Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
fixed grammar
Source Link
user150450
user150450

Heegaard splittingsplittings of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the genus $g$ surface $\Sigma_g$.

It turns out that any closed orientable $3$-manifold $Y$ can be obtained by gluing together two handlebodies $H_1$ and $H_2$ (such a decomposition is called Heegaard splitting):

  • $Y= H_1 \cup H_2$,
  • $\partial H_1 = \partial H_2 = \Sigma_g$.

The basic examples of such $3$-manifolds are

  • $S^3$,
  • $S^1 \times S^2$,
  • $S^1 \times S^1 \times S^1$,
  • Lens spaces $L(p,q) = S^3 / \mathbb Z_p$,
  • Brieskorn spheres $\Sigma(p,q,r) = \{ x^p + y^q +z^r = 0 \} \cap S^5 \subset \mathbb C^3$.

There are many references for Heegaard splittings forof the first four of these examples, for example chapter 1 of Saveliev's book: Lectures on the Topology of 3-Manifolds.

How about the Brieskorn spheres? Is there an easy way to think about their Heegaard splittings? How can we draw them? Is there any good reference for this?

Heegaard splitting of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the genus $g$ surface $\Sigma_g$.

It turns out that any closed orientable $3$-manifold $Y$ can be obtained by gluing together two handlebodies $H_1$ and $H_2$ (such a decomposition is called Heegaard splitting):

  • $Y= H_1 \cup H_2$,
  • $\partial H_1 = \partial H_2 = \Sigma_g$.

The basic examples of such $3$-manifolds are

  • $S^3$,
  • $S^1 \times S^2$,
  • $S^1 \times S^1 \times S^1$,
  • Lens spaces $L(p,q) = S^3 / \mathbb Z_p$,
  • Brieskorn spheres $\Sigma(p,q,r) = \{ x^p + y^q +z^r = 0 \} \cap S^5 \subset \mathbb C^3$.

There are many references Heegaard splittings for the first four of these examples, for example chapter 1 of Saveliev's book: Lectures on the Topology of 3-Manifolds.

How about the Brieskorn spheres? Is there an easy way to think about their Heegaard splittings? How can we draw them? Is there any good reference for this?

Heegaard splittings of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the genus $g$ surface $\Sigma_g$.

It turns out that any closed orientable $3$-manifold $Y$ can be obtained by gluing together two handlebodies $H_1$ and $H_2$ (such a decomposition is called Heegaard splitting):

  • $Y= H_1 \cup H_2$,
  • $\partial H_1 = \partial H_2 = \Sigma_g$.

The basic examples of such $3$-manifolds are

  • $S^3$,
  • $S^1 \times S^2$,
  • $S^1 \times S^1 \times S^1$,
  • Lens spaces $L(p,q) = S^3 / \mathbb Z_p$,
  • Brieskorn spheres $\Sigma(p,q,r) = \{ x^p + y^q +z^r = 0 \} \cap S^5 \subset \mathbb C^3$.

There are many references for Heegaard splittings of the first four of these examples, for example chapter 1 of Saveliev's book: Lectures on the Topology of 3-Manifolds.

How about the Brieskorn spheres? Is there an easy way to think about their Heegaard splittings? How can we draw them? Is there any good reference for this?

Source Link
user150450
user150450

Heegaard splitting of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the genus $g$ surface $\Sigma_g$.

It turns out that any closed orientable $3$-manifold $Y$ can be obtained by gluing together two handlebodies $H_1$ and $H_2$ (such a decomposition is called Heegaard splitting):

  • $Y= H_1 \cup H_2$,
  • $\partial H_1 = \partial H_2 = \Sigma_g$.

The basic examples of such $3$-manifolds are

  • $S^3$,
  • $S^1 \times S^2$,
  • $S^1 \times S^1 \times S^1$,
  • Lens spaces $L(p,q) = S^3 / \mathbb Z_p$,
  • Brieskorn spheres $\Sigma(p,q,r) = \{ x^p + y^q +z^r = 0 \} \cap S^5 \subset \mathbb C^3$.

There are many references Heegaard splittings for the first four of these examples, for example chapter 1 of Saveliev's book: Lectures on the Topology of 3-Manifolds.

How about the Brieskorn spheres? Is there an easy way to think about their Heegaard splittings? How can we draw them? Is there any good reference for this?