Why is the classification of topological spaces up to homeomorphism impossible or undesirable?
$begingroup$
In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).
Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?
But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.
So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.
general-topology algebraic-topology
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
$endgroup$
add a comment |
$begingroup$
In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).
Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?
But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.
So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.
general-topology algebraic-topology
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
$endgroup$
1
$begingroup$
"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
$endgroup$
– Arthur
Dec 11 '18 at 11:55
$begingroup$
I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
$endgroup$
– Billy Smith
Dec 11 '18 at 12:06
$begingroup$
I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:33
add a comment |
$begingroup$
In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).
Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?
But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.
So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.
general-topology algebraic-topology
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
$endgroup$
In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).
Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?
But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.
So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.
general-topology algebraic-topology
general-topology algebraic-topology
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
edited Dec 11 '18 at 11:54
Billy Smith
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
asked Dec 11 '18 at 11:36
Billy SmithBilly Smith
787
787
1
$begingroup$
"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
$endgroup$
– Arthur
Dec 11 '18 at 11:55
$begingroup$
I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
$endgroup$
– Billy Smith
Dec 11 '18 at 12:06
$begingroup$
I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:33
add a comment |
1
$begingroup$
"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
$endgroup$
– Arthur
Dec 11 '18 at 11:55
$begingroup$
I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
$endgroup$
– Billy Smith
Dec 11 '18 at 12:06
$begingroup$
I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:33
1
1
$begingroup$
"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
$endgroup$
– Arthur
Dec 11 '18 at 11:55
$begingroup$
"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
$endgroup$
– Arthur
Dec 11 '18 at 11:55
$begingroup$
I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
$endgroup$
– Billy Smith
Dec 11 '18 at 12:06
$begingroup$
I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
$endgroup$
– Billy Smith
Dec 11 '18 at 12:06
$begingroup$
I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:33
$begingroup$
I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
$endgroup$
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
$endgroup$
– Billy Smith
Dec 11 '18 at 13:53
$begingroup$
@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
$endgroup$
– freakish
Dec 11 '18 at 13:55
|
show 6 more comments
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$begingroup$
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
$endgroup$
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
$endgroup$
– Billy Smith
Dec 11 '18 at 13:53
$begingroup$
@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
$endgroup$
– freakish
Dec 11 '18 at 13:55
|
show 6 more comments
$begingroup$
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
$endgroup$
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
$endgroup$
– Billy Smith
Dec 11 '18 at 13:53
$begingroup$
@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
$endgroup$
– freakish
Dec 11 '18 at 13:55
|
show 6 more comments
$begingroup$
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
$endgroup$
Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).
Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.
Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
edited Dec 11 '18 at 13:28
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
answered Dec 11 '18 at 13:07
freakishfreakish
12.4k1630
12.4k1630
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
$endgroup$
– Billy Smith
Dec 11 '18 at 13:53
$begingroup$
@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
$endgroup$
– freakish
Dec 11 '18 at 13:55
|
show 6 more comments
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
$endgroup$
– Billy Smith
Dec 11 '18 at 13:53
$begingroup$
@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
$endgroup$
– freakish
Dec 11 '18 at 13:55
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics.
$endgroup$
– tomasz
Dec 11 '18 at 13:28
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory.
$endgroup$
– Arthur
Dec 11 '18 at 13:36
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question.
$endgroup$
– Billy Smith
Dec 11 '18 at 13:37
$begingroup$
@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
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– Billy Smith
Dec 11 '18 at 13:53
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@freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts?
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– Billy Smith
Dec 11 '18 at 13:53
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@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
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– freakish
Dec 11 '18 at 13:55
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@BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem.
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– freakish
Dec 11 '18 at 13:55
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"homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence?
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– Arthur
Dec 11 '18 at 11:55
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I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense.
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– Billy Smith
Dec 11 '18 at 12:06
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I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold: en.wikipedia.org/wiki/… Well, that's a shame. That part of my question still stands, then.
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– Billy Smith
Dec 11 '18 at 13:33