Homology of complex projective space

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology group and the topological space/family is complex projective space
Get more specific information about complex projective space | Get more computations of homology group

Statement

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.

$n$ Complex projective space $\mathbb{C}\mathbb{P}^n$ $H_0$ $H_1$ $H_2$ $H_3$ $H_4$ $H_5$ $H_6$ $H_7$ $H_8$
1 2-sphere $\mathbb{Z}$ 0 $\mathbb{Z}$
2 complex projective plane $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$
3 CP^3 $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$
4 CP^4 $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$ 0 $\mathbb{Z}$

Related invariants

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

Invariant General description Description of value for complex projective space $\mathbb{P}^n(\mathbb{C})$
Betti numbers The $k^{th}$ Betti number $b_k$ is the rank of the torsion-free part of the $k^{th}$ homology group. $b_0 = b_2 = b_4 = \dots = b_{2n} = 1$, all other $b_k$ values are zero.
Poincare polynomial Generating polynomial for Betti numbers $1 + x^2 + x^4 + \dots + x^{2n} = \frac{x^{2n + 2} - 1}{x^2 - 1}$
Euler characteristic $\sum_{k=0}^\infty (-1)^k b_k$ $n + 1$

Facts used

1. 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 $2n$.