# 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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### 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 $N=(M-\operatorname{int}\nu K)\cup_\partial (S^1\times D^2)$, ...
1 vote
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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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Lickorish theorem states that every closes, orientable, compact 3-manifold can be obtained by surgery on $S^3$. What do we know about surgery for non-compact manifolds? I.e. can we obtain $\mathbb{R}^... 2 votes 0 answers 221 views ### explanations about Heegaard diagrams Heegaard diagrams are used to describe a three-dimensional manifolds, a classical way to do so is to take$M={\displaystyle M\cong (H_{1}\cup H_{2})/{\sim }}$as a topological quotient of the union of ... 0 votes 0 answers 35 views ### Oriented but not Reversible manifolds? [duplicate] In Hirsch's Differential Topology, he defines a smooth manifold$M$to be reversable if it is orientable and admits an orientation-reversing diffeomorphism. I feel confused since I believe that any ... 1 vote 1 answer 95 views ### What is the topological space obtained by cutting$M\times N$along a copy of$M$or$N$? What is the topological space obtained by cutting$M\times N$along a copy of$M$or$N$(both closed topological spaces). E.g. if we cut a torus$\Bbb S^1\times \Bbb S^1$along a circle then the ... 4 votes 0 answers 57 views ### Reference needed for Novikov paper I'm having trouble finding this construction. Novikov (1964) constructed homotopy equivalences$f:N\rightarrow S^p\times S^q$for$p,q > 1$which are not homotopic to homeomorphisms. References? -... 2 votes 1 answer 236 views ### Confused about A. Kosinski's description of Surgery in his book "Differential Manifolds" So i was trying to get my head around it, but i still haven't managed to do so. I am currently reading A. Kosinski's Differential Manifolds. On p.112 he introduces Surgery on a$(\lambda-1)$-Sphere in ... 1 vote 1 answer 135 views ### Why are embedded spheres removed in the connected sum but not in the handle attachment of (smooth) manifolds? So i am currently studying differential manifolds and morse-theory. When i came across the connected sum, i learned that we glue two manifolds$M_1$and$M_2$along the boundaries of removed disks ... 2 votes 0 answers 98 views ### Generalized Schoenflies - formalizing step in proof? I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, which is a specific case of the generalized Schoenflies theorem: Every smoothly embedded$S^2\subset \...
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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 ... 