Low-dimensional topology generally refers to the study of 3 or 4 dimensional topological manifolds and knot theory.

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355 views

explicit “exotic” charts

can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
14
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1answer
200 views

Is the torus the union of two connected, simply-connected open sets?

Is the torus the union of two connected, simply-connected open sets? A routine computation with the Mayer-Vietoris sequence shows that if so, then their intersection must have exactly three ...
12
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8answers
524 views

Why is the 3D case so rich?

The Banach--Tarski theorem applies only in the case of three or more dimensions. In 3D, there are five regular solids, two of them being not at all obvious, and the 4D case is also interesting; but ...
12
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1answer
489 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
10
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2answers
1k views

What is the importance of the Poincaré conjecture?

The Poincaré conjecture is listed as one of the Millennium Prize Problems and has received significant attention from the media a few years ago when Grigori Perelman presented a proof of this ...
10
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1answer
317 views

Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
8
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1answer
509 views

For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
8
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2answers
182 views

In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...
8
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1answer
90 views

Is there a domain in $\mathbb{R}^3$ with finite non-trivial $\pi_1$ but $H_1=0$?

The exterior of the Alexander Horned Sphere has $H_1=0$ but $\pi_1\neq 0$, in fact, $\pi_1$ is infinite. (See Hatcher p.171-172). Is there an example of a domain (connected open set) in $\mathbb{R}^3$ ...
8
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1answer
522 views

Understanding the Hopf fibration

I'm taking a class in manifolds, and the Hopf fibration recently came up. I'm trying to get a handle on it, so I'm going to try and explain what I think is going on, and hopefully math.stackexchange ...
8
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2answers
219 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
7
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3answers
366 views

Putting Geometries on Knot Complements

I have two different, but related, questions about the type of geometry one can get on a knot complement. Quickly some notation: $K$ will be a non-trivial smooth knot - living in $S^3$ - and $M$ will ...
7
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3answers
440 views

Embedding compact (boundaryless?) n-manifolds in n-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
7
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1answer
229 views

How to Classify $2$-Plane Bundles over $S^2$?

I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since ...
7
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1answer
172 views

Homology class calculation

Let $SO(3)$ be the $3\times 3$ orthogonal group. Define a map $i\colon SO(3)\to SO(3)\times S^2$ by $A\mapsto (A,A^{-1} e_1)$, where $e_1$ is the unit normal vector $(1,0,0)\in S^2$. Then, $i$ ...
6
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4answers
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Representation of $S^{3}$ as the union of two solid tori

Well, I'm trying to prove that you can express the 3-dimensional sphere $S^{3}$ as the union of two solid tori. I tried first use that a solid tori is homeomorphic to $S^{1}$$\times$$D^{2}$ and use ...
6
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1answer
154 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 ...
6
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2answers
333 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
5
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3answers
134 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
5
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2answers
477 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
5
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1answer
56 views

Why are there always pairwise intersections in a Heegaard splitting?

Let $M=A\cup B$ be a Heegaard splitting, such that $\{\alpha_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $A$, and $\{\beta_i\}_{i=1}^g$ is a set of boundaries for meridian disks of $B$ ...
5
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1answer
150 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 ...
5
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1answer
185 views

Isotopy and homeomorphism

Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$. The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$? I am especially ...
5
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1answer
115 views

Classification of orientable non-closed surfaces

How does the classification of closed (compact, boundaryless) surfaces imply the classification of all orientable not-necessarily-compact surfaces with boundary? It seems to be that they are all ...
5
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0answers
119 views

$\operatorname{Spin}^c(n)$ is a Lie group?

Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.) Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
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0answers
123 views

Heegaard Splitting of Brieskorn homology 3-spheres

For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. I want to know ...
4
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2answers
239 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 ...
4
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2answers
221 views

Heegaard splitting of a 3-manifold with boundary

A Heegaard splitting of a closed orientable 3-manifold $M$ is $M=H \cup H'$, where $H$ and $H'$ are handlebodies. Is there any similar concept for orientable 3-manifolds with boundaries?
4
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2answers
98 views

Length of a Coastline

When B. Mandelbrot's typical example of measuring the length of a coastline is referenced, they mention how at every scale the length increases. In pure mathematics, I can imagine this quite well-- ...
4
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2answers
86 views

High-Dimensional Topology vs. Low-Dimensional Topology: What are the hard questions in the former?

This is a somewhat vague/non-technical question. I've heard a lot about how the topology of manifolds (smooth or otherwise) is simpler in dimension at least 5, due to the applicability of surgery ...
4
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1answer
65 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 ...
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2answers
85 views

Recommended books on knot invariants

I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd ...
4
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1answer
82 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
4
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1answer
166 views

Representation of (co)homology classes of $3$-manifolds by embedded surfaces

Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify $$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$ where (co)homology is meant with integer ...
4
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1answer
141 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 ...
4
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1answer
117 views

What knot groups are Abelian?

The knot group (the fundamental group of the complement of a knot) of the unknot is $\mathbb{Z}$ and the Hopf link is $\mathbb{Z}^2$, so those are knots (links) with Abelian knot group but are there ...
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Problem from Hempel — surfaces in Lens spaces

This isn't a homework problem, just working through Hempel's 3-Manifolds for my own benefit. One exercise is to show that the lens space $L(2k,q)$ contains a surface of Euler characteristic $2-k$. ...
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0answers
132 views

Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface. Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$. In the very ...
3
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1answer
182 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
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2answers
30 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
3
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1answer
168 views

Relation between the braid group and the mapping class group of the plane

According to the following link, page 248, the braid group modulo its center is isomorphic to the mapping class group of the $N$-times punctured plane, i.e. $B_N/Z(B_N)\cong M_N(\mathcal(R)^2)$. Could ...
3
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1answer
318 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
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1answer
66 views

Signature of $S^2 \times D^2$

Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain ...
3
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1answer
70 views

Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. ...
3
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1answer
148 views

Annulus Theorem

I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is Use the Schoenflies theorem ...
3
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1answer
91 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
3
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1answer
68 views

Orbit space of S3/S1 is S2

I'm having trouble finishing this homework assignment. I did the first part by showing that the orbits are invariant: every element from the same $(S^1(z_1, z_2) \in S^3/S^1)$ is mapped to the same ...
3
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1answer
54 views

Compact orientable surfaces embeddings

I wonder if we can embed a compact orientable surface of genus $g$ into another of genus $g'$, if $g < g'$. I already know that this is false if $g>g'$, because of the first homology groups. Any ...
3
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1answer
444 views

the restriction of a homeomorphism on a subset

Let $X$ be a topological space and $f:X\to X$ be a homeomorphism. Then the induced map $f_1:\pi_1(X,x)\to\pi_1(X,fx)(\cong \pi_1(X,x))$ is an isomorphism (automorphism up to conjugate). In the ...
3
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1answer
208 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 ...