Mathematics Stack Exchange
Mathematics Stack Exchange

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
82
votes
1 answer

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. Can someone clear the sentence to me?
Sepideh Bakhoda
  • 5,426
  • 1
  • 25
  • 43
17
votes
1 answer

Intuition for Exotic $\mathbb R^4$'s

Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like that he said it's just hard $4$-dimensional…
user153312
10
votes
0 answers

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 $\mathbb{Z}/2$ for large $n$.) It is well-known that it has…
Christian
  • 258
  • 1
  • 8
8
votes
2 answers

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in $x$-$z$ plane of $\mathbb{R}^3$. Assume each unknot…
user65175
6
votes
0 answers

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial M$ and $\partial N$ are homeomorphic. Suppose…
user65175
6
votes
1 answer

Surgery on manifold

In this article on surgery on manifolds it is explained that from an $n$-manifold $M$ an $n$-manifold $M'$ can be constructed by cutting out $S^p \times D^q$ and gluing in $D^{p+1}\times S^{q-1}$. If $n=3=p+q=1+2$ the manifold $M$ could be a solid…
snailspace
  • 827
  • 5
  • 11
5
votes
1 answer

Construction of lens spaces

I have a question about the surgery construction of lens spaces. Let $T=S^1 \times D$ be a solid torus. Let $T'$ be another torus. We fix a meridian $m$ and longitude $l$ of the torus. Then the lens space $L(p, q)$ is obtained by gluing two tori $T$…
user65175
5
votes
1 answer

Turn a torus inside out

Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle. Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian $\partial D^2\times \{*\} \subset T$ to a longitude…
user65175
5
votes
0 answers

Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; I just want to make sure that I understand the…
Carl
  • 51
  • 1
5
votes
1 answer

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 different from the connected sum?
Canonical Momenta
  • 453
  • 2
  • 8
5
votes
0 answers

$\mathbb{HP}^2$, 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-spheres? p.s. The description I found is…
wonderich
  • 5,529
  • 3
  • 21
  • 44
4
votes
1 answer

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This is defined here Surgery theory (wikipedia) It…
Ashley
  • 2,486
  • 11
  • 26
4
votes
0 answers

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By Lickorish's theorem any smooth closed connected…
Leo
  • 10,166
  • 4
  • 44
  • 76
4
votes
0 answers

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? --thanks
sarah
  • 41
  • 1
4
votes
1 answer

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 theory would have to require five dimensions and…
NotZack
  • 143
  • 5
1
2 3 4 5 6