On Wikipedia (https://en.wikipedia.org/wiki/Minimal_volume) it is said that the minimal volume

$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$

is a topological invariant, introduced by Gromov.

I have no doubt that this concept was introduced by Gromov, but I am having my doubts that this is really a topological invariant... That would mean that homeomorphic manifolds have the same minimal volume and that seems to good to be true... So, my question is: Is the minimal volume invariant under homeomorphisms?

I apologize if this question is too basic for mathoverflow... in that case I will reask it on math.stackexchage.

On Wikipedia, it is said that the minimal volume

$$\operatorname{MinVol}(M):=\inf\{\operatorname{vol}(M,g) :g\text{ a complete Riemannian metric with }|K_{g}|\leq 1\}$$

is a topological invariant, introduced by Gromov.

I have no doubt that this concept was introduced by Gromov, but I am having my doubts that this is really a topological invariant. That would mean that homeomorphic manifolds have the same minimal volume and that seems too good to be true. So, is the minimal volume invariant under homeomorphisms?

I apologize if this question is too basic for mathoverflow... in that case I will reask it on math.stackexchage.