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?
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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 BudneyCommented Mar 21, 2019 at 16:50
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$\begingroup$ @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 HajlaszCommented Mar 21, 2019 at 16:54
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$\begingroup$ De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think. $\endgroup$– Igor BelegradekCommented Mar 21, 2019 at 16:59
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$\begingroup$ 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 BudneyCommented Mar 21, 2019 at 20:24
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1 Answer
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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.
answered Mar 21, 2019 at 15:44
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$\begingroup$ Is there a more recent reference to this result? $\endgroup$ Commented Apr 1, 2019 at 14:25
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$\begingroup$ @PiotrHajlasz: I wondered the same in mathoverflow.net/questions/21848/… and I still do not know the answer (in dimensions $>3$). $\endgroup$ Commented Apr 1, 2019 at 19:41