1 - From Vector Calculus to Algebraic Topology

Summary

1A. Chains, Cochains and Integration

Homology theory reduces topological problems that arise in the use of the classical integral theorems of vector analysis to more easily resolved algebraic problems. Stokes’ theorem on manifolds, which may be considered the fundamental theorem of multivariable calculus, is the generalization of these classical integral theorems. To appreciate how these topological problems arise, the process of integration must be reinterpreted algebraically.

Given an n-dimensional region Ω, we will consider the set C_p( Ω) of all possible p-dimensional objects over which a p-fold integration can be performed. Here it is understood that 0 ≤ p ≤ n and that a 0-fold integration is the sum of values of a function evaluated on a finite set of points. The elements of C_p( Ω), called p-chains, start out conceptually as p-dimensional surfaces, but in order to serve their intended function they must be more than that, for in evaluating integrals it is essential to associate an orientation to a chain. Likewise the idea of an orientation is essential for defining the oriented boundary of a chain (Figure 1.1).

At the very least, then, we wish to ensure that our set of chains C_p(ω) is closed under orientation reversal: for each c ∈ C_p( Ω) there is also −c ∈ C_p( Ω).