Product of real projective three-dimensional space and 2-sphere

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Definition

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

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 3-sphere and real projective plane, 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, something which can easily be seen from the fact that they do not have isomorphic homology groups, or even from the fact that $\mathbb {R} \mathbb {P} ^{3}\times S^{2}$ is orientable whereas $S^{3}\times \mathbb {R} \mathbb {P} ^{2}$ is non-orientable.

Algebraic topology

Homology groups

The homology groups with coefficients in integers are as follows:

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

These can be computed by combining knowledge of the homology of real projective space, the homology of spheres, and the Kunneth formula.

Cohomology groups

The cohomology groups with coefficients in integers are as follows:

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

Homology-based invariants

Invariant	General description	Value for $\mathbb {R} \mathbb {P} ^{3}\times S^{2}$
Betti numbers	The $k^{th}$ Betti number $b_{k}$ is the rank of the $k^{th}$ homology group.	$b_{0}=b_{2}=b_{3}=b_{5}=1$, all other $b_{k}$ values are zero.
Poincare polynomial	Generating polynomial for Betti numbers	$1+x^{2}+x^{3}+x^{5}=(1+x^{3})(1+x^{2})$. See also Poincare polynomial of product is product of Poincare polynomials
Euler characteristic	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$, equal to the Poincare polynomial evaluated at -1.	0 (hence it is a space with Euler characteristic zero). This follows from the fact that it is a product of spaces with Euler characteristic zero, and the Euler characteristic of the circle is zero (see Euler characteristic of product is product of Euler characteristics). It can also be seen from the fact that Euler characteristic of odd-dimensional compact connected orientable manifold is zero.