18
votes
0answers
431 views

The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...
10
votes
1answer
1k views

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
10
votes
0answers
484 views

Once differentiable, piecewise degree three polynomials on triangulated planar domains

Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of ...
7
votes
1answer
390 views

Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
23
votes
1answer
1k views

Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...
17
votes
1answer
817 views

What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$? Background Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
4
votes
1answer
431 views

Milnor's isotopy invariant using spectral sequence?

I'm reading stalling's article "the augmented ideal in group ring" in Ann. Math. Studies 84(R. H. Fox memorial volume) In his final remark, he says that Milnor's link invariant could be interpreted ...
2
votes
1answer
928 views

The importance of Poincare Conjecture or SPC4?

As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture. Somebody says that we can understand part of the universe from Poincare conjecture. As a purely ...
11
votes
3answers
2k views

Permute Wada Lakes keeping the coastline intact? (still open in dim >2)

Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...
37
votes
13answers
5k views

What are some of the big open problems in 3-manifold theory?

From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ...
13
votes
1answer
746 views

$SL_2 R$ Casson invariant?

Casson's invariant is an invariant of a homology 3-sphere, obtained by ``counting" representations of the fundamental group into $SU(2)$. I was wondering if there is an analogous invariant counting ...
14
votes
1answer
836 views

Topological version of Bogomolov’s question

I'm quoting a question from p. 753 of Gromov's recent paper Singularities, Expanders and Topology of Maps: "Does there exist, for every closed oriented $n$-manifold $X_0$, a closed oriented ...
7
votes
3answers
931 views

Is Murasugi's conjecture still open?

Normalize the Alexander polynomial (in $t$) so that the positive and negative exponents are balanced. For example in the Conway normalization, make the substitution $z = t^{1/2} - t^{-1/2}$. The ...