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 $D^{n+1}\times S^{m-n-1}$.

87 questions
4
votes
0 answers

Dehn surgery for infinite cyclic cover

An infinite cyclic cover for the complement of a knot $K$ can be constructed by performing a suitable Dehn surgery that unknots $K$. For example, in the following picture we take $6_1$ and use Dehn surgery to eliminate the twist. The result is…
4
votes
0 answers

On Kervaire and Milnor's paper, "Groups of homotopy spheres: 1" (surgery on 3-manifolds)

Kervaire and Milnor's paper "Groups of homotopy spheres: 1" in Annals, is the condition "$k>1$" necessary in Lemma 6.3? If yes, why? (I couldn't find any part requiring the condition in their progress to obtain the lemma.) Lemma 6.3 is the…
4
votes
1 answer

Connected sum of submanifolds

Given two disjoint oriented knots $K_1, K_2 \subset S^3$, I think that there is a notion of knot connected sum $K_1 \sharp K_2 \subset S^3$ defined by picking a path between $K_1, K_2$. Is the connected sum independent of the path and does isotopy…
user39598
  • 1,418
  • 8
  • 20
4
votes
0 answers

Effect of surgery on $S^1 \times S^3$

Suppose we take some $p \in S^3$ and consider a surgery on $S^1 \times S^3$ removing a tubular neighborhood of $S^1 \times p$ and gluing in a $D^2 \times S^2$. What manifold do we get? I've been trying to visualize this, not to much avail.
3
votes
0 answers

How to get a Kirby diagram of $S^1 \times M^3$ if $M^3$ is given by a surgery diagram?

In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of $S^1 \times M^3$, given a Kirby diagram of $M^3$. Basically, one thickens the diagram, adds one 1-handle and connects it to the…
3
votes
1 answer

Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. First, consider a Heegaard decomposition of a…
3
votes
2 answers

the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of $\pi/6$, $\pi/6+2\pi/3$, $\pi/6+4\pi/3$ and then gluing…
3
votes
1 answer

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take unknots with integer framing. I am confused whether if…
3
votes
1 answer

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 boundary $\mathcal{M}$ and $\mathcal{N}$ and cuts out two open balls…
B.Hueber
  • 1,542
  • 4
  • 14
3
votes
1 answer

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 $M_-$ when orientation is reversed. the problem is to…
3
votes
2 answers

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 $\mathbb{S}^3$. The most common examples found…
3
votes
0 answers

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 coefficient -1. (which is a standard exercise in geometric…
cjackal
  • 2,113
  • 11
  • 14
3
votes
1 answer

Inverse of spherical fibration

I am reading Browder's 'Surgery on simply connected manifolds'. There is a discussion of spherical fibrations in I.4.5 (P.21) that goes roughly as follows: Fix a connected space $X$. A fibration $E \overset{\pi}{\to} X$ is called a spherical…
3
votes
0 answers

surgery of type $(\lambda,n-\lambda)$ on manifold, h-cobordism theorem by Milnor

I am reading Milnor's Lectures on h-cobordism theorem, and I am stuck on Milnor's definition on surgery of type $(\lambda,n-\lambda)$ on manifold, where the definition following can be found on page 31 of the book. Milnor uses this to describe the…
AG learner
  • 3,897
  • 2
  • 12
  • 30
3
votes
1 answer

To define Dehn surgery, should one allow orientation reversing diffeomorphisms or arbitrary ones?

So for the definition of Dehn surgery (also called rational/integer surgery), what is correct: Definition Let $K$ be a knot in an oriented $3$-manifold with a regular neighbourhood $N(K)\simeq S^1\times D^2$. Dehn surgery is the operation of…
w_w
  • 679
  • 3
  • 11