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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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 if there are analogous definitions for…
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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, $\mathcal{A}^{\circ}(α)$, contains no other critical point…
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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 $$ \tilde H_0(N^4,\mathbb{Z})=0, $$ $$ H_1(N^4,\mathbb Z)=\mathbb Z^2, $$ $$ H_2(N^4,\mathbb…
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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 point sets? In the sense that after surgery we…
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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 and the $V^{d_2}$ is the base, so the total space is…
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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?
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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: Any closed 3-manifold is a boundary of some…
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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 situation: Background: Define a cobordism between two…
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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 $S^1\times S^2$. This is somehow explained on Wikipedia,…
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(Un-)oriented manifold with (un-)oriented interfaces

Are there examples that An un-oriented manifold is glued from pieces of oriented manifolds [with boundaries], separated by interfaces [where boundaries are glued]? I suppose a Mobius strip is one example, but do we have any concrete 4-dimensional…
wonderich
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Fundamental Group of a Manifold with a genus G Heegaard Splitting

Struggling to answer this question from Lickorish's introduction to Knot Theory. If M has a 3-manifold with a genus g Heegaard splitting, then the fundamental group of M has a presentation with g generators and g relators I know that since the…
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Surgery on links

Suppose $K^m \subset Y^n$ is a link of spheres $S^m_1, \cdots, S^m_i$ with trivialized normal bundle in a smooth manifold $Y^n$. I can do surgery on $Y^n$ using $K^m$. Is it true that the resulting manifold $Y'$ is independent of the order in which…
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Crossing change by Dehn surgery versus by projection

I read the following proposition in "Crossing changes" by Martin Scharlemann. A crossing change for a knot $K:S^1\to S^3$ with crossing disk $D\subset S^3$ can be obtained by performing Dehn surgery along $\partial D$ with slope $\pm 1$. The…
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Surgery presentation for an abstract open book decomposition

Suppose $(\Sigma,\phi)$ is an abstract open book whose monodromy is expressed as a product of Dehn twists about boundary-parallel curves. Is there a standard way to produce a surgery presentation of the resulting 3-manifold? (I have such a procedure…
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Fundamental group of result of 0-Dehn surgery and meridian

Let $M_K$ be the result of $0$-Dehn surgery along a knot $K$ in $S^3$. Let $m$ be a meridian to $K$, and we view $m$ as a circle in $M_K$ (without changing the notation for it). The claim I wanted to prove is: Then the fundamental group $\pi_1(M_K)$…