Confused about the possibility of different differentiable structures.











up vote
3
down vote

favorite












In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.



He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.



My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept?










share|cite|improve this question


















  • 2




    "Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
    – Eric Towers
    Jun 30 '14 at 16:04










  • Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
    – Memeozuki
    Jun 30 '14 at 16:12










  • All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
    – Lee Mosher
    Jun 30 '14 at 18:31

















up vote
3
down vote

favorite












In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.



He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.



My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept?










share|cite|improve this question


















  • 2




    "Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
    – Eric Towers
    Jun 30 '14 at 16:04










  • Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
    – Memeozuki
    Jun 30 '14 at 16:12










  • All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
    – Lee Mosher
    Jun 30 '14 at 18:31















up vote
3
down vote

favorite









up vote
3
down vote

favorite











In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.



He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.



My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept?










share|cite|improve this question













In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.



He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.



My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept?







differential-geometry manifolds






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jun 30 '14 at 15:59









Memeozuki

433312




433312








  • 2




    "Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
    – Eric Towers
    Jun 30 '14 at 16:04










  • Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
    – Memeozuki
    Jun 30 '14 at 16:12










  • All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
    – Lee Mosher
    Jun 30 '14 at 18:31
















  • 2




    "Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
    – Eric Towers
    Jun 30 '14 at 16:04










  • Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
    – Memeozuki
    Jun 30 '14 at 16:12










  • All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
    – Lee Mosher
    Jun 30 '14 at 18:31










2




2




"Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
– Eric Towers
Jun 30 '14 at 16:04




"Maximal" usually means "cannot be further extended", but does not imply uniqueness. For instance a maximal ideal is one any extension of which is either itself or the whole ring. A ring may have many distinct maximal ideals.
– Eric Towers
Jun 30 '14 at 16:04












Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
– Memeozuki
Jun 30 '14 at 16:12




Thank you. I think I understand this concept then. But my question still remains in this case. Are all coordinate maps compatible with each other as a result of being diffeomorphisms? Would this not imply that there could only be one maximal atlas on a manifold by our construction regardless of dimension?
– Memeozuki
Jun 30 '14 at 16:12












All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
– Lee Mosher
Jun 30 '14 at 18:31






All coordinate maps within the same atlas are certainly compatible with each other. As my answer shows, there will always exist coordinate maps which are incompatible with a given atlas. Such coordinate maps may form the foundation of an entirely different maximal atlas for an entirely different differentiable structure.
– Lee Mosher
Jun 30 '14 at 18:31












1 Answer
1






active

oldest

votes

















up vote
3
down vote













You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $mathbb{R}$: $f(x)=x$; and $f(x) = sqrt[3]{x}$.






share|cite|improve this answer





















  • Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
    – An old man in the sea.
    Nov 19 at 10:30












  • That's correct.
    – Lee Mosher
    Nov 19 at 14:04











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: "69"
};
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',
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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f852386%2fconfused-about-the-possibility-of-different-differentiable-structures%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








up vote
3
down vote













You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $mathbb{R}$: $f(x)=x$; and $f(x) = sqrt[3]{x}$.






share|cite|improve this answer





















  • Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
    – An old man in the sea.
    Nov 19 at 10:30












  • That's correct.
    – Lee Mosher
    Nov 19 at 14:04















up vote
3
down vote













You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $mathbb{R}$: $f(x)=x$; and $f(x) = sqrt[3]{x}$.






share|cite|improve this answer





















  • Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
    – An old man in the sea.
    Nov 19 at 10:30












  • That's correct.
    – Lee Mosher
    Nov 19 at 14:04













up vote
3
down vote










up vote
3
down vote









You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $mathbb{R}$: $f(x)=x$; and $f(x) = sqrt[3]{x}$.






share|cite|improve this answer












You cite from Tu's book a statement that "for any chart on a manifold, the coordinate map is a diffeomorphism onto its image", but if you check that statement carefully I'm sure that it applies only to charts within the given maximal atlas. It is certainly possible to have incompatible coordinate charts that are not in a given maximal atlas, e.g. here are two incompatible coordinate charts on $mathbb{R}$: $f(x)=x$; and $f(x) = sqrt[3]{x}$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jun 30 '14 at 16:09









Lee Mosher

47.4k33681




47.4k33681












  • Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
    – An old man in the sea.
    Nov 19 at 10:30












  • That's correct.
    – Lee Mosher
    Nov 19 at 14:04


















  • Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
    – An old man in the sea.
    Nov 19 at 10:30












  • That's correct.
    – Lee Mosher
    Nov 19 at 14:04
















Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
– An old man in the sea.
Nov 19 at 10:30






Just to be sure I understand you example. Then each of those charts must belong to different maximal atlas for the same manifold ($mathbb{R}$), right?.
– An old man in the sea.
Nov 19 at 10:30














That's correct.
– Lee Mosher
Nov 19 at 14:04




That's correct.
– Lee Mosher
Nov 19 at 14:04


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • 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.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • 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.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f852386%2fconfused-about-the-possibility-of-different-differentiable-structures%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Ellipse (mathématiques)

Quarter-circle Tiles

Mont Emei