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Is there a moduli space $mathcalM$ of the closed, $n$-dimensional, simply connected manifolds?
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Is there a moduli space $mathcalM$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, NinmathcalM$ are identified if their embeddings in $mathbbR^infty$ (actually $mathbbR^max(2dim M,2dim N)$ suffices by the Whitney embedding theorem) are the same, i.e., if the images of their $C^k$-smooth embeddings are equal: $im(textEmb(M,mathbbR^infty))=im(textEmb(N,mathbbR^infty))$;
(ii) $mathcalM$ is itself a manifold which carries a natural Hausdorff topology induced by the metric $d(M,N)=sup_(p,q)in Mtimes Nd(textEmb(B(p;r),mathbbR^infty),textEmb(B(q;r),mathbbR^infty))$ where $B(p;r):=xin M: d(p,x)<r$ ("the distance between the two embeddings").
For instance, could one use the affine Grassmannian $Graff(n,V)$? I have also seen the use of the moduli space of a manifold in this series, defined as
$$ Psi_d(mathbbR^n+1) = left omegasubseteqmathbbR^n middlevert beginarrayl
texttopologically closed smooth dtext-dimensional\
textsubmanifold, i.e. closed as a subset of mathbbR^n, \
partialomega = emptyset.
endarrayright.
$$How well are these objects known and have they been studied?
Thoughts: If we consider $Graff(n,mathbbR^infty)$, then we can identify each point in it as the tangent bundle of one such manifold. Then we could possibly use the exponential map to give a description of this moduli space. I am curious if there is a known expression which exhausts all such manifolds.
Please note I am posting this here, as this was not received well on MO. Before you vote to close this, could you suggest a way to improve the question? I am not an expert, so it would be nice to receive constructive feedback instead of an immediate adverse response.
Thanks in advance! Any help would be appreciated.
manifolds grassmannian moduli-space
asked Jul 31 at 18:43
Multivariablecalculus
488313
 |Â
show 12 more comments
up vote
0
down vote
favorite
Is there a moduli space $mathcalM$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, NinmathcalM$ are identified if their embeddings in $mathbbR^infty$ (actually $mathbbR^max(2dim M,2dim N)$ suffices by the Whitney embedding theorem) are the same, i.e., if the images of their $C^k$-smooth embeddings are equal: $im(textEmb(M,mathbbR^infty))=im(textEmb(N,mathbbR^infty))$;
(ii) $mathcalM$ is itself a manifold which carries a natural Hausdorff topology induced by the metric $d(M,N)=sup_(p,q)in Mtimes Nd(textEmb(B(p;r),mathbbR^infty),textEmb(B(q;r),mathbbR^infty))$ where $B(p;r):=xin M: d(p,x)<r$ ("the distance between the two embeddings").
For instance, could one use the affine Grassmannian $Graff(n,V)$? I have also seen the use of the moduli space of a manifold in this series, defined as
$$ Psi_d(mathbbR^n+1) = left omegasubseteqmathbbR^n middlevert beginarrayl
texttopologically closed smooth dtext-dimensional\
textsubmanifold, i.e. closed as a subset of mathbbR^n, \
partialomega = emptyset.
endarrayright.
$$How well are these objects known and have they been studied?
Thoughts: If we consider $Graff(n,mathbbR^infty)$, then we can identify each point in it as the tangent bundle of one such manifold. Then we could possibly use the exponential map to give a description of this moduli space. I am curious if there is a known expression which exhausts all such manifolds.
Please note I am posting this here, as this was not received well on MO. Before you vote to close this, could you suggest a way to improve the question? I am not an expert, so it would be nice to receive constructive feedback instead of an immediate adverse response.
Thanks in advance! Any help would be appreciated.
manifolds grassmannian moduli-space
asked Jul 31 at 18:43
Multivariablecalculus
488313
1
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
2
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
2
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
2
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
2
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17
 |Â
show 12 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is there a moduli space $mathcalM$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, NinmathcalM$ are identified if their embeddings in $mathbbR^infty$ (actually $mathbbR^max(2dim M,2dim N)$ suffices by the Whitney embedding theorem) are the same, i.e., if the images of their $C^k$-smooth embeddings are equal: $im(textEmb(M,mathbbR^infty))=im(textEmb(N,mathbbR^infty))$;
(ii) $mathcalM$ is itself a manifold which carries a natural Hausdorff topology induced by the metric $d(M,N)=sup_(p,q)in Mtimes Nd(textEmb(B(p;r),mathbbR^infty),textEmb(B(q;r),mathbbR^infty))$ where $B(p;r):=xin M: d(p,x)<r$ ("the distance between the two embeddings").
For instance, could one use the affine Grassmannian $Graff(n,V)$? I have also seen the use of the moduli space of a manifold in this series, defined as
$$ Psi_d(mathbbR^n+1) = left omegasubseteqmathbbR^n middlevert beginarrayl
texttopologically closed smooth dtext-dimensional\
textsubmanifold, i.e. closed as a subset of mathbbR^n, \
partialomega = emptyset.
endarrayright.
$$How well are these objects known and have they been studied?
