Explicit determination of the open book in $S^3$
I'm familiar with the fundamental concepts of algebraic and differential topology.
How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: mathbb{C}^2 to mathbb{C}, f(z_1, z_2) = z_1z_2.$$
Any reference request or suggestion will be appreciated.
reference-request algebraic-topology differential-topology low-dimensional-topology
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
add a comment |
I'm familiar with the fundamental concepts of algebraic and differential topology.
How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: mathbb{C}^2 to mathbb{C}, f(z_1, z_2) = z_1z_2.$$
Any reference request or suggestion will be appreciated.
reference-request algebraic-topology differential-topology low-dimensional-topology
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
2
The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48
add a comment |
I'm familiar with the fundamental concepts of algebraic and differential topology.
How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: mathbb{C}^2 to mathbb{C}, f(z_1, z_2) = z_1z_2.$$
Any reference request or suggestion will be appreciated.
reference-request algebraic-topology differential-topology low-dimensional-topology
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
I'm familiar with the fundamental concepts of algebraic and differential topology.
How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: mathbb{C}^2 to mathbb{C}, f(z_1, z_2) = z_1z_2.$$
Any reference request or suggestion will be appreciated.
reference-request algebraic-topology differential-topology low-dimensional-topology
reference-request algebraic-topology differential-topology low-dimensional-topology
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
edited Dec 11 '18 at 21:01
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
asked Dec 11 '18 at 20:53
M. Giovanni Lucaretti
797
797
2
The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48
add a comment |
2
The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48
2
2
The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48
add a comment |
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The binding of the open book is a link of $z_1z_2 = 0$ near the origin, which is a Hopf link in $S^3$. I think the fibers $f^{-1}(theta)$ for $theta in S^1 subset Bbb C$ are Seifert surfaces of this link: when one goes from $z_1 z_2 = 0$ to $z_1 z_2 = epsilon$, one resolves the singularity at the origin (which entails two transverse disks intersecting at a point) by making it a Seifert surface of the boundary of the two disks, which forms a Hopf link.
– Balarka Sen
Dec 11 '18 at 21:59
If I'm not wrong, you can get the description in the paper of John etnyre.
– Anubhav Mukherjee
Dec 12 '18 at 0:48