31) in 1967 used much the same ideas to numerically approximate a Brouwer fixed point. One "follows" a "path" which leads from the boundary to some one or more of the fixed points. For some contemporary papers using similar methods, see (19), (23). This has been called the Newton method. B. C. Eaves (8) in 1972 developed a slightly different approach to the same end. His idea was to homotope one of a set of standard maps to the map in question; by foUowing the fixed points of the changing maps, the fixed-point-set of the map in question could be located. It is this version we call the homotopy continuation method and

Abstract: The Borsuk-Ulam theorem of topology is applied to a problem in discrete mathematics. A bisection of a necklace with k colors of beads is a collection of intervals whose union captures half the beads of each color. Every necklace with k colors has a bisection formed by at most k cuts. Higherdimensional generalizations are considered.

Abstract: A very important graph parameter is the chromatic number. For a given graph, it is the smallest number of colors for which a coloring of the vertices exists such that adjacent vertices receive different colors. The search for a proof of the four color theorem—stating that every planar map can be colored with four colors such that adjacent countries receive different colors (Fig. 2.1)—has certainly been one of the driving sources [Ore67, Saa72, Tho98] of graph theory for a long time. Presently, graph coloring plays an important role in several real-world applications and still engages exciting research. In this chapter we will present an important side story, the story of a conjecture formulated by Martin Kneser in 1955 that remained unsolved until 1977 (published in 1978 [Lov78]). The revolutionary method with which Laszlo Lovasz settled the notorious conjecture can be seen as the origin of the field with which this book deals. Guided by some deep insight, Lovasz associated a simplicial complex to a graph in such a way that the topology of the complex provides some information about the chromatic number of the graph, thereby transforming a discrete problem into a topological one. The main tool he employed was the Borsuk–Ulam theorem. His proof, and the efforts to understand it, have triggered a considerable amount of research. By now, Lovasz’s original proof has gone through many transformations and inspired alternative proofs even until very recently. We will touch upon most of the ideas involved in the several proofs that emerged over the last decades.

Abstract: : An S to the first power version of the Borsuk-Ulam Theorem is proved for a situation where Fix S to the first power may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critica points is established for a class of functionals invariant under an S to the first power symmetry. Minimax arguments from the calculus of variations serve as an important tool in establishing the existence of nonlinear vibrations of discrete mechanical systems as modelled by Hamilton's equations. In these arguments one obtains the solutions of the differential equations as critical points of an associated Lagrangian by minimaxing the Lagrangian over appropriate classes of sets. Intersection theorems such as are proved in this paper play a crucial role in this process. In addition to obtaining some intersection theorems this report illustrates their use by proving an existence theorem for multiple critical points of the functional invarianet under an S to the first power symmetry group.