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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 $p,q\ge 2$, 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.

Start with a singular disk whose boundary represents a prescribed nontrivial class in a boundary subsurface. Repeatedly lift it through double covering spaces of regular neighbourhoods of its image. The number of distinct image simplexes strictly increases and is bounded by the number of domain simplexes, so the tower ends. At the top, vanishing first mod-two homology forces every boundary component to be a sphere. An embedded disk can then be constructed and projected down the tower. Resolving double curves preserves a boundary class outside a chosen normal subgroup, giving an embedded compression disk at the bottom.

The two-fold branched cover of a knot is the double cover of $S^3$ branched along the knot. Its first homology has a nonsingular linking form and, when finite, has order $\det K$.

Away from the knot this is a two-sheeted covering space; at the knot it has the standard local model of a branched covering.

The Warsaw circle joins the two path components of a closed topologist's sine curve by an arc attached at their outer endpoints. It is path-connected and has trivial fundamental group, yet admits a continuous map to the circle with no continuous lift to its real-line covering space. Set the map equal to $1$ on the closed topologist's sine curve and wind once on the joining arc. A lift would be constant on each path component of the sine curve, with the same constant on both by continuity, but would change by $1$ along the joining arc. This proves that the map is not null-homotopic, by the homotopy lifting theorem for a covering map. Thus finiteness of the fundamental group alone cannot replace local path connectedness in the usual lifting criterion for a covering space.