Homology of complex projective space

Unreduced version with coefficients in integers
$$H_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

Reduced version with coefficients in integers
$$\tilde{H}_p(\mathbb{P}^n(\mathbb{C});\mathbb{Z}) = \left\lbrace\begin{array}{rl} \Z, & \qquad p \ \operatorname{even}, 2 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

Unreduced version with coefficients in an abelian group or module
For coefficients in an abelian group $$M$$, the homology groups are:

$$H_p(\mathbb{P}^n(\mathbb{C});M) = \left\lbrace\begin{array}{rl} M, & \qquad p \ \operatorname{even}, 0 \le p \le 2n\\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

Homology groups with integer coefficients in tabular form
We illustrate how the homology groups work for small values of $$n$$ (whereby the dimension of the corresponding complex projective space is $$2n$$). Note that for $$p > 2n$$, all homology groups are zero, so we omit those cells for visual clarity.

Related invariants
These are all invariants that can be computed in terms of the homology groups.

Facts used

 * 1) uses::CW structure of complex projective space

Proof
We use the CW-complex structure on complex projective space that has exactly one cell in every even dimension till $$2n$$. The cellular chain complex of this thus has $$\Z$$s in all the even positions till $$2n$$, and hence its homology is $$\Z$$ in all even dimensions till $$2n$$.