= Simultaneous antipodal quotient of two spheres
{title2=$(S^p\times S^q)/\langle(-1,-1)\rangle$}
For positive $p,q$, quotient $S^p\times S^q$ by $(x,y)\mapsto(-x,-y)$. This is a free double <covering space> action. Its <closed manifold> quotient is orientable exactly when $p+q$ is even, because the product <mapping degree> is $(-1)^{p+q+2}$. If $p,q\ge2$, the product is <simply connected> and the quotient has <fundamental group> $\mathbb Z/2$. For $p=q=2$, the quotient has integral <cohomology groups> $\mathbb Z$ in degrees $0,4$, $\mathbb Z/2$ in degrees $2,3$, and zero elsewhere. Every positive-degree <cup product> is zero. These conclusions follow from <Poincare duality>, the <Euler characteristic under a finite covering>, and the <universal coefficient theorem for cohomology>.
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