Product of 2-sphere and real projective plane

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

Definition

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

Topological space properties

Property	Satisfied?	Is the property a homotopy-invariant property of topological spaces?	Explanation	Corollary properties satisfied/dissatisfied
manifold	Yes	No	product of manifolds is manifold	satisfies: metrizable space, second-countable space, and all the separation axioms down from perfectly normal space and monotonically normal space, including normal, completely regular, regular, Hausdorff, etc.
path-connected space	Yes	Yes	path-connectedness is product-closed	satisfies: connected space, connected manifold, homogeneous space (via connected manifold, see connected manifold implies homogeneous)
simply connected space	No	Yes	Product of two spaces, one of which (the real projective plane) is not simply connected.	dissatisfies: simply connected manifold
rationally acyclic space	No	Yes	The 2-sphere is not rationally acyclic.	dissatisfies: acyclic space, weakly contractible space, contractible space
space with Euler characteristic zero	No	Yes	Product of two spaces, neither of which has Euler characteristic zero. Note that Euler characteristic of product is product of Euler characteristics
space with Euler characteristic one	No	Yes	The Euler characteristic is $2\times 1=2$.
compact space	Yes	No	Product of compact spaces, see Tychonoff's theorem	satisfies: compact manifold, compact polyhedron, polyhedron (via compact manifold), compact Hausdorff space, and all properties weaker than compactness

Algebraic topology

Homology groups

Further information: Kunneth formula, homology of spheres, homology of real projective space

The homology groups with coefficients in integers are as follows:

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