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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
53
votes
Accepted
Thurston's 24 questions: All settled?
A nice summary of the status of these problems may be found here:
Otal, Jean-Pierre, William P. Thurston: ``Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Jahresber. Dtsch. Ma …
49
votes
Accepted
Are there "principal" bundles $S^1 \to S^3 \to S^2$ other then Hopf's? (They would be necess...
The 3-sphere has infinitely many Seifert fibrations with generic
fiber a torus knot (including the unknot).
For a $(p,q)$ torus knot, the Hopf invariant will be $pq$ (up to sign).
To see this, note t …
35
votes
Level sets of Morse functions
No such collection exists for $n=2$. This follows the construction in my paper "Small 3-manifolds of large genus".
The result of the paper is that for any $g$, there are closed orientable irreducible …
34
votes
Accepted
Is every rational realized as the Euler characteristic of some manifold or orbifold?
Products of 2-orbifolds with manifolds will do the trick. There are 2-orbifolds of Euler characteristic $1/n$ (take a quotient of $S^2$ by a rotation of order $2n$). Then take a product with a manifol …
32
votes
Smallest tile to tessellate the hyperbolic plane
Binary Tiling
In fact, one can tile the hyperbolic plane with arbitrarily small tiles. There is a tiling of the hyperbolic plane (apparently due to Boroczky) by pentagons.
The horizontal edges ar …
31
votes
Accepted
Complex projective manifolds are homeomorphic if homotopy equivalent
For curves this follows from the classification of (2-dimensional topological) surfaces, and for simply-connected surfaces this follows from Freedman's theorem.
My former colleagues Anatoly Libgober a …
28
votes
Accepted
Complete knot invariant?
As Ryan says, this follows from Waldhausen's paper, when appropriately interpreted. Sufficiently large 3-manifolds are usually called "Haken" in the literature, and as Ryan says, they are irreducible …
26
votes
Accepted
isotopy inverse embeddings vs. diffeomorphisms
Here I'll prove that $M$ and $N$ must be diffeomorphic under your hypotheses (I'll assume that $M$ and $N$ are open, i.e. no boundary).
Consider the direct limit (see below) $X= M \overset{f}{\to} …
25
votes
Compactification theorem for differentiable manifolds ?
There are contractible 3-manifolds which cannot be embedded in any compact 3-manifold. Kister and McMillan constructed a variant of the Whitehead manifold $M'$ which is contractible but which cannot e …
25
votes
Elevator pitch for the Virtual Fibering Theorem
I suppose I'm obligated to make a stab at answering this :)
First, the virtual fibering question was asked by Thurston:
"Does every hyperbolic 3-manifold have a finite-sheeted cover which
fibers over …
24
votes
What are some of the big open problems in 3-manifold theory?
One of the unresolved questions about 3-manifolds is the generalized Smale conjecture, which roughly interpreted asks for the homotopy type of the space of diffeomorphisms of a 3-manifold. Smale origi …
23
votes
$3$-manifold that is a surgery on a knot
This is an extensively studied question and is far from being understood in general. Here are some other conditions beyond the fact that $H_1(M)$ is cyclic.
the fundamental group should have weight 1 …
22
votes
Independent evidence for the classification of topological 4-manifolds?
There's a somewhat different exposition in Freedman and Quinn's book. I think
the main difference is that they use gropes instead of Casson handles. Gropes are
made of embedded surfaces instead of sin …
21
votes
Accepted
Does this approach for the Poincaré conjecture work?
I had a quick look. Although I haven't found a specific error, as far as I can tell, he's not using the hypothesis of simple-connectivity anywhere in an essential way. Even though he posits this as a …
21
votes
Why should I care about the Jones polynomial?
There have been some topological applications of the Jones polynomial and its various generalizations. I believe that these applications increased the interest in these invariants by topologists.
On …