Are singularities part of the manifold?

[George Jones]

Years ago, there was a related discussion between vanesch and me, posts 53, 54, 58, 62, 63 in

https://www.physicsforums.com/showthread.php?p=1251928#post1251928.

I don't know how this is for the present discussion.

 

[martinbn]

I see what you're saying, and I don't disagree but micromass is speaking of something different from what I said. That should sort out the pandemonium of this already chaotic thread :)

Yes, sorry about this. I was just commenting on the original question, because I think that's what he was asking, and your example may give him the wrong impression.

 

[martinbn]

It is in question here. We put on $M$ the $x^3$-structure. I say that $\varphi:M\rightarrow \mathbb{R}:x\rightarrow x$ is a valid chart (and thus a homeomorphism), but not a diffeomorphism.

But it is not a chart of the same atlas. The way I understand the question is, can e chart map be homeomorphism but not a diffeomorphism, no other structures are considered.

 

[WannabeNewton]

Yes, sorry about this. I was just commenting on the original question, because I think that's what he was asking, and your example may give him the wrong impression.

Yeah but there's no need to apologize, this thread has just been really hard to sort out.

Years ago, there was a related discussion between vanesch and me, posts 53, 54, 58, 62, 63 in

https://www.physicsforums.com/showthread.php?p=1251928#post1251928.

Thanks George!

But it is not a chart of the same atlas. The way I understand the question is, can e chart map be homeomorphism but not a diffeomorphism, no other structures are considered.

Ok then I myself misunderstood the original question.

 

[micromass]

OK, let's throw in some references to end this:

From Lee's smooth manifolds:

Let $M$ be a topological $n$-manifold. A coordinate chart (or just a chart) on $M$ is a pair $(U,\varphi)$, where $U$ is an open subset of $M$ and $\varphi:U\rightarrow \tilde{U}$ is a homeomorphism from $U$ to an open subset $\tilde{U}=\varphi(U)\subseteq \mathbb{R}^n$.

So no smooth structure is required to be a chart.

If $M$ is a smooth manifold, any chart $(U,\varphi)$ contained in the given maximal smooth atlas will be called a smooth chart.

So charts don't need to be diffeomorphisms. While smooth charts are.

 

[micromass]

But it is not a chart of the same atlas. The way I understand the question is, can e chart map be homeomorphism but not a diffeomorphism, no other structures are considered.

Sure, it can be. A chart map is just defined to be a homeomorphism. The smooth structure is irrelevant.
If it turns out to be a diffeomorphism, then it's a smooth chart.

 

[PeterDonis]

The way I understand the question is, can e chart map be homeomorphism but not a diffeomorphism, no other structures are considered.

Since I asked the original question that spawned this subthread, I suppose I should clarify: I was not really trying to impose specific conditions on the question, I was just trying to understand why the distinction between homeomorphisms and diffeomorphisms is drawn at all in this connection. The various examples given show why: a chart map, by itself, does not *have* to be a diffeomorphism, only a homeomorphism. That's what I was trying to clarify; I wasn't concerned with any particular specific examples, only the general question.

 

[martinbn]

Since I asked the original question that spawned this subthread, I suppose I should clarify: I was not really trying to impose specific conditions on the question, I was just trying to understand why the distinction between homeomorphisms and diffeomorphisms is drawn at all in this connection. The various examples given show why: a chart map, by itself, does not *have* to be a diffeomorphism, only a homeomorphism. That's what I was trying to clarify; I wasn't concerned with any particular specific examples, only the general question.

Yes, that is how I understood the question, and the reason for my remark. A differentiable manifold is not just the topological space $M$, but a pair [itex](M\mathcal A)[/tex] and any question about differentiability of maps is in the context of the chosen atlas.

 

[PeterDonis]

A differentiable manifold is not just the topological space $M$, but a pair $(M, \mathcal A)$ and any question about differentiability of maps is in the context of the chosen atlas.

This seems to me to be a question of terminology. You're not saying micromass' example is invalid, period; you're just saying it doesn't fit within the definition you're using. I wasn't concerned with any specific set of definitions; I was just looking for examples. Yours, micromass', and WN's have all helped to clarify what's going on.

 

[martinbn]

Well, yes, I agree that what micro and wbn say is correct, I was just giving my take on this. It is indeed a terminological issue. Sorry if I wasn't expressing myself clearly. English is a hard language to write in.

 

[PeterDonis]

Well, yes, I agree that what micro and wbn say is correct, I was just giving my take on this. It is indeed a terminological issue. Sorry if I wasn't expressing myself clearly. English is a hard language to write in.

No problem, that's why we have math.