Singular homology of a torus

Question

I am trying to form an intuition of how singular homology captures the n-dimensional hole in a topological space by studying torus as an example. I read in Lee's Introduction to Topological Manifolds the following:

"The point of homology theory is to use singular chains to detect “holes.” The intuition is that any chain that closes up on itself (like a closed path) but is not equal to the “boundary value” of a chain of one higher dimension must surround a hole in X."

For the torus below if we, to the contrary, assume that there does exist a 2-chain whose boundary are the closed curves $\alpha$ and $\beta$ what contradictions we can derive from this ?

Answer 1

The definition of singular homology is not well attuned to answering questions like this since it is such a strange beast. To some extent, it seems like all we can say is that certain invariants of the torus we can relate to the singular homology would be contradicted. To name a few: its simplicial homology, cellular homology, fundamental group, Euler characteristic, etc. Perhaps this last one is the best answer for your question. The loop tracing out the two circles would be nullhomotopic via a tool called the Hurewicz homomorphism.

Answer 2

An easy invariant that would be contradicted is the fundamental group: via the Hurewicz homomorphism we have that the first homology is the abelianization of the fundamental group, which in this case can easily be computed to be $\mathbb{Z}^{2}$. In general proving things directly about singular chains (i.e. maps from a topological n-simplex) is nigh on impossible, but you can prove that singular homology can be computed via simplicial homology whenever you have $\delta$-complex (this is proven in Hatcher chapter 2 by using the long exact sequence in homology), and via cellular homology whenever you have a CW-structure. The intuition you're looking for can be derived from the definition of simplicial homology and computing some examples (like the torus), so I'd start there.