In Chapter 15 we saw how Riemann found the topological concept of
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
, 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
of a surface. Topology also interacts with algebra. In this chapter we focus on the
, 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,
, such that
. In 1904, Poincaré famously conjectured that a closed three-dimensional space with trivial fundamental group is topologically the same as the threedimensional sphere. This
was proved only in 2003, with the help of methods from differential geometry. Thus the interaction between geometry and topology continues.