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

$H_p({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},0\le p\le 2n\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\\ \end{array}$

Reduced version with coefficients in integers

${\tilde{H}}_p({\mathbb{P}}^n(\mathbb{C});\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},2\le p\le 2n\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\\ \end{array}$

Unreduced version with coefficients in an abelian group or module

For coefficients in an abelian group $M$, the homology groups are:

$H_p({\mathbb{P}}^n(\mathbb{C});M)=\{\begin{array}{cc}M,& p\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n},0\le p\le 2n\\ 0,& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\\ \end{array}$

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	$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k{b}_k$	$n+1$

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 $\mathbb{Z}$s in all the even positions till $2n$, and hence its homology is $\mathbb{Z}$ in all even dimensions till $2n$.