All Questions
Tagged with gt.geometric-topology handle-decomposition
20 questions
12
votes
1
answer
557
views
+200
Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary smooth, closed, connected, n-dimensional manifold for $n\geq 4$. For every such $M$, does there exist a closed (not necessarily connected!) codimension two submanifold $S \...
- 1,174
4
votes
1
answer
154
views
Kirby diagram of the complement of a subhandlebody of a smooth closed 4-manifold
Let $X$ be a smooth closed connected 4-manifold. It admits a handlebody structure, having a unique 0- and a unique 4-handle. We can express the handlebody structure as a Kirby diagram (https://en....
- 213
5
votes
2
answers
222
views
$\mathbb{CP}(2)$ from gluing boundary of 4-ball
Many manifolds can be obtained from gluing the boundary of a ball. For example, $\mathbb{RP}(2)$ is obtained from gluing the two edges of a bi-gon (2-ball). Or, lens spaces are obtained from a 3-cell ...
- 3,001
4
votes
1
answer
88
views
$\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducible 3-manifold produces a reducible 3-manifold
This question arises in my previous question.
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $\alpha\subseteq \partial M$ be a simple closed curve, which is ...
- 605
10
votes
3
answers
684
views
Doubles of 2-handlebodies
Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...
- 1,174
3
votes
1
answer
255
views
Handle attachment information from Morse function and triangulation
First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$.
For simplicity, let's restrict for now to the ...
- 159
1
vote
1
answer
181
views
An example of handle decomposition on modified $S^5$
I would like to give the following object, $M=S^5 \setminus \sqcup_{2 \text{ copies}} \text{int}(S^1\times D^4)$, a handle decomposition. It is then to be attached to another manifold. along the two ...
3
votes
1
answer
76
views
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 ...
- 173
6
votes
0
answers
391
views
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 ...
- 283
6
votes
1
answer
287
views
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 ...
- 173
4
votes
2
answers
313
views
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 $g$?
- 1,465
4
votes
1
answer
170
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....
2
votes
1
answer
287
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 ...
- 1,465
4
votes
2
answers
715
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,\...
- 5,565
7
votes
2
answers
412
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_{...
- 43.2k
7
votes
2
answers
604
views
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
1
answer
248
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 ...
- 365
3
votes
2
answers
360
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:
$...
- 2,283
2
votes
0
answers
327
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 ...
6
votes
2
answers
1k
views
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 ...
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