18
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ Do you agree with my edits? $\endgroup$ Commented Mar 29, 2023 at 18:42
  • 4
    $\begingroup$ Does Gromov need a link to his Wikipedia page on a Math Overflow post? $\endgroup$ Commented Mar 29, 2023 at 19:43

1 Answer 1

Reset to default
32
$\begingroup$

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".

Improve this answer
$\endgroup$
2
  • 5
    $\begingroup$ I changed the link to one that has the correct author, plus the advantage of being free. $\endgroup$ Commented Mar 29, 2023 at 21:11
  • 2
    $\begingroup$ My email to EMS produced results. The DOI link doi.org/10.1007/s000140050064 now leads to a page in which the correct author is shown $\endgroup$ Commented Apr 6, 2023 at 5:34

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.