# Correct definition of singular homology

Every time I look up something about singular homology I seem to find a different definition, so I just want to clear up a couple of things. Let $$X$$ be a topological space and $$\Delta^n$$ the $$n$$-th standard simplex. An $$n$$-simplex is then a continuous map $$\sigma \colon \Delta^n \to X$$. For the sake of simplicity (no pun intended), let us set $$X=\mathbb{C}\setminus\{0\}$$ and $$\sigma \colon [0,1] \cong \Delta^1 \to X, t \mapsto e^{2\pi i t}$$.

$$(1):$$ Some authors define the group of $$n$$-cycles $$C_n(X,\mathbb{Z})$$ as the free abelian group generated by all the $$n$$-simplices. Others (for example Wikipedia) as the free abelian group generated by the images of all $$n$$-simplices. Now, this second definition does not seem fitting to me: in our example, the double loop $$\sigma^2$$ around zero has the same image as the single loop $$\sigma$$, and we clearly do not want to identify the two maps, do we?

$$(2):$$ In any case, we are talking of a free abelian group of $$n$$-cycles. So all operations are formal, the only concrete things we have are the generators, i.e., the $$n$$-simplices. Hence, in the above example, a double loop around the origin is not, strictly speaking, the same as twice a single loop around the origin: $$\sigma^2 \neq 2\sigma$$. But now it seems to me that we do want to identify these, don't we?

$$(3):$$ Related to $$(2)$$. If, on the other hand, we identify cycles having the same image and boundary (hence $$C_n(X,\mathbb{Z})$$ is a non-trivial quotient of the free abelian group), then each $$0-$$cycle is $$0$$, because it is identified with its additive inverse. We do not want this, do we?

So, here it all is. I am very grateful for any comment about correct or erroneous reasonings and for any good references. Thanks in advance!

2. Although $\sigma+\sigma=\sigma^2"$, cycles with same image will not necessarily be equal in homology. For instance, $\sigma^2"$ and $\sigma$, where $\sigma$ is $t \mapsto e^{2\pi it}$ in $S^1$, have the same image, but are different in homology.
• @AndresMejia $[\sigma]$ is a generator for $H_1(S^1)$. $2 [\sigma] \neq [\sigma]$. – Aloizio Macedo Feb 25 '18 at 18:59