Isometries between spherical space formsFree actions of finite groups on products of even-dimensional spheresIsometry classification of spherical space formsRealizing a homology by a smooth immersionCombination theorems for discrete subgroups of isometry groupsGeodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometryWhich spherical space forms embed in $S^4$?Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups4-manifolds with finite fundamental group and spherical boundaryCan a hyperbolic manifold be a product?
Isometries between spherical space forms
Free actions of finite groups on products of even-dimensional spheresIsometry classification of spherical space formsRealizing a homology by a smooth immersionCombination theorems for discrete subgroups of isometry groupsGeodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometryWhich spherical space forms embed in $S^4$?Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups4-manifolds with finite fundamental group and spherical boundaryCan a hyperbolic manifold be a product?
$begingroup$
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 10 hours ago
TotoroTotoro
42827
$endgroup$
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 10 hours ago
TotoroTotoro
42827
$endgroup$
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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
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
4 hours ago
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 10 hours ago
TotoroTotoro
42827
$endgroup$
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 10 hours ago
TotoroTotoro
42827
asked 10 hours ago
TotoroTotoro
42827
asked 10 hours ago
TotoroTotoro
42827
asked 10 hours ago
TotoroTotoro
42827
asked 10 hours ago
TotoroTotoro
42827
42827
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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
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
4 hours ago
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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
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
4 hours ago
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
8 hours ago
$begingroup$
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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
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
4 hours ago
$begingroup$
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
4 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 9 hours ago
Igor BelegradekIgor Belegradek
19.2k143125
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f325973%2fisometries-between-spherical-space-forms%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 9 hours ago
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 9 hours ago
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 9 hours ago
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 9 hours ago
Igor BelegradekIgor Belegradek
19.2k143125
edited 9 hours ago
answered 9 hours ago
Igor BelegradekIgor Belegradek
19.2k143125
answered 9 hours ago
Igor BelegradekIgor Belegradek
19.2k143125
answered 9 hours ago
Igor BelegradekIgor Belegradek
19.2k143125
19.2k143125
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f325973%2fisometries-between-spherical-space-forms%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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
8 hours ago
$begingroup$
@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
8 hours ago
$begingroup$
De Rham's theorem is for every dimension. For 3d lens spaces there are other proofs, I think.
$endgroup$
– Igor Belegradek
8 hours ago
$begingroup$
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
4 hours ago