In Chapter 15 we saw how Riemann found the topological concept of genus to be important in the study of algebraic curves. In the present chapter we will see how topology became a major field of mathematics, with its own methods and problems. Naturally, topology interacts with geometry, and it is common for topological ideas to be noticed first in geometry. An important example is the Euler characteristic, which was originally observed as a characteristic of polyhedra, then later seen to be meaningful for arbitrary closed surfaces. Today, we tend to think that topology comes first, and that it controls what can happen in geometry. For example, the Gauss–Bonnet theorem seems to show that the Euler characteristic controls the value of the total curvature of a surface. Topology also interacts with algebra. In this chapter we focus on the fundamental group, a group that describes the ways in which flexible loops can lie in a geometric object. On a sphere, all loops can be shrunk to a point, so the fundamental group is trivial. On the torus, however, there are many closed loops. But they are all combinations of two particular loops, a and b, such that ab = ba. In 1904, Poincaré famously conjectured that a closed three-dimensional space with trivial fundamental group is topologically the same as the threedimensional sphere. This Poincaré conjecture was proved only in 2003, with the help of methods from differential geometry. Thus the interaction between geometry and topology continues.
Fundamental Group Euler Characteristic Total Curvature Simple Closed Curve Homology Sphere
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