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.

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.

A surgery on an $m$-dimensional manifold $M$ is the procedure of constructing a new $m$-dimensional manifold

$$M^{\prime} =cl.(M\setminus S^n\times D^{m-n})\cup_{S^n\times S^{m-n-1}}D^{n+1}\times S^{m-n-1}$$

by cutting out $S^n\times D^{m-n}\subset M$ and replacing it by $D^{n+1}\times S^{m-n-1}$.

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Perform 0-framed surgery, then remove neighbourhood of meridian. Is this the knot complement?

Let $K$ be a knot in $S^3$ and let $m$ be a meridian of $K$. Let $M_K$ be the 3-manifold obtained by performing 0-framed surgery on $K$. The meridian $m$ can also be viewed as a circle in $M_K$. Is $M_K\setminus (m\times D^2)$, i.e. the 3-manifold…
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The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move (adding the trivial knots with framing $\pm 1$)…
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Remove one ring of Borromean rings in 3-sphere: linked or unlinked?

We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings. Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times S^1$ region of one of the three rings from $S^3$…
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Surgery, framing and Dehn twist

Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$. How can I interpretate the following sentence? "We identify $U$ with $S^1 \times B^2$ so that $L$ is identified $S^1 \times 0$, where $0$ is the center of…
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How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of $L$ in $S^3$, which is $S^1 \times B^2$. We attach…
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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 $x$ is a fixed point on the boundary of $B_2$) has…
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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 references for Thom's isomorphism theorem, seems to be…
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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 $1$ critical point in the interior of $M$, such…
user928824
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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 two handlebodies $H_1, H_2$ modulo a relation…
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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 a manifold $M^m$. He says Surgery on a…
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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 $S^2\subset \mathbb{R}^3$ bounds a smooth 3-ball. The proof seems to…
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$\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$

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 directly? It is a figure 2 in this ref:…
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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, even in the beggining chapters of heegard…
Nick A.
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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 the case when $M$ has no holes, but beyond this I…
Doc
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When is a band connected sum equal to a connected sum of knots?

In $S^3$ we have a well defined connected sum operation of knots, $K_1 \# K_2$ and a band connected sum $K_1 \#_b K_2$, which is not well defined and depends on the position of the band $b$. What are the conditions on $b$, so that the band connected…