Is there a chain representing a generator of $H_q(S^q)$ consisting of only one simplex?
Here $S^q$ denotes the q-dimensional sphere and the answer can be given either for simplicial or singular homology.
The question makes sense only for q > 0, since $H_0(S^0) \cong \mathbb{Z} \oplus \mathbb{Z}$.
I know already that a generator can always be represented by a chain consisting of two simplices, e.g., gluing two q-simplices $\Delta_1^q, \Delta_2^q$ along their boundary in such a way that their difference $\Delta_1^q - \Delta_2^q$ is a cycle. Then this cycle represents also a generator of $H_q(S^q)$.
In the case q = 1, the answer is yes, since we can wrap the unit interval once around the circle such that the two endpoints of the interval are mapped to one point on the circle. So we get a cycle and one can also show that this cycle is homologous to $\Delta_1^1 - \Delta_2^1$, which is a generator of $H_1(S^1)$.
But what about the case q > 1?