Homology of real projective space

Example 3.3.20 (Real projective space). Real projective space \(\RP ^n\) has a CW-structure with one cell in each dimension \(0, 1, \ldots , n\). The attaching map \(\phi _k\colon S^{k-1} \to \RP ^{k-1}\) for the \(k\)-cell is the quotient map identifying antipodal points. One may compute using geometric methods that the composite \(\Delta \colon S^{k-1} \xrightarrow {\phi _k} \RP ^{k-1} \to \RP ^{k-1}/\RP ^{k-2} \simeq S^{k-1}\) has degree \(1 + (-1)^k\): the two preimages of a generic point contribute with the same sign when \(k\) is even and opposite signs when \(k\) is odd. Thus the cellular boundary maps are \[ d_k \;=\; \begin {cases} 0 & \text {if } k \text { is odd}, \\ 2 & \text {if } k \text { is even}. \end {cases} \] This gives \[ H_k(\RP ^n) \;\cong \; \begin {cases} \Z & \text {if } k = 0, \\ \Z /2\Z & \text {if } 0 < k < n \text { and } k \text { is odd}, \\ \Z & \text {if } k = n \text { and } n \text { is odd}, \\ 0 & \text {otherwise}. \end {cases} \]