Product of 3-sphere and real projective plane

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Definition

This topological space is defined as the Cartesian product (equipped with the product topology) of the 3-sphere $S^{3}$ and the real projective plane $\mathbb {R} \mathbb {P} ^{2}$. It is denoted $S^{3}\times \mathbb {R} \mathbb {P} ^{2}$ or $\mathbb {R} \mathbb {P} ^{2}\times S^{3}$.

Interesting feature

This topological space is one of the simplest examples of a connected manifold such that there exists another connected manifold, namely product of real projective three-dimensional space and 2-sphere, such that both manifolds have the property that their corresponding homotopy groups are isomorphic to each other but the manifolds themselves are not homotopy-equivalent spaces.

Algebraic topology

Homology groups

The homology groups with coefficients in integers are as follows:

$H_{p}(S^{3}\times \mathbb {R} \mathbb {P} ^{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&p=0,3\\\mathbb {Z} /2\mathbb {Z} ,&p=1,4\\0,&p=2,p\geq 5\\\end{array}}$

Cohomology groups

The cohomology groups with coefficients in integers are as follows:

$H^{p}(S^{3}\times \mathbb {R} \mathbb {P} ^{2};\mathbb {Z} )=\lbrace {\begin{array}{rl}\mathbb {Z} ,&p=0,3\\\mathbb {Z} /2\mathbb {Z} ,&p=2,5\\0,&p=1,4,p\geq 6\\\end{array}}$

Homology-based invariants

Invariant	General description	Description of value for torus $S^{3}\times \mathbb {R} \mathbb {P} ^{2}$
Betti numbers	The $k^{th}$ Betti number $b_{k}$ is the rank of the $k^{th}$ homology group.	$b_{0}=b_{3}=1$, all other $b_{k}$s are zero
Poincare polynomial	Generating polynomial for Betti numbers	$1+x^{3}$
Euler characteristic	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$, equal to the Poincare polynomial evaluated at -1.	0. This can also be seen from the fact that Euler characteristic of product is product of Euler characteristics and one of the factors, $S^{3}$, has Euler characteristic zero.