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Reference request fundamental group of surface of genus $g$ and $n$ boundary components
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Let $Sigma_g,n$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $partial_i$. I would like to find a reference for the following group presentation of the fundamental group and the first homology group:
$pi_1(Sigma_g,n,x)
=langle alpha_1,ldots,alpha_g,beta_1,ldots,beta_g, partial_1,ldots, partial_n rangle Big/ Big(prod_i=1^g alpha_ibeta_ialpha_i^-1beta_i^-1prod_j=1^n partial_jBig) \
H_1(Sigma_g,n,mathbbZ)
=pi_1(Sigma_g,n,x)/[pi_1(Sigma_g,n,x),pi_1(Sigma_g,n,x)]
=mathbbZ^2g+nBig/ Big(prod_j=1^n partial_jBig)$
Does anyone know a reference? It would be nice if the reference would also include a picture with the $alpha_i$ and $beta_i$.
If $n=0$ the result can for example be found in Hatchers algebraic topology, page 51. Of course, if we know the first fact, the second one is quite clear.
algebraic-topology surfaces fundamental-groups group-presentation
asked Jul 17 at 12:29
mathstackuser
61011
add a comment |Â
up vote
2
down vote
favorite
Let $Sigma_g,n$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $partial_i$. I would like to find a reference for the following group presentation of the fundamental group and the first homology group:
$pi_1(Sigma_g,n,x)
=langle alpha_1,ldots,alpha_g,beta_1,ldots,beta_g, partial_1,ldots, partial_n rangle Big/ Big(prod_i=1^g alpha_ibeta_ialpha_i^-1beta_i^-1prod_j=1^n partial_jBig) \
H_1(Sigma_g,n,mathbbZ)
=pi_1(Sigma_g,n,x)/[pi_1(Sigma_g,n,x),pi_1(Sigma_g,n,x)]
=mathbbZ^2g+nBig/ Big(prod_j=1^n partial_jBig)$
Does anyone know a reference? It would be nice if the reference would also include a picture with the $alpha_i$ and $beta_i$.
If $n=0$ the result can for example be found in Hatchers algebraic topology, page 51. Of course, if we know the first fact, the second one is quite clear.
algebraic-topology surfaces fundamental-groups group-presentation
asked Jul 17 at 12:29
mathstackuser
61011
The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $Sigma_g,n$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $partial_i$. I would like to find a reference for the following group presentation of the fundamental group and the first homology group:
$pi_1(Sigma_g,n,x)
=langle alpha_1,ldots,alpha_g,beta_1,ldots,beta_g, partial_1,ldots, partial_n rangle Big/ Big(prod_i=1^g alpha_ibeta_ialpha_i^-1beta_i^-1prod_j=1^n partial_jBig) \
H_1(Sigma_g,n,mathbbZ)
=pi_1(Sigma_g,n,x)/[pi_1(Sigma_g,n,x),pi_1(Sigma_g,n,x)]
=mathbbZ^2g+nBig/ Big(prod_j=1^n partial_jBig)$
Does anyone know a reference? It would be nice if the reference would also include a picture with the $alpha_i$ and $beta_i$.
If $n=0$ the result can for example be found in Hatchers algebraic topology, page 51. Of course, if we know the first fact, the second one is quite clear.
algebraic-topology surfaces fundamental-groups group-presentation
asked Jul 17 at 12:29
mathstackuser
61011
Let $Sigma_g,n$ be the compact, oriented surface of genus $g$ with $n$ disjoint open discs removed, where the boundary circles are called $partial_i$. I would like to find a reference for the following group presentation of the fundamental group and the first homology group:
$pi_1(Sigma_g,n,x)
=langle alpha_1,ldots,alpha_g,beta_1,ldots,beta_g, partial_1,ldots, partial_n rangle Big/ Big(prod_i=1^g alpha_ibeta_ialpha_i^-1beta_i^-1prod_j=1^n partial_jBig) \
H_1(Sigma_g,n,mathbbZ)
=pi_1(Sigma_g,n,x)/[pi_1(Sigma_g,n,x),pi_1(Sigma_g,n,x)]
=mathbbZ^2g+nBig/ Big(prod_j=1^n partial_jBig)$
Does anyone know a reference? It would be nice if the reference would also include a picture with the $alpha_i$ and $beta_i$.
If $n=0$ the result can for example be found in Hatchers algebraic topology, page 51. Of course, if we know the first fact, the second one is quite clear.
algebraic-topology surfaces fundamental-groups group-presentation
asked Jul 17 at 12:29
mathstackuser
61011
edited Jul 24 at 8:57
asked Jul 17 at 12:29
mathstackuser
61011
asked Jul 17 at 12:29
mathstackuser
61011
asked Jul 17 at 12:29
mathstackuser
61011
61011
The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11
add a comment |Â
The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11
The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11
add a comment |Â
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The fundamental group of a surface with some positive number of punctures is free, on $2g+n-1$ punctures. (It deformation retracts onto a wedge of circles. Then you're just trying to identify how to write one boundary component in terms of the existing generators.
– Mike Miller
Jul 17 at 18:59
Do you know a reference for that? And is the claim true?
– mathstackuser
Jul 24 at 8:58
No, but here is a sketch: present $Sigma_g$ as the quotient of a $4g$-gon by identifying sides, and then delete some points in the interior. If you delete one point, you deformation retract onto the boundary of the fundamental polygon (which you may identify as a wedge of $2g$ circles). Deleting more points, you have to add an extra arc. I suggest drawing a picture.
– Mike Miller
Jul 24 at 9:10
(Your claim looks true to me but I didn't look at the notation extremely carefully.)
– Mike Miller
Jul 24 at 9:11