A Glance at Topology

Abstract

Topology is precisely that mathematical discipline which allows a passage from the local to the global. René Thom (1923-2002) Topology studies the qualitative behavior of mathematical and physical objects. The following results discussed in the preceding chapter are related to topology: • deformation invariance of the integral of holomorphic functions, • Cauchy's residue theorem, • properties of the winding number, • Liouville's theorem, • analytic continuation of holomorphic functions, • Abelian integrals and Riemann surfaces. Topology was created by Poincaré (1854-1912) at the end of the 19th century and was motivated by the investigation of Riemann surfaces and the qualitative behavior of the orbits of planets, asteroids, and comets in celestial mechanics. 1 Topology studies far-reaching generalizations of the results summarized above. 5.1 Local and Global Properties of the Universe Since ancient times, scientists have made enormous efforts to understand • the macrocosmos-our universe-and • the microcosmos-the world of elementary particles. A unified theory for the four fundamental forces in nature (i.e., strong, weak, electromagnetic , and gravity) has the task to combine the macrocosmos with the micro-cosmos. In Fig. 5.1, two 1-dimensional models of the universe are pictured. These two models possess the same local structure near the earth, but the global structures are completely different. To illustrate this, consider a spaceship starting on earth and moving the same direction all the time. In a closed universe, the spaceship may return to earth, whereas this is impossible in an open universe. Note that in reality, the universe is expanding. This means that the radius of the closed universe in Fig. 5.1(a) is expanding in time. At the time of the Big Bang, the universe was concentrated at one point. The study of the global behavior of geometric objects is the subject of topol-ogy. We expect that the topology of the global universe influences the physics of elementary particles. As an example, consider a photon of wave length λ. If the 1 H. Poincaré, Analysis situs,