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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 ...
fromscratch's user avatar
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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 ...
The_Rookie's user avatar
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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 ...
zeta space's user avatar
2 votes
1 answer
253 views

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 ...
Hamilijaona's user avatar
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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 $x$ ...
Andi Bauer's user avatar
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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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Homeomorphism type of Dehn filling depends only on the isotopy class of meridian

I am reading Dehn filling which is defined as follows in this lecture note(page 20): Let $M$ be a $3$-manifold and $T\subseteq \partial M$ be an embedded torus. For a homeomorphism $\varphi\colon \...
Random's user avatar
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Connected sum of one manifold WITH boundary with a manifold WITHOUT boundary

For manifolds with boundary, there are two different types of "connected sums". On the one hand, there is the notion of "boundary connected sum", where one takes two manifolds with ...
B.Hueber's user avatar
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Fundamental group of Homeo($\mathbb{R}^n$)

My question is easy to formulate: What is known about the homotopy groups of Homeo($\mathbb{R}^n$)? Especially, what is its fundamental group? (A guess would be $\mathbb{Z}$ for $n=2$ and $\mathbb{Z}/...
Christian's user avatar
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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 ...
horned-sphere's user avatar
2 votes
1 answer
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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 ...
Prerak Deep's user avatar
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For a knot $K\subset S^3$, the inclusion $\partial \nu K\to (S^3-\text{int}(\nu K))$ induces a surjection on $\pi_1$

Let $K$ be a knot (embedded circle) in $S^3$ and let $M$ be obtained from $S^3$ by $0$-surgery on $K$. A meridian of $K$ in $S^3$ can be viewed as a circle $C$ in $M$. Consider the product manifold $X=...
blancket's user avatar
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Knot Complement of Knot Sum and Fibered Knot Sum

Let $K_1,K_2$ be two knots and $X_1, X_2$ be their knot complements. Let $K_3$ be sum of two knots and $X_3$ the resp. complement. I am tempted to thinking that there is some relationship between $X_i$...
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2 votes
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Elementary Morse Cobordism of Diffeomorphic Boundary Components

Let $(M,V,V')$ be a smooth manifold triads. I would like to find a Morse cobordism which is elementary, i.e. there exists Morse function $f:M\to[0,1]$ such that $f^{-1}(0)=V, f^{-1}(1)=V'$ and of only ...
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1 vote
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Making precise Dehn filling

Dehn surgery along a knot is a well-known construction: choose a regular neighbourhood $N(K)$ of a knot $K \subset S^3$, let $X_K := S^3 - N(K)$ and choose an essential simple closed curve $\alpha$ on ...
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If $\phi : \partial{M} \times [0,1) \to M $ is an embedding prove that $\phi(\partial{M} \times [0,a])$ is closed in $M$.

Let $M$ be a compact manifold and let $\phi:\partial{M}\times [0,1) \to M $ be an embedding whose image is open in $M$. How to prove that $ \phi(\partial{M} \times [0,a]) $ is closed in $M$ for any $a&...
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Dehn surgery for non-compact manifolds

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}^...
Alonso Perez-Lona's user avatar
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238 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 ...
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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 ...
TheWildCat's user avatar
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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 ...
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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? -...
sarah's user avatar
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2 votes
1 answer
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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 ...
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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 ...
Zest's user avatar
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2 votes
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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 \...
Hempelicious's user avatar
1 vote
1 answer
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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 ...
user avatar
1 vote
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Exercice Quasiconformal Surgery (4.2.3)

I was tring to do this exercise from Branner and Fagella book on Quasiconformal Surgery. Suppose $P$ is a polynomial with a superattracting fixed point, say $\alpha$, whose immediate basin, $\mathcal{...
Monkeydsuka's user avatar
3 votes
1 answer
142 views

Determining surfaces by self-gluing after removing interiors of two disks.

