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:

$H_{p}(\mathbb {R} \mathbb {P} ^{2}\times \mathbb {R} \mathbb {P} ^{2};\mathbb {Z} )=\left\lbrace {\begin{array}{rl}\mathbb {Z} ,&p=0\\\mathbb {Z} /2\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} ,&p=1\\\mathbb {Z} /2\mathbb {Z} ,&p=2,3\\0,&p\geq 4\\\end{array}}\right.$

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	$\mathbb {Z} \times \mathbb {Z}$
$k\geq 2$	$k^{th}$ 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	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$, 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.