Singular homology

Remarks

A continuous map $f \colon X \to Y$ induces $f_* \colon H_n(X) \to H_n(Y)$ by $\sigma \mapsto f \circ \sigma$, and homotopic maps induce the same homomorphism. Homology with coefficients in an abelian group $A$ uses $S_n(X) \otimes A$.

Example: let $X$ be a point. There is exactly one singular $n$-simplex $\sigma_n$ for each $n$, and $\partial_n \sigma_n = \sum_{i=0}^n (-1)^i \sigma_{n-1}$, which is $\sigma_{n-1}$ for even $n \ge 2$ and $0$ for odd $n$. So $\partial_n$ is an isomorphism for even $n \ge 2$ and zero for odd $n$, which gives $H_0(X) \cong \Z$ and $H_n(X) = 0$ for $n \ge 1$.

Non-example: using all functions $\Delta^n \to X$, continuous or not, in place of singular simplices still gives a chain complex, but not singular homology. Any two points of $X$ are then joined by a $1$-simplex, so its $H_0$ is $\Z$ for every nonempty $X$. For $X$ a discrete space with two points, singular homology gives $H_0(X) \cong \Z^2$.