Generator of singular homology of n-sphere I am learning singular homology theory right now. The homology of n-sphere is computed by Mayer-Vietoris argument. Intuitively, for example the class represented by a loop is the generator of $H_1(S^1)$. Is there a way to show that this is truly the case in singular homology? 

You have a long exact sequence 
$$\dots→\tilde H_1(A)\oplus \tilde H_1(B)→\tilde H_1(A+B)→\tilde H_0(A∩B)→\tilde H_0(A)⊕\tilde H_0(B)→\dots$$ where $H_n(A+B)$ is the homology group for the chain complex
$$\dots→C_n(A+B)→C_{n-1}(A+B)→\dots$$
where $C_n(A+B)$ consists of chains whose simplices are each in $A$ or in $B$, which are small open neighborhoods around the upper and lower semicircle such that $A∩B$ is the disjoint union of two arcs.
The inclusion $C_1(A+B)\hookrightarrow C_1(S^1)$ induces an isomorphism $\tilde H_1(A+B)\cong \tilde H_1(S^1)$ whose inverse is induced by the map $\rho:C_1(S^1)→C_1(A+B)$, which turns each simplex $\sigma$ into a chain of smaller simplices, each of which has image in $\text{Im}(\sigma)∩A$ or in $\text{Im}(\sigma)∩B$. For a precise definition see Hatcher's Algebraic Topology, Proposition 2.21.
If we apply $\rho$ to the loop $\sigma$ around $S^1$, we get the first barycentric subdivision of $σ$, which is $σ_1-σ_2$, where $σ_1$ is the semi-circle from $-1$ to $1$ with non-negative second coordinate, and $σ_2$ is the semi-circle from $-1$ to $1$ with non-positive second coordinate.
Now $σ_1-σ_2$ is a generator of $H_n(A+B)$, as we can see by chasing the cycle $σ_1-σ_2$ through the definition of the connecting homomorphism $H_n(A+B)→H_{n-1}(A∩B)$:
$σ_1-σ_2$ is the image of $(σ_1,σ_2)$ in $C_n(A)\oplus C_n(B)$, which has boundary $(1-(-1),1-(-1))$, hence $\partial[σ_1-σ_2]=[1-(-1)]$. Since this difference is the generator of $\tilde H_{0}(A∩B)$, $σ_1-σ_2$ is a generator of $\tilde H_1(A+B)$. It follows that $σ$ is a generator of $\tilde H_1(S^1)$
