Advanced Mathematics

5.3Geometric Topology

Purpose of Section

To introduce the idea of topological equivalence between figures (i.e. curves, surfaces, …) and the idea of a homeomorphism between them. We show how the Euler characteristic, previously defined for connected, planar graphs and convex polyhedra, comes into its own as an aid in determining if two figures are topologically equivalent.

5.3.1 Introduction

It is instructive to compare a geometer with a topologist. For a geometer, the pattern of ridges on your fingers and palms are too amorphous for serious analysis. There are no lines, circles, or other geometric shapes which geometers love. On the other hand, a topologist is not restricted to the rigid shapes of Euclidean geometry, but studies more general patterns and classifies them according to specific topological rules. See Figure 5.49.

Figure 5.49 Topological fingerprints.

When a topologist looks at the ridges on your fingers and palms, the topologist sees nearly parallel curves, which is nature's way of preferring order and continuity, but when ridges collide things get complicated. Ridges come together in a variety of interesting, often unexpected ways, and it is the job of a topologist to classify the ways this can occur. You may not have had an interest in topology at this stage in your life, but the ridges on your fingers provide a lesson in basic topology.

The study ...