# 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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### 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 $T_{L}$...
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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 ...
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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. . ...
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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 ...
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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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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 $H_*(X) \simeq A \otimes ... • 1,944 5 votes 0 answers 174 views ### L-theory periodicity Let$\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})$for ... • 1,023 5 votes 1 answer 408 views ### L-theory of additive category Reading some articles in the field, I found the following statement: Proposition: Let$\mathcal{B}$be an additive category and$\mathcal{A}$a full additive subcategory of$\mathcal{B}$. If$\mathcal{...
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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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### 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 ...
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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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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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-...
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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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### 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 ...
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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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### 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 ...
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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_{\...
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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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### Wall self-intersection invariant for odd-dimensional manifolds?

I am trying to convince myself that a naïve definition of the Wall self intersection number should not work for odd-dimensional manifolds. Namely, let $X^{2n-1}$ be a smooth oriented closed manifold ...
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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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### 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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