Homology of $S^n$ for all $n\ge 0$

For \(n\ge1\), the sphere \(S^n\) has a CW structure with one \(0\)-cell and one \(n\)-cell, with no cells in intermediate dimensions. Its cellular chain complex therefore gives \[ H_k(S^n;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=0,\\ \mathbb Z,&k=n,\\ 0,&\text{otherwise}. \end{cases} \] Equivalently, \[ \widetilde H_k(S^n;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=n,\\ 0,&k\ne n. \end{cases} \]

For \(n=0\), \(S^0\) consists of two points, so \[ H_0(S^0;\mathbb Z)\cong\mathbb Z\oplus\mathbb Z, \qquad H_k(S^0;\mathbb Z)=0\quad(k>0), \] and \(\widetilde H_0(S^0)\cong\mathbb Z\).