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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Reducing the spin bordism group to its Pontryagin dual of the torsion subgroup

The spin bordism group for the classifying space $BG$ of group $G$ is denoted as $\Omega^{Spin}_d(BG)$. $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, E. H. Brown, Jr. and F. P. Peterson, The structure of the Spin…
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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 $S_n$. I just want to ask if this is correct or…
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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. My professor used the notation when talking me…
user062295
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Homology orientation induced by manifolds

To define Seiberg-Witten invariants one needs homology orientation. So for a closed oriented smooth (let us say as well simply connected) 4-manifold $M$, a homology orientation is an orientation of $H_2(M;\mathbb{R})$. I read that for example knot…
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Surgery to unlink $S^1$ and $S^2$ in $S^4$

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 $S^2$ maintain unknotted)? Can I do the surgery to first (1)…
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Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} \times S^1$ in $S^2$. =For $d=3$= Can we embed…
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Smooth structure after surgery

I'm having some trouble understanding how surgery again produces a smooth manifold. My understanding of surgery is something like this: start with a smooth manifold $M$ of dimension $m$ and, suppose we have an embedding $S^k \times \mathbb{D}^{m-k}…
anonymous
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Surgery and boundary

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 $m$ copies of the 2-handle $B^2\times B^2$ to $B^4$…
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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 $G$-action on $H^{k-1}(BG)$ with $e(X)=e$. However,…
Anonymous - a group
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Inverse operation of Dehn surgery

Suppose we have two closed oriented 3-manifolds $M$ and $N$. Suppose $N$ is obtained by a Dehn surgery operation on a knot $K$ in $M$, so $N=(M-\operatorname{int}\nu K)\cup_\partial (S^1\times D^2)$, with coefficient $p/q$, so that the meridian of…
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Apparent contradiction with gluing 2-handle to a 4-manifold all being isotopic.

In Hirsch's Differential Topology Chapter 8 Theorem 2.3, it says : Let $f, g:\partial Q\approx \partial P$ be isotopic diffeomorphisms. Then $P\cup_f Q\approx P\cup_g Q$. Here $\approx$ means diffeomorphic. Now the question I have is the following.…
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Knot Complement of Knot Sum and Fibered Knot Sum

Let $K_1,K_2$ be two knots and $X_1, X_2$ be their knot complements. Let $K_3$ be sum of two knots and $X_3$ the resp. complement. I am tempted to thinking that there is some relationship between $X_i$'s (for example, trying to show $X_3$ can be…
user928824
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Making precise Dehn filling

Dehn surgery along a knot is a well-known construction: choose a regular neighbourhood $N(K)$ of a knot $K \subset S^3$, let $X_K := S^3 - N(K)$ and choose an essential simple closed curve $\alpha$ on the torus $\partial X_K$. Then surgery along $K$…
Minkowski
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What is the topological space obtained by cutting $M\times N$ along a copy of $M$ or $N$?

What is the topological space obtained by cutting $M\times N$ along a copy of $M$ or $N$ (both closed topological spaces). E.g. if we cut a torus $\Bbb S^1\times \Bbb S^1$ along a circle then the result is a cylinder. If the answer depends on $M$…
C.F.G
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Why are embedded spheres removed in the connected sum but not in the handle attachment of (smooth) manifolds?

So i am currently studying differential manifolds and morse-theory. When i came across the connected sum, i learned that we glue two manifolds $M_1$ and $M_2$ along the boundaries of removed disks such that we obtain the quotient manifold $M_1 \#…
Zest
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