Given a surface $M$, after removing the interiors of two discs on the surface and then self-gluing along the boundary you obtain two surfaces: $M_+$ which is when orientation is preserved, and $M_-$ ...
LunarGyrial 's user avatar
7 votes
3 answers
766 views

Obtaining the three torus via Dehn surgery

It is a well known theorem from the '60 (Lickorish-Wallace) that any closed orientable three dimensional smooth manifold can be obtained performing a sequence of integral Dehn surgeries along knots in ...
Overflowian's user avatar
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Reduced homology group $H_k(S^4 - N^4, \mathbb Z)=H^{4-k}(S^4 - N^4,\mathbb Z)=H_{k-1}(N^4, \mathbb Z)=\mathbb Z^2$?

Let $N^4$ be a 4-dimensional $D^2 \times T^2 = D^2 \times S^1 \times S^1$. (let us denote $\tilde H$ as the reduced homology or homology group) I know that $$ \tilde H_0(N^4,\mathbb{Z})=0, $$ $$ H_1(...
annie marie cœur's user avatar
4 votes
1 answer
456 views

Kirby calculus on E8 plumbing

I was trying to show that the 4-manifold described in Kirby diagram as a E8-plumbing (see the diagram below) has the same boundary as the 2-handlebody on the left-handed trefoil with surgery ...
cjackal's user avatar
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Handle Decomposions for $2$ Manifolds

I have a question about the notation for "$n$-handles" with respect to a decomposion of handlebodies. At german wikipedia page https://de.wikipedia.org/wiki/Henkel-Zerlegung I found a statement ...
user267839's user avatar
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1 vote
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Killing homology below middle dimension with equivariant surgery

Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$. Is there a way to kill $a$ with equivariant surgery and keep the same fixed ...
piotrmizerka's user avatar
6 votes
1 answer
353 views

Connected Sum Surgery

Is there any relationship between the connected sum operation and surgery theory? Is it possible to use surgery theory to "sew" two manifolds together and if so how is doing it by that approach ...
Canonical Momenta's user avatar
2 votes
1 answer
282 views

semi-direct product between manifolds

question 1: Are there mathematical definition of the semi-direct product between manifolds $$ M^{d_1} \rtimes V^{d_2}? $$ For example, is it defined as a fibration such that $M^{d_1}$ is the fiber ...
annie marie cœur's user avatar
2 votes
0 answers
73 views

$\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ [closed]

I was stuck by reading this figure: It looks that $\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ are somehow related. Are there some easier explanations from math ...
annie marie cœur's user avatar
5 votes
0 answers
140 views

$\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds

I am looking for some explanation how $\mathbb{HP}^2$, exotic 7-spheres, and Bott manifolds are related? And how the construction of a Bott manifold is related to $\mathbb{HP}^2$ and exotic 7-...
wonderich's user avatar
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Reference for Dehn Surgery

I'm looking for an introductory reference for the basics on Dehn surgery on links. Does anybody have any recommendations?
Kai Nakamura's user avatar
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Any closed 3-manifold is a boundary of some compact 4-manifold.

I knew this is true: Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$. See this post: https://mathoverflow.net/q/63373/27004/ I heard this statement is true: ...
wonderich's user avatar
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1 vote
1 answer
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Limitive result in constructing cobordisms for 3-manifolds

I'm just disovering cobordism theory and piecing together the subject from various resources, and the concept of explicitly constructing cobordisms between 3-manifolds is confusing me. Here's my ...
Doc's user avatar
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3 votes
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Comprehensive textbook in surgery theory

I am currently reading Prasolov & Sossinski: Knots, Links, Braids and 3-Manifolds but I have a hard time understanding some of the more intuitive argument on the chapters of surgery in 3 manifolds,...
Nick A.'s user avatar
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2 votes
1 answer
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$M \# M \cong M$ in the noncompact case

I recently saw this question and its generalisation, and it made me wonder about the non-compact case: is there ever a case when $M \# M \cong M$ for $M$ non-compact? Clearly this would only ever be ...
Doc's user avatar
  • 321
1 vote
1 answer
151 views

Surgery on $S^3$

Assume the embedding of $S^0$ in $S^3$ extends to an (orientation preserving) embedding of $S^0 \times D^3$ in $S^3$. Show that the manifold which is the result of surgery is diffeomorphic to $S^1\...
user557's user avatar
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4 votes
1 answer
286 views

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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