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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
  • 68.9k
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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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} …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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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 …
Ian Agol's user avatar
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