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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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 ...
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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 ...
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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(\...
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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 ...
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Classification of fake (quaternionic, octonionic) projective spaces
If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
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A search for a sequence of $6$-manifolds
How to construct closed, orientable, smooth, simply-connected $6$-manifolds such that $H^{*}(M,\mathbb{Z}) \cong \mathbb{Z}[a]/(a^{4})$ (Where $a$ is a generator of degree 2) satisfying $p_{1}(M) = n ...
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Is there a notion of a chain complex with corners?
Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.
On the topological side, although in the PL ...
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Smooth structures on the connected sum of a manifold with an Exotic sphere
What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth ...
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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 ...
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Do the results of (1/n)-surgery determine the link?...
Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...
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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 ...
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Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$
In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...
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Elements of infinite order in the topological mapping class group
Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
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Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
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How to compute $[CP^2, G/PL]$?
Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...
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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 ...
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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 ...
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motivation of surgery
an $n$-surgery on m dim manifold M is to cut out $S^n\times D^{m-n}$and replace it by $D^{n+1}\times S^{m-n-1}$.
I want to know how this is invented?
I do know that the effect of passing a critical ...
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Any PL-homology-manifold is homotopy equivalent to a manifold
Is it true that any compact piecewise linear homology manifold is homotopically equivalent to a (smooth?) manifold of the same dimension?
Let me say bit more since my question was wrongly understood.
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topological type of smooth manifolds with prescribed homotopy type and pontryagin class
Can someone help explain the following result:
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Thank ...
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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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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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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 ...
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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 ...
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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, ...
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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?
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Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
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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^...
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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"...
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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:...
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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 ...
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Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1
I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...
user22741
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Diffeomorphism type of the added sphere in simply connected surgery
A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...
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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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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 ...
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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:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...
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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 $...
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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, ...
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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 ...
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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 ...
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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 ...
user150450
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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 ...
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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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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 $...
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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{...
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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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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 ...
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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 ...
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Chirality and Anti-Chirality of links in 3 and in 5 dimensions
We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory:
https://en.wikipedia.org/wiki/Chiral_knot
My ...
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Manifolds whose diffeomorphism group has the homotopy type of a manifold itself
I have a very stupid question.
Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=...
