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# Questions tagged [surgery-theory]

In geometric topology, surgery theory is used to produce one finite-dimensional manifold from another in a 'controlled' way. Originally developed for differentiable (smooth) manifolds, surgery techniques also apply to piecewise linear and topological manifolds. 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 related to handlebody decompositions.

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### 4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation: Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
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### About connected cobordism and surgery

I need to find various ways of performing two surgeries on a collection of circles so that the resulting 2-dimensional cobordism (the trace of the surgeries) is connected. How can I find these ? up ...
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### Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s

Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
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### About two surgeries on a collection of circles

in this paper (A. Hatcher, W. Thurston, A presentation for the mapping class group of a closed orientable surface) page 4 we have : There are five essentially distinct possibilities, corresponding to ...
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### Attaching a 2-handle to a once-twisted unlink in the boundary of the 4-ball

Consider the 3-sphere $S_3$ with an unlink loop $L$ whose tubular neighborhood is identified with the solid torus $B_2\times S_1$ with one twist, i.e., such that the image of $x\times S_1$ (where $x$ ...
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### Expository material on the Gromov-Lawson surgery theorem

I am looking for an expository text on the paper "The classification of simply connected manifolds of positive scalar curvature" by Gromov and Lawson, in particular on the proof of Theorem A....
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### Survey or good reference of taut foliations

I am interested in the topology of foliations. In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows. I guess that A. Candel and L. Conlon, Foliations I (...
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### Surgery along knots and connected sum

Denote $S^3_{p/q}(K)$ by performing $p/q$-surgery along a knot $K$ in $S^3$. Let $K$ and $J$ be two arbitrary oriented non-trivial knots in $S^3$. Is there a nice relation between surgery on the ...
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### A question on the proof of Theorem 1 in Milnor's "Killing Homotopy Groups"

Theorem 1 of Milnor's paper "A procedure for killing homotopy groups of differentiable manifolds" states that two manifolds are in the same cobordism class if and only if they can be ...
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### Regarding the surgery construction in "A procedure for killing homotopy groups of differentiable manifolds" by Milnor

In the first section of "A procedure for killing homotopy groups of differentiable manifolds", Milnor gives the surgery construction as follows. Let $W$ be an $n=p+q+1$ dimensional manifold. ...
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### Kind of "associativity" of certain connected sum involving both manifolds with and without boundary

Consider two compact, oriented and connected manifolds $\mathcal{M},\mathcal{N}$ with possibly non-empty connected boundaries $\partial\mathcal{M}$ and $\partial\mathcal{N}$. Now, in some project, I ...
137 views

### Is spin cobordism an invariant for surgery of codimension $q\ge3$?

Recall that a surgery of codimension $q$ on an $n$-manifold $X$ is a modification of $X$ of the following type. Let $\Sigma^{n-q}\subset X$ be a smoothly embedded $(n-q)$-sphere with a trivialized ...
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### L-theory periodicity

Let $\mathcal{A}$ be an additive category. I have two questions: Is there a conceptual explanation why $L(\mathcal{A})$ is 4-periodic, in the sense that $L_{i}(\mathcal{A})=L_{i+4}(\mathcal{A})$ for ...
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Reading some articles in the field, I found the following statement: Proposition: Let $\mathcal{B}$ be an additive category and $\mathcal{A}$ a full additive subcategory of $\mathcal{B}$. If $\mathcal{... 12 votes 1 answer 509 views ### Roadmap for L-Theory Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ... 3 votes 1 answer 117 views ### Effect of a Lutz twist on Euler number I already asked this question on the Math Stack Exchange but did not get an answer. I am currently working through Geiges proof of the Martinet-Lutz theorem, which can be found here, and am trying to ... 5 votes 0 answers 71 views ### Fiber product formulae for surgery obstructions Is there a formula for the surgery obstruction of a fiber product of maps assuming the fiber product is also a homotopy equivalence? In more detail, suppose that$X \to Y$and$Z \to Y$are homotopy ... 6 votes 1 answer 252 views ### Dehn surgery along primitive knot in 3-dimensional handlebody I'm studying the article "An alternative proof of Lickorish–Wallace theorem" (doi link) and I got stuck in a problem. Let$H_g$be a 3 dimensional handlebody of genus$g$, a primate curve in ... 2 votes 0 answers 188 views ### Surgery for algebraic varieties I have a number of vague questions that I wasn't sure whether they are suitable to ask or not, but I decided to ask! According to this result, any two birational varieties can be constructed by a ... 5 votes 0 answers 280 views ### Are there alternate descriptions of `elementary cobordisms'? Let$M^d$,$N^d$be cobordant$d$-manifolds. Then$M^d \sqcup \bar{N}^d = \partial W^{d+1}$for some$(d+1)$-manifold$W$. This cobordism can be implemented via an elementary set of 'moves' called ... 6 votes 0 answers 157 views ### Visualize how the 5d Dold manifold and Wu manifold are cobordant via a 6d manifolds with boundaries Are there simple intuitions and arguments to visualize why the following two 5-manifolds are cobordant to each other with the oriented structures? (They can be two boundaries of 6-dimensional oriented ... 7 votes 3 answers 222 views ### Normal invariants I am having a hard time finding examples of computations of normal invariants of surgery theory (or more generally the set of homotopy classes of maps$[X,G/O]$). Does anybody have good references? 0 votes 0 answers 69 views ### Bipartedly slice links and their surgeries A link L in$S^3$is said to be strongly slice if$L=∂D$,where$D$is a disjoint union of smoothly and properly embedded disks in$B^4$. A link$L$in$S^3$is called bipartedly slice if$L = L_1 \cup ... 1 vote
In this post, we discuss the relation between the concordance of knots in $S^3$ and the integral homology cobordism. Following its notation, assume that knots $K_0$ and $K_1$ in $S^3$ are concordant. ... 