Singular homology: $n$-singular chains are to be thought abstractly or geometrically?
Emily Wong 
I'm starting to explore Singular Homology and after a little reading, this question jumped to mind:
The $n$-chains, which are elements of the free abelian group $C_{n}(X)$ generated by the $n$-singular simplexes, should be thought purely abstractly as these formal linear combinations of functions, or this definition is just mathematical formalism and the reader should try to understand this geometrically, like somehow 'summing' the geometrical objects which are the images of the standard geometrical simplexes sitting inside $X$?
If the latter, then how to make sense of these 'sums' of geometrical objects?
Thanks a lot. Any help or reference is appreciated.
$\endgroup$ 21 Answer
$\begingroup$You should think of them both abstractly and geometrically.
A chain consists of a finite $\mathbb{Z}$-linear combination of simplices. Geometrically, then, you can picture all the images of the simplices that appear in the sum at once (there is only a finite number of them). These look like topological oriented points, lines, triangles, etc., in $X$. (Note, however, that two distinct simplices (maps $\Delta^n \to X$) can have the same image in $X$.) The coefficients represent multiplicities, and a minus sign in front of a simplex means that its orientation is reversed. The algebraic boundary really corresponds to the geometric boundary: the minus signs on the algebraic side are necessary to orient the faces consistently, so that the geometric representation of the algebraic boundary is exactly the geometric boundary, with its induced orientation.
This setup should naturally align with your intuition. For instance, you can picture a $1$-cycle $\sigma$ as an oriented loop in $X$. If this simplex appears with coefficient $2$, then you can think of it as looping around the circle twice. The algebraic operation $2\sigma - 3\sigma = -\sigma$ amounts to saying that "two loops forward followed by three loops backward is the same as a single loop backward".
As another example, you may glue coherently oriented simplices along their boundaries. The figure below comes from Bredon.The $2$-chain $\sigma$ on the right is to be interpreted as a sum of eight $2$-simplices. Intuitively, $\sigma$ is just a triangulation of a closed annulus, so the boundary of all these triangles added together should yield the boundary of an annulus, namely, two circles. The outer (resp. inner) $1$-cycle is labeled $c_0$ (resp. $c_1$). As is evident from the picture, and also true if you compute it algebraically, the boundary of this chain simply becomes $\partial \sigma = c_1 - c_0$, as the inner faces of the triangles cancel each other out. Note that the orientation of $c_0$ is the opposite of the induced boundary orientation, which explains the sign in front of $c_0$.
