Questions tagged [handle-decomposition]
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36 questions
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On the Morse handle decomposition of $T^3$ and the attaching maps of the $2$-handles
Consider the motion space of a linkage of 3 rods of equal length = 1 with the first vertex anchored to the origin (see attached image). Take the standard negative height function
$$f(\theta_i)=-\sum_{...
- 73
9
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1
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548
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Null-homologous Gluck twist when $\pi_1\neq 0$
Let $M$ be a closed (smooth) 4-manifold and let $S\subseteq M$ be an embedded 2-sphere with trivial normal bundle. The Gluck twist $M_S$ along this sphere has the following properties relating it to $...
- 1,377
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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 \...
- 1,224
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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....
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$\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 ...
- 3,105
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$\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 ...
user519738
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Is there a canonical "minimal" Morse decomposition?
Classical Morse theory gives a handle decomposition of a finite dimensional manifold $M$ for every choice of Morse function $f: M \to \mathbb R$. Is there always a Morse function that induces a "...
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A sufficient condition for a collection of open sets of a manifold to contain all open sets
Question
Let $k\geq 0$ be an integer and let $M$ be a topological $n$-manifold. Let $\mathcal{U}$ be a set of open sets of $M$ which satisfies the following closure properties:
(1). Let $U\subset M$ ...
- 2,556
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Betti numbers of non-orientable $3$-manifolds
Let $M^3$ be a compact $3$-manifold with boundary $\partial M$.
If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...
- 303
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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. ...
- 1,224
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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 ...
- 191
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Multi-connected sum decomposition of $n$-manifolds
A connected sum decomposition of a closed $n$-manifold $M^n$:
$M^n = M_1^n \# M_2^n$, is to view $M^n$ as two closed $M_1^n$ and $M_2^n$, joined
by a neck $I\times S^{n-1}$.
Similarly, a $k$-connected ...
- 4,874
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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 ...
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Attaching a 2-handle to a once-twisted unlink in the boundary of the 4-ball
Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...
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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 ...
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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 ...
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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 ...
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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 ...
- 1,358
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450
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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$?
- 1,495
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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....
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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=\...
- 3,105
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136
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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 ...
- 5,685
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317
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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 (...
- 5,685
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369
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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 ...
- 1,495
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2
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818
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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,\...
- 5,685
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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_{...
- 44.8k
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
- 614
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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:
$L ...
- 2,343
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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 ...
- 5,685
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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 ...
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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 ...
- 3,128
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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 ...
user6204