For what restrictions on topological spaces $A$ and $B$ does the following implication hold?
$$\Sigma A \cong \Sigma B \implies A\cong B$$
where $\Sigma$ is the unreduced suspension.
This should be equivalent (by simply removing the suspension points) to asking when:
$$A\times \mathbb{R} \cong B\times \mathbb{R} \implies A\cong B$$
and is thus related to 2, 3 and 4. (Unfortunately I don't understand the counterexample by Sergei Ivanov in 2)
In 3 Igor Belegradek says:
I guess, I just like to advertize that fact that if two closed manifolds
become homeomorphic after multiplying by RR , then they are h-cobordant. :)
Thus, together with the h-cobordism theorem6, it follows that for closed simply connected manifolds the implication holds.
This is all completely new to me, is it right?