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.

Filter by
Sorted by
Tagged with
2 votes
0 answers
42 views

Dehn surgery on $RP^2 \times S^1$

A standard example of Dehn surgery is obtaining $S^3$ from $S^2 \times S^1$. Consider a unknot $L$ wrapping the non-trivial cycle $S^1$ in $S^2 \times S^1$. We drill out a tubular neighborhood $T_{L} $...
4 votes
0 answers
162 views

Possible Euler characteristics of manifolds with tangential structures

Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
4 votes
0 answers
111 views

Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
1 vote
0 answers
132 views

Higher dimensional Seifert surfaces and link numbers of higher knots

In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots. Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
2 votes
0 answers
52 views

Tangential normal invariant isomorphism

Recently, I was reading the paper "Finite Group Actions on Kervaire Manifold" by Crowley, Hambolton. But I am having problem understanding a definition. Here it is, In page 15-16 they are ...
4 votes
1 answer
208 views

Gluing a manifold along its boundary, via chain complexes

Given closed oriented $n$-manifolds $M, M', M''$ and bordisms $W, W'$ with $\partial W = M \sqcup - M'$ and $\partial W' = M' \sqcup - M''$, we can collar-glue them to obtain a bordism from $M$ to $M''...
55 votes
3 answers
6k views

Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
6 votes
1 answer
441 views

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 ...
4 votes
1 answer
282 views

On the proof of the surgery step in Wall's book

This question regards a part of the proof of the so called surgery step, in Wall's book "surgery on compact manifolds", Theorem 1.1. Setting $M^m$ smooth manifold, $X$ CW complex, $\phi :M\...
2 votes
0 answers
109 views

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 ...
3 votes
0 answers
82 views

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 ...
2 votes
0 answers
114 views

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 $x$ ...
2 votes
0 answers
154 views

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....
4 votes
0 answers
135 views

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 (...
8 votes
1 answer
438 views

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 ...
3 votes
1 answer
220 views

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 ...
8 votes
1 answer
441 views

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. ...
2 votes
2 answers
368 views

Does the coefficient of the meridian determine the coefficient of the longitude?(on Dehn surgery)

I'm studying Dehn surgery, and it says that the coefficient $(p,q)$ which says how the meridian curve on solid torus is attached will determine the entire resulting manifold. I'm wondering whether the ...
3 votes
1 answer
146 views

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 ...
5 votes
1 answer
150 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 ...
24 votes
2 answers
3k views

Proofs of Kirby's theorem

Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained ...
8 votes
0 answers
150 views

Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?

For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above? The Two Summands Conjecture states that surgery ...
1 vote
0 answers
168 views

Mapping class group and surgery theory

Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\...
4 votes
0 answers
352 views

Kirby's theorem for 4-manifolds

In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...
4 votes
0 answers
229 views

Being a product - from homology to topology

The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras. Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...
23 votes
1 answer
1k views

Mapping class groups in high dimension

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Homeo{Homeo}$Let $M$ be a $1$-connected, closed, smooth manifold with $\dim(M)>4$ and let us set $\MCG(M)=\pi_0(\...
5 votes
0 answers
173 views

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 ...
5 votes
1 answer
406 views

L-theory of additive category

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
610 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
132 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
74 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
273 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 ...
23 votes
2 answers
832 views

Vanishing of characteristic numbers vs vanishing of characteristic classes

A famous result by Thom states that Oriented Bordism classes are determined by characteristic numbers; specifically, two closed manifolds are orientedly bordant if and only if they have the same ...
2 votes
0 answers
202 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
327 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
177 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
236 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
76 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 ...
7 votes
3 answers
597 views

Generalized Schoenflies - formalizing step in proof?

[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.] I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, ...
1 vote
0 answers
186 views

Concordance, surgery and homology cobordism

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. ...
8 votes
0 answers
403 views

Integer surgeries along links yielding lens spaces

Does there exist an integer $N$ such that any lens space $L(p,q)$ can be obtained by integer surgery from $S^3$ along a link $L$ with at most $N$ components? EDIT: I have worked out the comment by ...
5 votes
1 answer
364 views

Dehn surgery on $S^3$ along a Hopf link with rational surgery coefficients

Is there an exhaustive list of conditions satisfied by rational surgery coefficients assigned to the components of the Hopf link in $S^3$ such that the resulting 3-manifold by Dehn surgery acting on $...
5 votes
2 answers
301 views

Negative surgeries on negative knots

This question is two-fold. The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of ...
11 votes
0 answers
227 views

Torus trick without surgery theory

It follows from surgery theory that in dimension $\geq 5$ every closed PL manifold homotopy equivalent to a torus has a finite cover which is PL homeomorphic to a torus. This is an important ...
1 vote
1 answer
421 views

Integer surgery on $S^3$

I know that any compact orientable 3-manifold can be obtained from the three sphere $S^3$ by an integer surgery. I am not sure why the surgery operation is completely determined by Where we map ...
1 vote
0 answers
135 views

Confused about A. Kosinski's description about surgery in his book "differential manifolds"

Please excuse me, if MO is not the proper place for this question. I aksed the same question on M.SE https://math.stackexchange.com/questions/3511134/confused-about-a-kosinskis-description-of-surgery-...
5 votes
2 answers
403 views

Rational surgery and attaching $2$-handles

It is well-known fact that integral Dehn surgeries on $3$-sphere $S^3$ are viewed as the result on the boundary of attaching $2$-handles $B^2 \times B^2$ to the $4$-ball $B^4$. Is there an analogue ...
7 votes
1 answer
770 views

What are these 3-manifolds from surgery?

I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^...
9 votes
1 answer
293 views

Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.), Robion Kirby ...
2 votes
0 answers
92 views

Realizing an amalgamated product of groups by splitting a closed manifold along a codimension 1 submanifold

In the paper "A splitting theorem for manifolds" by S.E. Cappell, https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf the following "inverse" of the Seifert-van Kampen theorem for closed ...