Product of two 2-spheres

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Definition

This topological space is defined as the Cartesian product of two copies of the 2-sphere $S^{2}$, equipped with the product topology. It is denoted $S^{2}\times S^{2}$.

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	Yes	Yes	simple connectedness is product-closed	satisfies: simply connected manifold
rationally acyclic space	No	Yes	Product of spaces neither of which is rationally acyclic.	dissatisfies: acyclic space, weakly contractible space, contractible space
space with Euler characteristic zero	No	Yes	Product of two spaces, one of which (the circle) 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 4
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: homology of product of spheres

The homology groups with coefficients in integers are as follows:

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

The homology groups with coefficients in any module $M$ over a ring are as follows:

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

Cohomology groups

Further information: cohomology of product of spheres

The cohomology groups with coefficients in integers are as follows:

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

The homology groups with coefficients in any module $M$ over a ring are as follows:

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

Homology-based invariants

Invariant	General description	Description of value for $S^{2}\times S^{1}$	Comment
Betti numbers	The $k^{th}$ Betti number $b_{k}$ is the rank of the torsion-free part of the $k^{th}$ homology group.	$b_{0}=b_{4}=1$, $b_{2}=2$, $b_{1}=b_{3}=0$, all higher $b_{k}$s are zero.
Poincare polynomial	Generating polynomial for Betti numbers, i.e., the polynomial $\sum b_{k}x^{k}$	$(1+x^{2})^{2}=1+2x^{2}+x^{4}$	See also Poincare polynomial of product is product of Poincare polynomials. The Poincare polynomial for $S^{m_{1}}\times S^{m_{2}}\times \dots \times S^{m_{r}}$ is $\prod _{i=1}^{r}(1+x^{m_{i}})$
Euler characteristic	$\sum _{k=0}^{\infty }(-1)^{k}b_{k}$, obtained by evaluating Poincare polynomial at -1.	4	Follows from Euler characteristic of product is product of Euler characteristics. In particular, the Euler characteristic of a product space is zero if any of the factor spaces has Euler characteristic zero.