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.
44 questions with no upvoted or accepted answers
11
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0
answers
237
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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 ...
- 637
9
votes
0
answers
257
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Building examples of elements of $\Omega_4(\xi)$ via surgery theory: how to do it?
When computing special bordism groups, I often need to determine existence of (singular) smooth $4$-manifolds with fixed fundamental group and certain properties like the spin behaviour (i.e. being ...
- 2,018
8
votes
0
answers
151
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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 ...
- 6,962
8
votes
0
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445
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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 ...
- 1,314
7
votes
0
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355
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Making diffeomorphism of submanifolds boring
This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds"...
- 1,981
7
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180
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Partial converse to Novikov's conjecture
In Jim Davis's paper "Manifold aspects of the Novikov conjecture" (Surveys on surgery theory, vol 1, pages 195-224) he writes down (Theorem 6.5) a sort-of converse to the Novikov conjecture. He writes:...
- 371
6
votes
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184
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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 ...
- 10.5k
6
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179
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Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)
In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...
- 1,329
6
votes
0
answers
133
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What is the monoid of skew-symmetric trilinear forms on finite abelian groups?
I am interested in triple cup product operations on the cohomology ring $H^*(Y;\Bbb Z/p^r)$ of 3-manifolds. Trying to extract the algebra, I am led to the following question.
Let's fix a prime power $...
- 9,580
6
votes
0
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197
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Regarding a proof in the surgery theorem by Gromov and Lawson
I have a question regarding a proof in the article The classification of simply connected manifolds of positive scalar curvature written by Gromov and Lawson. The precise reference is:
Gromov, ...
- 386
6
votes
0
answers
88
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Surgery on $M\times S^1$
I've encountered such a question and I don't know if it's trivial or not.
Given a simply connected closed $n$-manifold $M$ ($n\geq 4$), $\pi_1(M\times S^1)\cong\mathbb{Z}$ and it can be killed by ...
- 123
6
votes
0
answers
199
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Surgering locally flat tori in 4-manifolds
Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching ...
- 2,283
5
votes
0
answers
184
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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 ...
- 855
5
votes
0
answers
77
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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 ...
- 1,628
5
votes
0
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350
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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 ...
- 545
4
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202
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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
116
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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. . ...
- 141
4
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0
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172
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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 (...
- 73
4
votes
0
answers
377
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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 ...
- 5,230
4
votes
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answers
249
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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 ...
- 2,152
4
votes
0
answers
181
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Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants
It is know that Borromean rings can be detected by Milnor invariant
$$
\bar{\mu}(\gamma_1,\gamma_2,\gamma_3)=
\# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK}
\sum_{\...
- 2,147
3
votes
0
answers
86
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Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s
Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
- 335
3
votes
0
answers
406
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A user guide to the theory on Corks
I am trying to digest the meanings of the corks from the both:
algebraic topology
and
geometry topology
perspectives.
Studying corks is important for understanding the exotic phenomenon of 4-...
- 10.5k
3
votes
0
answers
80
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Quartic link in a 5-sphere
In this post I would like to propose a quartic link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})...
- 10.5k
3
votes
0
answers
104
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A link of four 2-tori $T^2$ in $S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...
- 10.5k
3
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0
answers
106
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A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$
Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold:
$$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
- 10.5k
3
votes
0
answers
192
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Surgery to unlink $S^p$ and $S^q$ in $S^d$
We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres.
I am looking for a surgery to unlink $S^p$ and $...
- 755
2
votes
0
answers
55
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 ...
2
votes
0
answers
117
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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 ...
- 119
2
votes
0
answers
130
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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$ ...
- 3,001
2
votes
0
answers
175
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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....
- 147
2
votes
0
answers
214
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 ...
- 5,901
2
votes
0
answers
102
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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 ...
- 21
2
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0
answers
170
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Triple link in a 5-sphere -- Proposal
In this post I would like to propose a triple link in a 5-sphere.
Let us start with the following gluing into a 5-sphere:
$$S^5=(D^2_{} \times T^3_{}) \cup_{T^4} ({S^5 \smallsetminus D^2 \times T^3})$...
- 10.5k
2
votes
0
answers
123
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Surgery and Curvature on Foliation
Let $X$ be an oriented closed smooth $4$-manifold. Suppose that $TM$ admits a foliation $\mathcal F$ of dimension two, and admits a positvescalar curvature.
Q: If we do the surgery on $X$ to reduce ...
- 1,915
2
votes
0
answers
109
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Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms
We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).
Given an embedded torus $T$ with trivial normal ...
- 10.5k
2
votes
0
answers
99
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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 $G$...
- 21
1
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161
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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 $...
- 593
1
vote
0
answers
183
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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 $\...
- 5,230
1
vote
0
answers
248
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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. ...
user160180
1
vote
0
answers
169
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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-...
- 173
1
vote
0
answers
76
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Does a homotopy sphere that bounds a highly connected manifold also bound a parallelizable manifold?
Suppose that the homotopy sphere $\Sigma^{n}$ can be realized as the boundary of a smooth $(n+1)$-dimensional cobordism that is $(n-1)/2$-connected for $n$ odd (respectively, $(n-2)/2$-connected for $...
- 165
0
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0
answers
78
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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 ...
user160180
0
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0
answers
199
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Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
- 1