Is the minimal volume a topological invariant?

Question

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.

Answer 1

Minimal volume is not a homeomorphism invariant. It is shown in [L. Bessières, Un théorème de rigidité différentielle, Comm. Math. Helv. 73 443-479 (1998)] that the minimal volume of the connected sum of an exotic $7$-sphere and a closed hyperbolic manifold $M$ can be larger than $\mathrm{MinVol}(M)$. An online exposition can be found in section 3 of http://bremy.perso.math.cnrs.fr/smf_sec_18_07.pdf.

As to what Wikipedia says, some people use the phrase "topological invariant" to mean "diffeomorphism invariant". Here "topological" is contrasted with "geometric".