Homology of complex projective space

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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 \Zs in all the even positions till 2n, and hence its homology is \Z in all even dimensions till 2n.