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

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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 ...
2
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0answers
72 views

Intersection form on manifolds with boundary

It is a "basic fact" that the intersection form of a closed oriented 4k-dimensional manifold is unimodular. (Could anyone point me to a reference to a proof of this fact?) What can be said about the ...
2
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4answers
251 views

Higher Dimenional Tic Tac Toe

Here we have a problem that seems very intuitive, but is hard to define mathematically. In Tic Tac Toe, we can find an equivalent of the game in any number of dimensions, it seems. The trick is to ...
5
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3answers
132 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 ...
2
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1answer
80 views

Density of continuous knots in the plane transversal to some circles

This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C) Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
3
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1answer
146 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 ...
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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0answers
58 views

History of ' low-dimensional geometry '

I want to have a brief history about the low-dimensional manifolds and geometric structures on manifolds specially on low-dimensional manifolds .where I can read about thus ?
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0answers
100 views

intersection on manifold with boundary and relative long exact sequence in homology

Let $(M,\partial M)$ be a connected compact oriented 3-manifold with torus boundary. Let $j: M \to (M, \partial M)$ and $i: \partial M \to M$ be inclusions. We get a long exact sequence ...
3
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1answer
193 views

3-manifolds fibering over the circle and mapping tori

If $S$ is a closed connected surface and $\varphi \in \mathrm{Diff}(M)$, then we can build the mapping torus $M_\varphi = \dfrac{S \times [0,1]}{(x,0)\sim (\varphi(x),1)}$. Then we have that $ ...
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 ...
3
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0answers
65 views

Classification of Bieberbach groups

Does anybody know if there exists a list of the four dimensional Bieberbach groups presented by generators and relations on the web?. I know there exists the book Crystallographic Groups of ...
2
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1answer
91 views

intersection form of $CP^2$

I am trying to understand why the intersection form of $CP^2$ is <1>. First we introduce {[x:y:z], x=0} as a generator of second homology and then we say that it has one intersection with {[x:y:z], ...
4
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2answers
235 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 ...
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1answer
41 views

Homotopy topology question

Let f be a map from the real projective plane to the torus. Show that f must be homotopic to a constant map. This is a qual problem. Any help would be appreciated. Thank you.
5
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1answer
149 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 ...
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1answer
131 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
12
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1answer
478 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 ...
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0answers
78 views

Trivial eigenvalues

Many nonlinear manifold learning methods used for dimensional reduction (isomap, diffusion maps, local preserving projections, etc) solve the symmetric eigenvalue problem and then use the eigenvectors ...
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 ...
5
votes
1answer
183 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 ...
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, $i$ ...
4
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0answers
110 views

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
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
131 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 ...
2
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1answer
85 views

Characteristic cohomology class of 4-manifold with boundary

I have a question about Characteristic cohomology class of 4-manifolds. $X^4$ denotes the compact 4-manifold with boundary. I'm mainly concerned with $\partial X$ is nonempty. If $X^4$ is closed, we ...
2
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3answers
133 views

Finding the rank of subgroups of free groups?

How do you find the rank of a subgroup(of finite index) of a free group? I was thinking of looking at the fundamental group of a graph.
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4answers
1k views

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 ...
4
votes
1answer
140 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 ...
7
votes
3answers
421 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 ...
2
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1answer
109 views

Lift of a homeomorphism $f$ between two (hyperbolic) surfaces $X,Y$

Let $X,Y$ be two hyperbolic Riemann surfaces (i.e. they have universal cover the upper half plane $\mathbb{H}$). Let $\pi_X:\mathbb{H}\to X, \pi_Y:\mathbb{H}\to Y $ be the corresponding covering maps. ...
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1answer
62 views

Why can we consider only torus switch rather than arbitrary Dehn filling in Lickorish theorem?

I understand that any orientable $3$-manifold can be obtained by doing Dehn surgery on $S^3$ along a set of circles sitting in it; but why can we further assume the slop to be $0$, i.e. we can obtain ...
2
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1answer
85 views

Which manifolds have a circle as their boundary?

The boundary of a disk or of a Möbius band is a circle. Which other manifolds share that property?
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1answer
505 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 ...
10
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1answer
316 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. ...
0
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1answer
133 views

What's isotopy of framings?

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus.At pp.100-101,the authors say "isotopy classes of framings."I don't know how to determine isotopy to framings.Please tell me ...
3
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1answer
314 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 ...
4
votes
2answers
216 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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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 ...
5
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0answers
122 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 ...
3
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1answer
190 views

The teardrop and the spindle are bad orbifolds

I should prove that the teardop $S^2(p)$ (the orbifold with underlying surface $S^2$ and a single cone point of order $p>1$) and the spindle $S^2(p,q)$ (the orbifold with underlying surface $S^2$ and ...
8
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1answer
509 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 ...
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1answer
149 views

What is this class of meromorphic functions?

Let $\hat{\mathbb{R}}$ be $\mathbb{R} \cup \{ \infty \}$, $\hat{\mathbb{I}} = i \hat{\mathbb{R}}$ and $\hat{\mathbb{C}}$ be the Riemann sphere. Let $P$ be the set of all meromorphic functions from ...
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0answers
49 views

limit set of Kleinian grouos with closed manifolds as quotient

I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
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0answers
69 views

Relations between Kleinian groups and quotient manifolds

In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of ...
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1answer
64 views

Extending domain of a meromorphic function from non-compact to compact domain

Let $f$ be a meromorphic function with $\text{dom} (f)=\mathcal{M}$, where $\mathcal{M}$ is a non-compact Riemann surface. If $\mathcal{M}'= \mathcal{M} \cup \{\infty \}$ is the one-point ...
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2answers
184 views

When does an embedded $2$-torus bound a solid torus in $3$-manifolds?

This is a simple version of the question asked here. Let $M$ be a compact 3-manifold without boundary, $\mathbb{T}^2$ be the standard 2-torus, and $i:\mathbb{T}^2\to M$ be an embedding. (*) Assume ...
3
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1answer
175 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 ...
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2answers
145 views

Relationship between the $2$-plane bundles over $S^2$ and $\mathbb{Z}$

I want to follow up on this answer by asking a few more questions (posting directly on the question didn't seem to "bump" the thread). I was trying to read the referenced text (Husemoller's Fiber ...
7
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1answer
228 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 ...