Algebraic Topology; Hatcher 2.23 Example 2.23 from Hatcher starts by stating...
Let us find explicit cycles representing generators of the infinite cyclic groups $H_n(D^n,\partial D^n)$ and $\tilde{H}(S^n)$.  Replacing $(D^n, \partial D^n)$ by the equivalent pair $(\Delta^n,\partial \Delta^n)$, we will show by induction on $n$ that the identity map $i_n:\Delta^n\to \Delta^n$, viewed as a singular $n$-simplex, is a cycle generating $H_n(\Delta^n,\partial \Delta^n)$.  That it is a cycle is clear since we are considering relative homology.  When $n=0$ it certainly represents a generator.
Now, the  part I don't understand is the easy part, the stuff in bold.  How is the identity map a cycle and a generator when n=0?  Is there an example of a map from $\Delta^n\to \Delta^n$ that is not a cycle?
 A: Viewing it as a singular simplex, the identity $i_n: \Delta^n \to \Delta^n$ is a cycle by the definition of singular homology: its boundary $\partial i_n$ is sum with the alternating signs of the inclusions $\Delta^{n-1} \to \Delta^{n}$ of the $n-1$-simplex onto the faces of the $n$-simplex, so it can be viewed as a singular simplex in $\partial \Delta^n$. 
This is zero in relative homology, because it is already zero on the relative chain level: indeed, the relative homology is defined by a homology of a chain complex fitting in the following short exact sequence of chain complexes:
$$
0 \to C(\partial \Delta^n) \to C(\Delta^n) \to C(\Delta^n, \partial \Delta^n) \to 0
$$
$\partial i_n \in C_n(\Delta^n)$ is mapped to zero in $ C(\Delta^n, \partial \Delta^n)$, because it belongs to $C(\partial \Delta^n)$ as well, which is the kernel of the map $C(\Delta^n) \to C(\Delta^n, \partial \Delta^n)$.
For $n = 0$, the $\Delta^0$ is a single point space $*$, so the inclusion $i_0: \Delta^0 \to \Delta^0$ represents a generator of $H_0(\Delta^0, \partial \Delta^0) = H_0(*, \emptyset) = H_0(*)$ by the calculation of $H_0(X)$ of any space earlier in Hatcher (Hatcher proves that $H_0(X)$ is a free group generated by the classes of maps $* \to X$ into different path-connected components).
