Example 3.3.14. Complex projective space \(\CP ^n\) has a filtration whose successive strata are the affine cells \(\C ^0,\C ^1,\ldots ,\C ^n\), and hence a CW-structure with one cell in each even dimension \(0, 2, 4, \ldots , 2n\). Since no two cells are in adjacent dimensions, Exercise 3.3.13(3) gives \[ H_k(\CP ^n) \;\cong \; \begin {cases} \Z & \text {if } k = 0, 2, 4, \ldots , 2n, \\ 0 & \text {otherwise.} \end {cases} \]
Generated from the authoritative LaTeX source.