Simplicial homology of $S^1$, the torus, and the punctured torus

Calculate the simplicial homology groups of \(S^1\).

Calculate the simplicial homology groups of the torus using the \(\Delta\)-complex specified by the source diagram. The diagram itself is not present in the retained Markdown extraction, so its precise simplex labels remain unrecovered.

Let \(T_0^2\) be the torus with the interior of a disk removed. Use the Mayer–Vietoris sequence for \[ T^2=T_0^2\cup D^2, \qquad T_0^2\cap D^2\simeq S^1, \] to calculate the homology groups of \(T_0^2\).