SIngular complex $C_n(X)$ where $X$ is a CW-complex with cells of dimension less than n

Question

I wonder if the singular complex $C_n(X)=0$ where $X$ is a CW-complex with cells of dimension less than $n$. I know this is true for simplicial case. But what about singular complexes?

Answer 1

$C_n(X)$ is free abelian generated by every map $\Delta^n \to X$ in the universe. There are lots of maps no matter $n$ nor $X$, e.g. constants. (Unless, I guess, if $X$ is empty or a point.)