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

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

Reduced version with coefficients in integers

${\tilde {H}}_{p}(\mathbb {P} ^{n}(\mathbb {C} );\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&\qquad p\ \operatorname {even} ,2\leq p\leq 2n\\0,&\qquad \operatorname {otherwise} \\\end{array}}\right.$

Unreduced version with coefficients in an abelian group or module

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

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

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