# 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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## 1 Answer

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.

To summarize, you can "look" for homologous elements by seeing if you can represent their difference as the boundary of a chain of one dimension higher.

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.

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