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The Geometry of Physics by Theodore Frankel

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CHAPTER 13

Betti Numbers and De Rham’s Theorem

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When can we be certain that a closed form will be exact?

THE lack of simple connectivity is but one measure of topological complexity for a space. In this chapter we shall deal with others, the Betti numbers, and their relations with the potentials for closed exterior forms initiated in Chapter 5. This subject is a part of the discipline called algebraic topology.

13.1. Singular Chains and Their Boundaries

What does Stokes’s theorem say for a Möbius band?

13.1a. Singular Chains

The standard (euclidean) p-simplex in p is the convex set Δpp generated by the p + 1 points

Figure 13.1

We shall ...

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