Isometries between spherical space forms
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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?
gt.geometric-topology
asked Mar 21 at 15:00
TotoroTotoro
45327
$endgroup$
add a comment |
$begingroup$
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?
gt.geometric-topology
asked Mar 21 at 15:00
TotoroTotoro
45327
$endgroup$
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
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@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
add a comment |
$begingroup$
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?
gt.geometric-topology
asked Mar 21 at 15:00
TotoroTotoro
45327
$endgroup$
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?
gt.geometric-topology
gt.geometric-topology
asked Mar 21 at 15:00
TotoroTotoro
45327
asked Mar 21 at 15:00
TotoroTotoro
45327
asked Mar 21 at 15:00
TotoroTotoro
45327
asked Mar 21 at 15:00
TotoroTotoro
45327
asked Mar 21 at 15:00
TotoroTotoro
45327
45327
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
add a comment |
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
1
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
$begingroup$
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
$begingroup$
@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
$begingroup$
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
$begingroup$
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
add a comment |
1 Answer
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$begingroup$
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.
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
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add a comment |
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$begingroup$
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.
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
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 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
edited Mar 21 at 15:52
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
answered Mar 21 at 15:44
Igor BelegradekIgor Belegradek
19.2k143125
19.2k143125
add a comment |
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$begingroup$
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