Tagged Questions
4
votes
1answer
47 views
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 ...
0
votes
0answers
34 views
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
...
4
votes
2answers
101 views
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 ...
5
votes
1answer
96 views
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 ...
1
vote
1answer
69 views
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 ...
0
votes
0answers
84 views
slam-dunk operation of surgery diagram
I study surgery theory on 3-manifolds using the text book written by Gompf and Stipsicz. I can't understand the slam-dunk operation.
Let $K_{1}$ be the meridian of a knot $K_{2}$ in $S^3$, and $T$ is ...
2
votes
3answers
88 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.
4
votes
1answer
98 views
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 ...
3
votes
1answer
195 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 ...
3
votes
1answer
135 views
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 ...
3
votes
0answers
286 views
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 ...
2
votes
1answer
141 views
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 ...
5
votes
1answer
121 views
Exotic Manifolds from the inside
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 ...
14
votes
2answers
276 views
explicit “exotic” charts
can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
