Tagged Questions

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

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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. ...
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Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n$ be a framed link in $S^3$. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
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Cohomology after Dehn surgery

For $f:S\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a $0/1$-type Dehn surgery along $f$: First remove from $M$ a solid torus which is a tubular neighbourhood of the knot $f$; ...
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Seeking a good book or site describing the 3-sphere

would you be able to recommend a good book/chapter or a web site on visualization / structural elements / projection of the 3-sphere. I am trying to locate a good information source on this subject, ...
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Solid Ball Bearing 4 Dimensional?

Is a solid ball bearing a 4 dimensional object, but all we see is one particular level surface? When we look at a solid ball bearing, we can only see the outside of it, we cannot see inside even ...
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A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface (...
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Is my understanding of space correct?

I'm learning more about dimensions in multivariable calc, and have been able to make connections by studying level curves and level surfaces. I've learned that a function of 2 variables is really a 2 ...
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handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
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Pathway to low dimensional topology.

I am currently studying Artin Algebra and Topology by Munkres. I have previously studied analysis. After reading few lecture notes on knot theory, I have developed a genuine interest in Low ...
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I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as ...
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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 $\mathbb{R}^3$ ...
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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 components....
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Self-contained text on characteristic classes

I am looking for a clear, self-contained text (either a book or lecture notes) that deals with characteristic classes, starting from the very basics (fiber bundle, principal bundle etc.), and ...
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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 ...
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Coloring knot diagram obtained from colored one by applying crossing change to one crossing

Suppose $K_1$ is a knot diagram colored by a dihedral quandle $R_n$ of order $n$, By applying crossing change (exchanging over and under arcs) to one crossing in $K_1$, we obtain a new diagram let us ...
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Looking for a basic reference on propagators (in Topology)

I am looking for a basic (preferably self-contained) reference where I can read about propagators (as they appear in Topology), and in particular Morse propagators. Thanks!
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double decker set in a surface-knot

Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch ...
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Splittings of 4 manifolds

I am wondering why a copy of $CP^2$ may be split off from $CP^2\mathbin\#9\overline{CP^2}$ to leave $\overline M_{E_8}\mathbin\#\overline{CP^2}$(«The wild world of 4-manifolds» by Alexandru Scorpan, ...
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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 ...
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How is it possible calculate Volumes from Adding together Areas?

You can't find the volume for a $2D$ object, but by finding the area of say a square $A$ and multiply by any height $H$ you get the volume of cube with base $A$ and height $H$. Here's my question. ...
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Knot complement conjecture in solid tori

Has the knot complement conjecture been proven for knots in solid tori?
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How do you specify a link to a blind combinatorialist?

Regular projections of links look like graphs in the plane. So I'm wondering if it would be possible to specify a link up to isotopy with purely combinatorial data about this graph. If so, what kind ...
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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 $B$ (...
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Set of generators of the commutator subgroup of a surface group

Good morning, I am having a hard time trying to describe the commutator subgroup of a surface group. Namely, if $S$ is a compact orientable surface and $G$ its fundamental subgroup, let's recall that ...
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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 (...
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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 ...