# 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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### 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 ...
3answers
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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 ...
2answers
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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 ...
1answer
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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-...
3answers
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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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### 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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### 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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### 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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### Surgery of $S^3$

I have been troubled by this seemingly simple question recently. How do we easily visualize the statement: Surgery of $S^3$ over a trivial unknot gives $S^1 \times S^2$? All I can think of for ...
3answers
591 views

### 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. ...
1answer
822 views