# Questions tagged [handle-decomposition]

The handle-decomposition tag has no usage guidance.

22
questions

3
votes

1
answer

52
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### Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest .
Consider a ...

6
votes

0
answers

227
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### Questions about a paper by Laudenbach and Poénaru

I am working on the 1972 paper A Note on 4-Dimensional Handlebodies by F. Laudenbach and V. Poénaru, and I had two questions. I will use their notations to simplify things, since the paper is very ...

6
votes

1
answer

220
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### Dehn surgery along primitive knot in 3-dimensional handlebody

I'm studying the article "An alternative proof of Lickorish–Wallace theorem" (doi link)
and I got stuck in a problem.
Let $H_g$ be a 3 dimensional handlebody of genus $g$, a primate curve in ...

3
votes

1
answer

136
views

### Connected sum of algebraic curves, handlebody decomposition, and induction on genus

Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...

4
votes

2
answers

225
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### The handlebody decomposition of S^1 bundles over surfaces?

What is the most natural handlebody decomposition of $F_g \times S^1$, if $F_g$ is an orientable closed surface of genus $g$?

4
votes

1
answer

154
views

### Complement of Donaldson divisors in dimension 4

Let $(X,\omega)$ be a symplectic 4-manifold such that $\omega$ has a rational cohomology class. I am interested in Donaldson divisors (surfaces) $D$ in $(X,\omega)$ whose complement is a 1-handle body....

9
votes

3
answers

356
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### Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...

3
votes

0
answers

95
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### What's a completely computational/syntactical model for handle decompositions of manifolds?

Simplicial sets, CW complexes
Simplicial sets can be described completely algebraically, by specifying a family of sets, and maps between them satisfying certain relations. This description can be ...

4
votes

0
answers

216
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### Cap product for (co)homology from handle decompositions/Kirby diagrams

Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (...

2
votes

1
answer

233
views

### Different Heegaard splittings of a 3-manifold

I want to study same 3-manifolds with different Heegaard splitings.
Of course one has stabilization, but even with the same genus, we have different Heegaard splittings.
If we encode a 3-manifolds by ...

4
votes

2
answers

558
views

### Are there Kirby diagrams with 3-handles?

Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...

7
votes

2
answers

350
views

### Handle decompositions using only 1-handles

Let $\Sigma$ be an oriented, compact, connected 2-manifold with boundary. Assume that its boundary is equipped with a disjoint union decomposition into two non-empty parts:
$$\partial\Sigma=\partial_{...

7
votes

2
answers

496
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### Generalizations of the handle trading techniques

As Theorem 8.1 in "Lectures on the h-cobordism theorem (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no ...

6
votes

0
answers

384
views

### Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...

6
votes

1
answer

206
views

### Does there always exist a sequence of handle moves between handle decompositions that does not increase index? (+ ref. request)

Reference request: Firstly, I'm looking for a proof of the following well-known result about handle decompositions:
($\ast$) Given two handle decompositions of a smooth $n$-manifold $M$, there ...

3
votes

0
answers

179
views

### Is complex surface in CP(3) a two handlebody?

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take
\begin{equation}
S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\}
\end{...

15
votes

2
answers

1k
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### Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in ...

3
votes

2
answers

320
views

### Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that:
$...

10
votes

1
answer

636
views

### What are Kirby diagrams of candidate exotic 4-manifolds?

It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard $S^4$. The same is true for the 4-torus and several other manifolds. Handle ...

2
votes

0
answers

278
views

### Uniqueness of the Smooth Structure on a Handle Attachment [closed]

I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it.
Let $M^m$ be a smooth manifold with boundary. We may ...

3
votes

2
answers

1k
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### Self-indexing Morse functions on non-compact manifolds

Hi,
given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has $f(c) = \operatorname{index}(c)$) - a proof of this may ...

5
votes

2
answers

1k
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### Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus

Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or ...