It is natural to ask if it is possible for the mapping cone $X\cup\_\alpha CA$
to be homeomorphic to the mapping cone $X\cup\_\beta CB$ with $A$ and $B$
nonhomeomorphic. Is there a standard go-to example for this?


I have vague memories that there are manifolds $M$ and $N$ that are not homeomorphic, but
$M\times \mathbb{R} \cong N \times \mathbb{R}$, and it seems like it might be a mere hop, skip, and a jump from there to an example.



Nonhomeomophic spaces with homeomorphic mapping cones
The [double suspension theorem](https://en.wikipedia.org/wiki/Double_suspension_theorem) says that if $Y$ is a homology $3$-sphere, then its double suspension $\Sigma^2 Y$ is homeomorphic to $S^5$. If we take $Y$ to be the Poincaré sphere, then $\Sigma Y$ is not a topological manifold, since the suspension points are not manifold points, and in particular $\Sigma Y$ is not homeomorphic to $S^4$. Taking these two spaces as $A$ and $B$ and maps to a point as $\alpha$ and $\beta$ gives a fairly well-known example.


