Singular Homology

Abstract

For each n ≥ 0, we now construct the homology functors H _n : Top → Ab that we used in Chapter 0 to prove Brouwer’s fixed point theorem. The question we ask is whether a union of n-simplexes in a space X that “ought” to be the boundary of some union of (n + 1)-simplexes in X actually is such a boundary. Consider the case n = 0; a 0-simplex in X is a point. Given two points x₀, x₁ ∈ X, they “ought” to be the endpoints of a 1-simplex; that is, there ought to be a path in X from x₀ to x₁. Thus, H₀(X) will bear on whether or not X is path connected.