# Visualizing homologous elements

For the fundamental group it's easy to visualize when two loops are homotopic. I was wondering if there are any ways to look at the equivalent problem for homology? I guess this might be tricky for singular homology, but are there nice ways to think about this for say simplicial and cellular homology?

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By definition, two $k$-chains $a$ and $b$ are homologous if they represent the same homology class; that is to say that $a-b = \partial C$ for some $k+1$-chain $C$. Let's restrict our attention to surfaces for the moment, for simplicity, and let's set $k=1$ just to see how this goes. A $2$-chain will just be some two-dimensional subsurface, possibly with boundary. For instance, $C$ might be a cylindrical subsurface sitting inside the torus (a quarter-donut, if you will). The boundary of $C$ in this case will consist of two loops, and (having properly oriented everything) these two loops will be homologous, precisely because together they bound a subsurface.
Similarly, in some simply-connected space, like $\mathbb{R}^2$, say, all $1$-chains are homologous to zero, since we can always find a $2$ chain to "fill in" any holes; if we have a map of the circle, we can always extend it to a map of the $2$-disk. But of course, we can't always fill in holes in spaces with more interesting topology; i.e. a loop going around the hole in a torus can't be filled in, hence this will not represent zero in homology.
The introduction to Chapter 2 of Hatcher's $\textit{Algebraic Topology}$ has a nice discussion motivating the transition from homotopy to homology that you might enjoy looking at.