Mixing
van Kampen theorem for fundamental groupoid of $X$ relative to $A$
$begingroup$
Let $X$ be a manifold with submanifold $A subseteq X$. Let $Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For example, if $X$ is the upper hemisphere of a sphere and $A cong S^1$ is the equator, then $Pi_{1}(X,A) cong S^1$, and for any two objects there is a single morphism.
Question: Is there any theory developed for such groupoids? I'm looking for some kind of van Kampen theorem allowing me to express $Pi_{1}(X,A)$ in terms of $Pi_{1}(A)$ and $Pi_{1}(X setminus A)$.
algebraic-topology fundamental-groups groupoids
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
$endgroup$
add a comment |
$begingroup$
Let $X$ be a manifold with submanifold $A subseteq X$. Let $Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For example, if $X$ is the upper hemisphere of a sphere and $A cong S^1$ is the equator, then $Pi_{1}(X,A) cong S^1$, and for any two objects there is a single morphism.
Question: Is there any theory developed for such groupoids? I'm looking for some kind of van Kampen theorem allowing me to express $Pi_{1}(X,A)$ in terms of $Pi_{1}(A)$ and $Pi_{1}(X setminus A)$.
algebraic-topology fundamental-groups groupoids
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
$endgroup$
add a comment |
$begingroup$
Let $X$ be a manifold with submanifold $A subseteq X$. Let $Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For example, if $X$ is the upper hemisphere of a sphere and $A cong S^1$ is the equator, then $Pi_{1}(X,A) cong S^1$, and for any two objects there is a single morphism.
Question: Is there any theory developed for such groupoids? I'm looking for some kind of van Kampen theorem allowing me to express $Pi_{1}(X,A)$ in terms of $Pi_{1}(A)$ and $Pi_{1}(X setminus A)$.
algebraic-topology fundamental-groups groupoids
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
$endgroup$
Let $X$ be a manifold with submanifold $A subseteq X$. Let $Pi_{1}(X,A)$ denote the homotopy classes of paths with endpoints lying in $A$. This is a Lie groupoid with set of objects $A$. For example, if $X$ is the upper hemisphere of a sphere and $A cong S^1$ is the equator, then $Pi_{1}(X,A) cong S^1$, and for any two objects there is a single morphism.
Question: Is there any theory developed for such groupoids? I'm looking for some kind of van Kampen theorem allowing me to express $Pi_{1}(X,A)$ in terms of $Pi_{1}(A)$ and $Pi_{1}(X setminus A)$.
algebraic-topology fundamental-groups groupoids
algebraic-topology fundamental-groups groupoids
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
asked Jul 1 '15 at 8:04
unknownymousunknownymous
601310
601310
add a comment |
add a comment |
1 Answer
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active
oldest
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Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
$endgroup$
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
$endgroup$
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
add a comment |
$begingroup$
Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
$endgroup$
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
add a comment |
$begingroup$
Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
$endgroup$
Yes sure!
There is the book by R. Brown, "Topology and Groupoids", which treats exactly this: http://groupoids.org.uk/topgpds.html
Look also here: http://groupoids.org.uk/nonab-a-t.html
On the nlab (if you didn't know the site, take a look!) there are also these articles that you may find interesting!
http://ncatlab.org/nlab/show/van+Kampen+theorem
http://ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
edited Jan 13 at 11:36
Ronnie Brown
12.1k12939
12.1k12939
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
answered Jul 1 '15 at 9:47
geodudegeodude
4,1161343
4,1161343
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
add a comment |
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
$begingroup$
See also my answer mathoverflow.net/questions/39818/…, and presentations on my my preprint page pages.bangor.ac.uk/~mas010/brownpr.html .
$endgroup$
– Ronnie Brown
Jul 1 '15 at 21:23
add a comment |
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