Let $\sigma: \Delta^n \to X$ be a singular simplex such that $\partial(\sigma)=0$. I was wondering if given $k\geq 0$ there is a way of producing another singular simplex $$\theta: \Delta^n \to X \enspace \text{such that } \theta=k\sigma \in H_n(X)$$.
This seems like possible to do since we can go through a loop $k$-times and naively one should be able to subdivide the simplex in a clever way and just apply the map to the subdivision.
