3
$\begingroup$

Let $S^n/\Gamma_i\,(i=1,2)$ be a $n$-dimensional spherical space form, where $\Gamma_i \subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.

Suppose $S^n/\Gamma_1$ is diffeomorphic to $S^n/\Gamma_2$, can we show they are isometric?

$\endgroup$
  • 1
    $\begingroup$ A big chunk of the details are in Thurston's "Three-Dimensional Geometry and Topology". As Igor mentioned, the primary technicality issue is Lens spaces, and that part isn't as developed in Thurston's book. Bonahon has a lovely proof in that context, and you can also find Bonahon's proof in Hatcher's 3-manifolds notes (on his web page). $\endgroup$ – Ryan Budney Mar 21 at 16:50
  • $\begingroup$ @RyanBudney Thursron's book and Hatcher's notes are about the three dimensional manifolds while the question is for any $n$. I see something missing here. $\endgroup$ – Piotr Hajlasz Mar 21 at 16:54
  • $\begingroup$ De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think. $\endgroup$ – Igor Belegradek Mar 21 at 16:59
  • $\begingroup$ Whoops, I just imagined this was the $n=3$ case. The general case is similar, although the Bonahon proof is particular to $n=3$. $\endgroup$ – Ryan Budney Mar 21 at 20:24
8
$\begingroup$

Yes, diffeomorphic spherical space forms are isometric. This famous result of Georges de Rham can be found in [de Rham, G. Complexes à automorphismes et homéomorphie différentiable. Ann. Inst. Fourier Grenoble 2 (1950), 51–67 (1951)]. The main tool is Reidemeister's torsion. For lens spaces the result was proved by W. Franz in 1935.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.