# Tagged Questions

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 ...

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### 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 ...
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### The identification $G=\Omega^\infty \Sigma^\infty S_0$ of the stable group of self homotopy equivalences of spheres with the suspension spectrum

My topology professor told me in a discussion that the suspension spectrum $colim \Omega_n \Sigma_n S_0$ is the same as the monoid $G$ where $G=colim G_n$ where $G_n$ are self homotopy equivalences of ...
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### What is the classifying space G/Top?

I simply can't find the definition(except in one book on surgery where a definition was not actually given but instead they alluded to what the definition is) and I have spent an hour and half looking....
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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 ...
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### Surgery to unlink $S^1$ and $S^2$ in $S^4$ [closed]

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
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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 ...
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### 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 ...
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### 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 ...
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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 $\... 1answer 81 views ### Why is the oriented$G$-homotopy type of a$G$-complex uniquely determined by the periodicity generator? Say we have a periodicity generator$e \in H^k(BG)$. I can show that we then have a$(k-1)$-dimensional$G$-complex$X$with free$G$-action. It's also not that difficult to see that it has trivial$G$... 1answer 331 views ### Surgery on trivial knots I know a theorem that any closed orientable 3 manifold can be obtained from the sphere$S^3$by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ... 0answers 101 views ### Does the boundary of a handle decomposition obtain a handle decomposition? Let$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 ... 1answer 2k views ### 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. ... 2answers 78 views ### 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^... 1answer 155 views ### 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$D^{p+1}\times S^{q-1}$. ... 1answer 354 views ### 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 ... 0answers 78 views ### 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$...
Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
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; ...