# Tagged Questions

218 views

### Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
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### Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. ...
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### handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
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### Splittings of 4 manifolds

I am wondering why a copy of $CP^2$ may be split off from $CP^2\mathbin\#9\overline{CP^2}$ to leave $\overline M_{E_8}\mathbin\#\overline{CP^2}$(«The wild world of 4-manifolds» by Alexandru Scorpan, ...
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### Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
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### How to make a $C^1$ knot into a $C^\infty$ knot

Suppose I have a $C^1$ imbedding $f: S^1 \rightarrow S^3$. From the point of view of knot theory, what's the "best" way to get a $C^\infty$ curve that "looks like" or is "equivalent to" $f$? For ...
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### intersection on manifold with boundary and relative long exact sequence in homology

Let $(M,\partial M)$ be a connected compact oriented 3-manifold with torus boundary. Let $j: M \to (M, \partial M)$ and $i: \partial M \to M$ be inclusions. We get a long exact sequence ...
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### How does Thurston's geometrisation conjecture imply Poincaré's conjecture?

I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
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### Cancelling 3-handles in Kirby diagrams

Recently been trying to understand the proofs of Gompf and Akbulut that certain 4-manifolds are $S^4$ (these 2 papers: Gompfs paper in Topology Vol. 30 Issue. 1, Akbulut). In which they use a clever 2 ...
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### How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
133 views

### Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
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### Prime decomposition of 3-manifolds

Let $H_g$ be a three dimensional handlebody bounded by a genus $g$ surface. Let $M_g$ be a manifold obtained by gluing two copies of $H_g$ via an orientation reversing homeomorphism of the surface of ...
316 views

### How does handle attachment work in Morse Theory

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I can't understand the 2nd-paragraph of p.101, where they explain framings on the attaching sphere. In particular I cannot ...
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### What's the shining sparkle in the Sphere Inside-Out problem?

I've just seen the wonderfully done movie Sphere Inside Out, one about the Smale's paradox. And the first question came in mind is that, why it has to be so ugly? Why turning an ultimately simple ...
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### Useful topology on space of smooth structures on $\mathbb R^4$?

Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there. $Q_0$: Is there any use for a topology on the (continuum ...
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### 4-Manifolds of which there exist no Kirby diagrams

In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin ...
As we know, an exotic $\mathbb{R}^4$ is a manifold which is homeomorphic, but not diffeomorphic to the standard $(\mathbb{R}^4,id)$, and there are even very explicit descriptions of them (Kirby ...
can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?