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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Homology group and homotopy group of the standard twin

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$. What are the homology group $$H_n(M)$$ and homotopy group…
miss-tery
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Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times D^{m-n})=S^n \times D^{m-n} \cup D^{n+1}\times…
Sepideh Bakhoda
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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 definitions: Let $\mathcal{M}$ be a (sufficiently nice)…
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Redundacy in gluing map of a Heegaard Splitting

I am reading about the handlebody group which is the subgroup of all those mapping classes that extend to the handlebody from here. On page 12 the author defines an equivalence relation on mapping classes as follows: $\psi$ and $\phi$ are said to be…
HARSH
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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 $\varphi\colon \partial(\Bbb S^1\times \Bbb D^2)\to T$ define the…
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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 $X=S^1\times M$. $X$ contains an embedded torus…
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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 $a>0$. I was studying adjunction spaces, and, I…
Uncool
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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 $\mathbb{R}^2 \times S^1$ from surgery on $S^3$,…
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Oriented but not Reversible manifolds?

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 orientable manifold should admit one such…
TheWildCat
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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 that every closed orientable $2$-manifold has…
KarlPeter
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Regarding Surgery and Orientation

Is this true for any rational homology 3-sphere Y (or any 3-manifold where $$K \subset Y$$ is null-homologous)? $$Y_{\frac{p}{q}} (K) = (-Y)_{- \frac{p}{q}} (m(K))$$ where m(K) is the mirror of K in -Y. Any reference or a sketch of proof will be…
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Calculating intersection numbers of surfaces in example from knot surgy

There is a section in an explanation about knot surgery which I do not understand in "Knot surgery revisited" by Fintushel, p.203. Let $X$ be a simply-connected compact $4$-manifold. Let $K$ be a knot in $S^3$. Let $M_K$ be the $3$-manifold obtained…
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