Thoughts: If we consider $Graff(n,mathbbR^infty)$, then we can identify each point in it as the tangent bundle of one such manifold. Then we could possibly use the exponential map to give a description of this moduli space. I am curious if there is a known expression which exhausts all such manifolds.
Please note I am posting this here, as this was not received well on MO. Before you vote to close this, could you suggest a way to improve the question? I am not an expert, so it would be nice to receive constructive feedback instead of an immediate adverse response.
Thanks in advance! Any help would be appreciated.
manifolds grassmannian moduli-space
asked Jul 31 at 18:43
Multivariablecalculus
488313
Is there a moduli space $mathcalM$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, NinmathcalM$ are identified if their embeddings in $mathbbR^infty$ (actually $mathbbR^max(2dim M,2dim N)$ suffices by the Whitney embedding theorem) are the same, i.e., if the images of their $C^k$-smooth embeddings are equal: $im(textEmb(M,mathbbR^infty))=im(textEmb(N,mathbbR^infty))$;
(ii) $mathcalM$ is itself a manifold which carries a natural Hausdorff topology induced by the metric $d(M,N)=sup_(p,q)in Mtimes Nd(textEmb(B(p;r),mathbbR^infty),textEmb(B(q;r),mathbbR^infty))$ where $B(p;r):=xin M: d(p,x)<r$ ("the distance between the two embeddings").
For instance, could one use the affine Grassmannian $Graff(n,V)$? I have also seen the use of the moduli space of a manifold in this series, defined as
$$ Psi_d(mathbbR^n+1) = left omegasubseteqmathbbR^n middlevert beginarrayl
texttopologically closed smooth dtext-dimensional\
textsubmanifold, i.e. closed as a subset of mathbbR^n, \
partialomega = emptyset.
endarrayright.
$$How well are these objects known and have they been studied?
Thoughts: If we consider $Graff(n,mathbbR^infty)$, then we can identify each point in it as the tangent bundle of one such manifold. Then we could possibly use the exponential map to give a description of this moduli space. I am curious if there is a known expression which exhausts all such manifolds.
Please note I am posting this here, as this was not received well on MO. Before you vote to close this, could you suggest a way to improve the question? I am not an expert, so it would be nice to receive constructive feedback instead of an immediate adverse response.
Thanks in advance! Any help would be appreciated.
manifolds grassmannian moduli-space
asked Jul 31 at 18:43
Multivariablecalculus
488313
edited Jul 31 at 23:28
asked Jul 31 at 18:43
Multivariablecalculus
488313
asked Jul 31 at 18:43
Multivariablecalculus
488313
asked Jul 31 at 18:43
Multivariablecalculus
488313
488313
1
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
2
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
2
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
2
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
2
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17
 |Â
show 12 more comments
1
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
2
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
2
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
2
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
2
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17
1
1
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
2
2
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
2
2
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
2
2
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
2
2
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17
 |Â
show 12 more comments
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1
The question seems to be unchanged from MO. You might be able to write a better question if you first think a while about the comments you got on MO.
– Lee Mosher
Jul 31 at 18:58
2
I haven't seen the MO posting, but here are some thoughts. It sounds like you want is to understand "the set (moduli space) of (closed?) simply-connected $n$-manifolds." For this, you'll want to ask some concrete, focused questions. Just to start: Is there a natural topology on that set? Is the set itself a manifold in any sense? You also need to specify the equivalence relation. That is: Do you regard two manifolds as "the same" if they're diffeomorphic, or merely homeomorphic, or merely homotopy equivalent? Also: Do all your manifolds live in euclidean space, or are they abstract?
– Jesse Madnick
Jul 31 at 19:21
2
What people want you to ask instead is, "is there a moduli space of the closed, n-dimensional, simply connected manifolds that has (insert desired properties)." Sure, you can use the affine Grassmannian, both collections have the same cardinality. But how do you want the topology of the moduli space to capture 'closeness' of two manifolds? Especially, what is your definition of two simply connected, closed n-manifolds being 'close'? This is fundamental to your question. Gromov Hausdorff?
– John Samples
Jul 31 at 21:09
2
Yes, the topology of $BtextDiff(M)$ is known to some degree and studied: this is closely related to mapping class groups, which are certainly very interesting. Notice that e.g. $Bbb R subset Bbb R^infty$ is "topologically closed", and so the moduli space in the lecture series is different, as it includes some noncompact manifolds; and in fact the topology is such that the resulting space is connected (non-diffeomorphic manifolds may be connected). The keyword as a starting place is Madsen-Tillman-Weiss theorem; Hatcher has some notes on this. There are ways to calculate the homology...
– Mike Miller
Jul 31 at 23:17
2
...of this second kind of moduli space, and ways to relate the homology in low degrees to the homology of $sqcup_M BtextDiff(M)$. This is part of the MTW theorem.
– Mike Miller
Jul 31 at 23:17