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](https://en.wikipedia.org/wiki/Mikhael_Gromov_(mathematician)).


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.



Is the minimal volume a topological invariant?
Minimal volume is not a homeomorphism invariant. It is shown in [L. Bessières, [Un théorème de rigidité différentielle](https://dx.doi.org/10.5169/seals-55112), 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".


