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 ...

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94
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1answer
9k views

What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
63
votes
24answers
27k views

Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
63
votes
4answers
4k views

Which manifolds are homeomorphic to simplicial complexes?

This question is only motivated by curiousity; I don't know a lot about manifold topology. Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. ...
50
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2answers
1k views

The topological analog of flatness?

Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module. Briefly the question is: what is the topological analog of this? Many ...
47
votes
3answers
6k views

The story about Milnor proving the Fáry-Milnor theorem

This question is similar to a previous one about "urban legends", but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, ...
47
votes
1answer
4k views

The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
47
votes
2answers
2k views

Gromov's list of 7 constructions in differential topology

At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order ...
40
votes
3answers
2k views

Explicit metrics

Every surface admits metrics of constant curvature, but there is usually a disconnect between these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces. Can ...
39
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8answers
4k views

Questions about analogy between Spec Z and 3-manifolds

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed ...
38
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4answers
2k views

Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...
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 ...
37
votes
3answers
2k views

Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
35
votes
1answer
4k views

Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times \mathbb R^2$ such that the projection $D^4\to S^2$ is an open map? Here $D^n$ denotes closed $n$-ball. An open map D⁴ → S². It ...
34
votes
7answers
2k views

Why should I care about Heegaard-Floer theory?

I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these ...
33
votes
3answers
1k views

Does Euclidean space have a compact factor?

Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point? Bing's Dogbone space is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with ...
33
votes
2answers
1k views

Knot security (When to trust your life with a knot)

This question is related to a a question about self-tightening knots. I am supervising a senior thesis and my student is interested in knots. My student is also a rock climber and has an ...
32
votes
4answers
1k views

To which extent can one recover a manifold from its group of homeomorphisms

Question. Suppose that $M$ is a closed connected topological manifold and $G$ is its group of homeomorphisms (with compact-open topology). Does $G$ (as a topological group) uniquely determine $M$? ...
32
votes
0answers
763 views

Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...
31
votes
9answers
2k views

In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?

In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that ...
31
votes
6answers
2k views

Status of PL topology

I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
31
votes
4answers
2k views

Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school. Caveat: I'm not a ...
31
votes
2answers
2k views

Meaning/Origin of Seiberg-Witten Equations/Invariants

Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of where they came from. We take an ...
30
votes
11answers
3k views

Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
30
votes
3answers
4k views

Mazur's unpublished manuscript on primes and knots?

The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur. I'm not absolutely sure this is the one I mean, but in his paper, ...
30
votes
2answers
2k views

Drawing of the eight Thurston geometries?

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries? I am imagining something akin to the standard picture (of a sphere, ...
30
votes
3answers
2k views

Independent evidence for the classification of topological 4-manifolds?

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...
29
votes
9answers
5k views

Nice proof of the Jordan curve theorem?

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof. What is the simplest known proof today? Is there an intuitive ...
28
votes
9answers
3k views

Classification problem for non-compact manifolds

Background It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic). I'm also under the impression that there is ...
28
votes
5answers
1k views

Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?

Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)? If not true in general, is there any condition ...
28
votes
0answers
1k views

Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
27
votes
6answers
4k views

Triangulating surfaces

I've had a few undergraduate students ask me for references for the classical fact (due to Rado) that closed topological surfaces can be triangulated. I know two sources for this, namely Ahlfors's ...
27
votes
4answers
1k views

Extending a diffeomorphism of S^2 to D^3

A fundamental result in 3-dimensional smooth topology, which in computer jargon we might refer to as "a primitive", is the statement that any ($C^\infty$) diffeomorphism of S2 extends to a ...
27
votes
1answer
624 views

Open immersions of open manifolds

For concreteness, I will work in the category of smooth manifolds, but my question makes sense in topological and PL category as well. Recall that a manifold $M$ is called open if every connected ...
26
votes
5answers
2k views

Usefulness of using TQFTs

What is a topological feature, that a (some) tqft (e.g. in 3 or 4 dim) sees but homology/cohomology/homotopy groups dont? Or: what is an example where using classical theories is hard, but using a ...
26
votes
3answers
2k views

Is there a reset sequence?

There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone ...
26
votes
3answers
1k views

Embeddings of $S^2$ in $\mathbb{CP}^2$

Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line? Note: I suspect ...
26
votes
2answers
989 views

Does every finitely presentable group have a presentation that simultaneously minimizes the number of generators and number of relators?

This should probably be an easy question, but I don't know how to answer it: Suppose G is a finitely generated presentable group. Suppose a is the absolute minimum of the sizes of all generating sets ...
26
votes
0answers
928 views

What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
25
votes
1answer
984 views

Square roots of $\mathbb R^{2n}$

Recently, Richard Dore asked us if $\mathbb R^3$ is the cartesian square of some space, and Tyler Lawson answered beautifully in the negative. The even powers of $\mathbb R$ were left out in that ...
25
votes
2answers
2k views

What else is Seiberg-Witten Theory equal to?

In low-dimensional topology there have been a bunch of invariants defined, and Seiberg-Witten Theory seems to make its appearance in [a lot of] them: 1) Heegaard Floer homology = SW Floer homology ...
24
votes
5answers
5k views

Poincaré Conjecture and the Shape of the Universe

Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?
23
votes
8answers
2k views

Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background: Gauss invented "Gauss curvature" to measure how surface curves. Riemann gives an ingenious generalization ...
23
votes
3answers
2k views

In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra. He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...
23
votes
2answers
1k views

Why are there no wild arcs in the plane?

On math.stackexchange it was asked whether all arcs in the plane are ambient-isotopic. I suggested that one could prove this by appealing to the Schönflies theorem, which you can do as long as you can ...
23
votes
2answers
929 views

A Pachner complex for triangulated manifolds

A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves". A ...
23
votes
1answer
473 views

Diffeomorphisms of finite order not in the image of a circle action

Does there exist a closed smooth manifold $M$ and a diffeomorphism $f\colon M \to M$ such that: $f$ is isotopic to the identity, $f$ is of finite order, $f^n=ID$, and $f$ is not contained in the ...
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 ...
23
votes
2answers
581 views

What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
22
votes
1answer
2k views

Math and Wormholes

Hopefully, MathOverflow is the correct place for this. I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
22
votes
2answers
2k views

Why is the volume conjecture important?

The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem ...