Understanding singular homology

Question

On the Wikipedia page for singular homology:

Consider first the set of all possible singular n-simplices $\sigma_n(\Delta^n)$ on a topological space X. This set may be used as the basis of a free abelian group, so that each $\sigma_n(\Delta^n)$ is a generator of the group.

This makes sense to me, but I'm not sure how they form a group. What is the group operation? Is there a notion of adding/subtracting simplexes that I don't know about? If so, the wikipedia page also states that the simplices are the generators of the group, so what are all the other elements?

Answer 1

You take formal linear combinations of the generating simplices. For example, $$3\sigma + 2\mu$$ is an element of that group, where $\sigma,\mu$ are two $n$-simplices. The group operation is addition, so $$(3\sigma + 2\mu) + \sigma = 4\sigma + 2\mu$$ and $$(2\sigma) + (14\mu) = 2\sigma + 14\mu\ .$$

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In other words, you are taking the $\mathbb Z$-module $$\bigoplus_{\sigma\ n\text{-simplex}}\mathbb{Z}\sigma = \operatorname{span}_\mathbb{Z}\{\sigma\ n\text{-simplex}\}$$ spanned by all the $n$-simplices and looking at its abelian group structure.

Answer 2

Since you have a free abelian group, the group operation is formal addition i.e. the sum of $\sigma_1$ and $\sigma_2$ is just $\sigma_1 + \sigma_2$. The generators of this group are singular $n$-simplices, which means that a generic element looks like a formal sum of these simplifies i.e $$\sum_{i=1}^n a_i \sigma_i$$ Adding and subtracting simplices is just that, formally adding and subtracting them. The signs on the simplices correspond to the orientation we take on them, and the addition is formal. You can think of it as a union in the image (it’s usually how it’s drawn) but that isn’t technically true. You can think of multiplying by a scalar as just saying “take $n$ copies of this simplex”.