Generator of singular homology of n-sphere

Question

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?

Answer 1

One way is to use the isomorphism with the simplicial homology of $\Delta\text{-complexes}$ where the isomorphism is induced by direct inclusion of $\Delta$ chains into singular chains, and the generator of $H_1^\Delta(S^1)$ is a loop. This also works for any space and dimension with a $\Delta\text{-complex}$ structure. Definitions and proofs are in Section 2.1 of Hatcher.

Another is to show that $H_1$ is the abelianization of the fundamental group (the proofs work directly with loops), and so generator loops of the fundamental group are automatically generators of the homology. Direct proof is in Section 2.A of Hatcher.

Answer 2

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 $\tilde H_1(S^1)$