Product of two real projective planes

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Definition

This topological space is defined as the Cartesian product, equipped with the product topology, of two copies of the real projective plane $\mathbb{R}{\mathbb{P}}^2$. It is denoted $\mathbb{R}{\mathbb{P}}^2\times \mathbb{R}{\mathbb{P}}^2$.

Algebraic topology

Homology groups

The homology groups with coefficients in integers are as follows:

$H_p(\mathbb{R}{\mathbb{P}}^2\times \mathbb{R}{\mathbb{P}}^2;\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z},& p=0\\ \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z},& p=1\\ \mathbb{Z}/2\mathbb{Z},& p=2,3\\ 0,& p\ge 4\\ \end{array}$

These are computed using the homology of real projective space and the Kunneth formula.

Homotopy groups

$k$	Homotopy group/set ${\pi }_k$	Value for $\mathbb{R}{\mathbb{P}}^2\times \mathbb{R}{\mathbb{P}}^2$
0	set of path components	one-point space
1	fundamental group	Klein four-group
2	second homotopy group	$\mathbb{Z}\times \mathbb{Z}$
$k\ge 2$	$k^{th}$ homotopy group	${\pi }_k(S^2)\times {\pi }_k(S^2)$

Homology-based invariants

Invariant	General description	Value for $\mathbb{R}{\mathbb{P}}^2\times \mathbb{R}{\mathbb{P}}^2$
Betti numbers	The $k^{th}$ Betti number $b_k$ is the rank of the $k^{th}$ homology group.	$b_0=1$, all higher $b_k$s are zero.
Poincare polynomial	Generating polynomial for Betti numbers	1. This follows from Poincare polynomial of product is product of Poincare polynomials, and the fact that the Poincare polynomial of $\mathbb{R}{\mathbb{P}}^2$ is 1.
Euler characteristic	$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k{b}_k$, equal to the Poincare polynomial evaluated at -1.	1. Follows from Euler characteristic of product is product of Euler characteristics, also by evaluating Poincare polynomial. Hence, this is a space with Euler characteristic one.