# Questions tagged [handle-decomposition]

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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 $U\subset M$ ...
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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_{...
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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{...
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 ...
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: $... 10 votes 1 answer 711 views ### What are Kirby diagrams of candidate exotic 4-manifolds? It is an open problem whether there exist smooth manifolds homeomorphic, but not diffeomorphic to the standard$S^4$. The same is true for the 4-torus and several other manifolds. Handle ... 2 votes 0 answers 293 views ### Uniqueness of the Smooth Structure on a Handle Attachment [closed] I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it. Let$M^m$be a smooth manifold with boundary. We may ... 3 votes 2 answers 1k views ### Self-indexing Morse functions on non-compact manifolds Hi, given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has$f(c) = \operatorname{index}(c)$) - a proof of this may ... 5 votes 2 answers 1k views ### Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus Given a Heegaard splitting of genus$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 ... 