Mixing
How many connected components for the intersection $S cap GL_n(mathbb R)$ where $S subset M_n(mathbb R)$ is a linear subspace?
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Let $S subset M_n(mathbb R^n)$ be a linear subspace. Is there a way to determine how many connected components there are for $S cap GL_n(mathbb R)$? Let us assume the intersection is nonempty. $GL_n(mathbb R)$ has two connected components. Does this intersection have two connected components or possibly more?
linear-algebra abstract-algebra general-topology connectedness
asked Jul 21 at 6:02
user9527
925525
add a comment |Â
up vote
12
down vote
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Let $S subset M_n(mathbb R^n)$ be a linear subspace. Is there a way to determine how many connected components there are for $S cap GL_n(mathbb R)$? Let us assume the intersection is nonempty. $GL_n(mathbb R)$ has two connected components. Does this intersection have two connected components or possibly more?
linear-algebra abstract-algebra general-topology connectedness
asked Jul 21 at 6:02
user9527
925525
Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
1
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44
add a comment |Â
up vote
12
down vote
favorite
up vote
12
down vote
favorite
Let $S subset M_n(mathbb R^n)$ be a linear subspace. Is there a way to determine how many connected components there are for $S cap GL_n(mathbb R)$? Let us assume the intersection is nonempty. $GL_n(mathbb R)$ has two connected components. Does this intersection have two connected components or possibly more?
linear-algebra abstract-algebra general-topology connectedness
asked Jul 21 at 6:02
user9527
925525
Let $S subset M_n(mathbb R^n)$ be a linear subspace. Is there a way to determine how many connected components there are for $S cap GL_n(mathbb R)$? Let us assume the intersection is nonempty. $GL_n(mathbb R)$ has two connected components. Does this intersection have two connected components or possibly more?
linear-algebra abstract-algebra general-topology connectedness
asked Jul 21 at 6:02
user9527
925525
edited Jul 21 at 6:16
asked Jul 21 at 6:02
user9527
925525
asked Jul 21 at 6:02
user9527
925525
asked Jul 21 at 6:02
user9527
925525
925525
Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
1
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44
add a comment |Â
Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
1
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44
Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
1
1
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44
add a comment |Â
1 Answer
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It may have more. Consider $Ssubseteq M_2(newcommandRRmathbbRRR)=newcommandset[1]left#1rightsetnewcommandbmatbeginpmatrixnewcommandematendpmatrixbmat a & b\c&demat : a,b,c,dinRR$
defined by the equations
$a=d$, $b=c$. This gives a two dimensional subspace. $det$ restricted to this subspace has the form $a^2-b^2$, so the intersection of the complement of $GL_2(RR)$ with $S$ is two intersecting lines ($a=b$ and $a=-b$), which divides the plane into four pieces. Hence $Scap GL_2(RR)$ has four connected components in this case.
I have no ideas as to how to compute the number of connected components in general, however I thought this might be useful.
answered Jul 21 at 6:27
jgon
8,54611435
Nice example. Thanks.
– user9527
Jul 21 at 20:22
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
It may have more. Consider $Ssubseteq M_2(newcommandRRmathbbRRR)=newcommandset[1]left#1rightsetnewcommandbmatbeginpmatrixnewcommandematendpmatrixbmat a & b\c&demat : a,b,c,dinRR$
defined by the equations
$a=d$, $b=c$. This gives a two dimensional subspace. $det$ restricted to this subspace has the form $a^2-b^2$, so the intersection of the complement of $GL_2(RR)$ with $S$ is two intersecting lines ($a=b$ and $a=-b$), which divides the plane into four pieces. Hence $Scap GL_2(RR)$ has four connected components in this case.
I have no ideas as to how to compute the number of connected components in general, however I thought this might be useful.
answered Jul 21 at 6:27
jgon
8,54611435
Nice example. Thanks.
– user9527
Jul 21 at 20:22
add a comment |Â
up vote
6
down vote
accepted
It may have more. Consider $Ssubseteq M_2(newcommandRRmathbbRRR)=newcommandset[1]left#1rightsetnewcommandbmatbeginpmatrixnewcommandematendpmatrixbmat a & b\c&demat : a,b,c,dinRR$
defined by the equations
$a=d$, $b=c$. This gives a two dimensional subspace. $det$ restricted to this subspace has the form $a^2-b^2$, so the intersection of the complement of $GL_2(RR)$ with $S$ is two intersecting lines ($a=b$ and $a=-b$), which divides the plane into four pieces. Hence $Scap GL_2(RR)$ has four connected components in this case.
I have no ideas as to how to compute the number of connected components in general, however I thought this might be useful.
answered Jul 21 at 6:27
jgon
8,54611435
Nice example. Thanks.
– user9527
Jul 21 at 20:22
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
It may have more. Consider $Ssubseteq M_2(newcommandRRmathbbRRR)=newcommandset[1]left#1rightsetnewcommandbmatbeginpmatrixnewcommandematendpmatrixbmat a & b\c&demat : a,b,c,dinRR$
defined by the equations
$a=d$, $b=c$. This gives a two dimensional subspace. $det$ restricted to this subspace has the form $a^2-b^2$, so the intersection of the complement of $GL_2(RR)$ with $S$ is two intersecting lines ($a=b$ and $a=-b$), which divides the plane into four pieces. Hence $Scap GL_2(RR)$ has four connected components in this case.
I have no ideas as to how to compute the number of connected components in general, however I thought this might be useful.
answered Jul 21 at 6:27
jgon
8,54611435
It may have more. Consider $Ssubseteq M_2(newcommandRRmathbbRRR)=newcommandset[1]left#1rightsetnewcommandbmatbeginpmatrixnewcommandematendpmatrixbmat a & b\c&demat : a,b,c,dinRR$
defined by the equations
$a=d$, $b=c$. This gives a two dimensional subspace. $det$ restricted to this subspace has the form $a^2-b^2$, so the intersection of the complement of $GL_2(RR)$ with $S$ is two intersecting lines ($a=b$ and $a=-b$), which divides the plane into four pieces. Hence $Scap GL_2(RR)$ has four connected components in this case.
I have no ideas as to how to compute the number of connected components in general, however I thought this might be useful.
answered Jul 21 at 6:27
jgon
8,54611435
answered Jul 21 at 6:27
jgon
8,54611435
answered Jul 21 at 6:27
jgon
8,54611435
answered Jul 21 at 6:27
jgon
8,54611435
8,54611435
Nice example. Thanks.
– user9527
Jul 21 at 20:22
add a comment |Â
Nice example. Thanks.
– user9527
Jul 21 at 20:22
Nice example. Thanks.
– user9527
Jul 21 at 20:22
Nice example. Thanks.
– user9527
Jul 21 at 20:22
add a comment |Â
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Maybe with homology?..
– Chris Custer
Jul 21 at 6:13
Not really sure (that's why the question mark). Isn't the zeroth homology group the one whose rank is the number of connected components.
– Chris Custer
Jul 21 at 6:35
1
May be $2^n$ is the maximum? Consider the space $D$ of diagonal matrices. In any path component of $Dcap GL_n$ no diagonal entry can go to zero, so in a path component the signs of the $n$ entries are fixed.
– Jyrki Lahtonen
Jul 30 at 21:33
@JyrkiLahtonen: Is it possible to give conditions on the subspace such that the intersection has precisely two connected components? Please take a look at this question if you had time math.stackexchange.com/questions/2858989/…
– user9527
Jul 30 at 21:44