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# Questions tagged [surgery-theory]

Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 4.

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### minimal cuts and pastes to make a t-shirt?

(I have not found an exercise about this in do Carmo or Struik, and I think it could go a long way towards building intuition about curvature and develop skills in computing special 2D areas, as well ...
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### Understanding the proof of Fenn-Rourke Theorem

Fenn-Rourke Theorem states that Framed links can be transformed into each other by Kirby moves if and only if they can be done by Fenn- Rourke moves. I'm trying to understand the proof of it in V.V....
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### Why $\pm 1$ surgery on homology sphere again yeilds homology sphere?

Let $L$ be a framed link in an integral homology 3-sphere $M$. I read in this paper that if $L$ is algebraically split (pairwise linking number is zero) and unit-framed (framing $\pm 1$), then by ...
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### Realizing homotopy classes as thickened spheres

Suppose that we are given a smooth, closed, connected homology sphere $M^n \subset \mathbb{R}^{n+1}$ with $n \geq 6$. I want to kill some elements in the fundamental group of $M$ by surgeries. As far ...
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### References needed for Dehn surgery and Kirby calculus

I learned from Colin Adams's book, $\textit{the knot book}$, that every compact connected three manifold comes from Dehn surgery on a link in $S^3$, and if two different Dehn surgery yield the same ...
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### Dehn surgery on pseudomanifold to make bonafide manifold

Consider four intersecting open cylinders arranged in the unit cube where the caps of the cylinders are of arbitrariy small radius and 'look' globally as if they coincide with the vertices of the unit ...
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### Orientation of Dehn surgery manifold

Suppose we have a 3-manifold $M$ obtained by Dehn surgery along a given framed link on $S^3$. Then it has a natural orientation which comes from the standard orientation of $S^3$. Is it true that $-M$ ...
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### Surgery on smooth four-manifold preserves the intersection form

In these notes (p. 190), it is claimed that (possibly with superfluous hypotheses): Claim. Let $M$ be a closed connected oriented smooth 4-manifold and let $c\colon S^1\hookrightarrow M$ be an ...
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### Notation in surgery theory (maths)

sary Hello everyone! In reading an article on surgery theory I found an expression of the type $M \cup_{\phi} N$ where $M$ and $N$ are manifolds(or more generally topological spaces), and $\phi$ some ...
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### Showing that a 4-manifold obtained by attaching a 2-handle is simply-connected

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation: Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
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### Outcome of a concrete surgery operation in 3D

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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### Inverse operation of Dehn surgery

Suppose we have two closed oriented 3-manifolds $M$ and $N$. Suppose $N$ is obtained by a Dehn surgery operation on a knot $K$ in $M$, so $N=(M-\operatorname{int}\nu K)\cup_\partial (S^1\times D^2)$, ...
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### Manifolds with Compressible Boundary

I recently stumbled over the following terminology, but since I am not really familiar with geometric topology I having a hard time to understand it correctly. So, lets start with the following ...
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### Apparent contradiction with gluing 2-handle to a 4-manifold all being isotopic.

In Hirsch's Differential Topology Chapter 8 Theorem 2.3, it says : Let $f, g:\partial Q\approx \partial P$ be isotopic diffeomorphisms. Then $P\cup_f Q\approx P\cup_g Q$. Here $\approx$ means ...
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### Milnor's Lectures on h-cobordism theorem: Lemma 6.2

In the book, Lemma 6.2 (stated below) talks about a corollary of the Thom's isomorphism theorem and Tubular neighbourhood theorem. The proof of the lemma is not provided by the author. And the ...
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### Do higher-dimensional knots have interesting knot polynomials?

Knot polynomials are one of the most common tools that allow us to distinguish between two given knots. And when I say 'knots' I mean one-dimensional knots embedded within $S^3$. I would like to know ...
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### Why would surgery theory require $5$ dimensions?

In the Wikipedia page for geometric topology it says "The Whitney trick requires $2+1$ dimensions, hence surgery theory requires $5$ dimensions". I am having trouble with understanding why surgical ...
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