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

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two-holed torus mapping class

Are the following two elements in the mapping class group of the two-holed torus equivalent? $\phi=ac^3ac^5bd$ and $\psi=(ab)^{11}d$ where letters denote positive Dehn twists about the curves in the ...
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Homeomorphic surfaces embedded in 4-space

A surface-knot is a closed connected surface embedded in the Euclidean 4-space $\mathbb{R}^4$. We consider the projection of the surface-knot into $\mathbb{R}^3$ with the singularity set contains of ...
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Regular neighbourhoods of non-orintable surfaces in $S^4$

Suppose that $F \subset S^4$ is a non-orientable surface. Let $N \subset S^4$ be a regular neighbourhood of $F$. Clearly, the boundary of $N$ is a circle bundle over $F$. Which is its Euler number? My ...
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Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
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Components of the space of immersions 2-manifold into $\mathbb R^3$

Let $M$ be a $2$-sphere with $g$ handles. Consider the space of maps $M\to \mathbb R^3$, which are immersions [i.e. smooth maps with nondegenerate differential in each point $x\in M$], with ...
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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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What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

What are some interesting and easy-to-understand (for non-differential geometers) facts about subobjects of $\mathbb{R}^4$ that are not only false in $\mathbb{R}^3$, but also specific to the structure ...
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Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...
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Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
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Platonic Hopf. Given the vertices of a tetrahedron circumscribed by unit sphere, find the stereographic projection of inverse Hopf fibers

I am trying to find the equations in $\mathbb{R}^3$ for the fibers of the four vertices of the tetrahedron circumscribed by the unit sphere. I want to find $s\circ h^{-1}$, where $s$ is the ...
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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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Kirby-like diagrams for $M^n$ when $n > 4$

Are there any attempts on constructing Kirby-like diagrams for representing manifolds $M^n$ with $n > 4$. What are the references on that ? I think you run out of dimension in which you can draw ...
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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 ...
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113 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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Reference for low-dimensional topology

I have read topology and algebraic topology by Munkres and I want to start low-dimensional topology. What is a good reference for stating low-dimensional topology?
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What closed 3-manifolds have fundamental group $\Bbb Z$?

For certain small groups, it is easy (and desirable) to classify closed (and orientable if necessary) 3-manifolds with that group as their fundamental group. (Essentially due to Waldhausen is that for ...
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Showing every knot has a regular projection using differential topology

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 imbedding. For ...
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The composition of a tricolorable knot with another knot is always tricolorable

Prove that the composition of a tricolorable knot and another knot (except the unknot, whether tricolorable or not) is tricolorable. I understand that the composition of two tricolorable knots ...
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Does the Dirac belt trick work in higher dimensions?

If the Dirac belt is in 4-space, is it still true that when the belt is initially given a 360 degree twist then it cannot be untwisted? I assume this is so because SO(n) is not simply connected, but ...
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If $M$ is a nonorientable $3$-manifold, why is $H_1(M, \mathbb{Z})$ infinite? [duplicate]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is nonorientable and $\partial M$ is empty, then how do I see that $H_1(M, \mathbb{Z})$ is infinite?
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The shape of a room that's bigger on the inside

I have an advanced facility with many floors containing experimental technology. One of these floors goes on forever. What shape is it? More specifically: within the room, you can move as far as you ...
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$3$-manifold has same homology groups as a $3$-sphere. [closed]

Let $M$ be a compact connected $3$-manifold with boundary $\partial M$. If $M$ is orientable, $\partial M$ is empty, and $H_1(M; \mathbb{Z}) = 0$, does it follow that $M$ has the same homology groups ...
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Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as ...
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Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
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Homotopically trivial $2$-sphere on $3$-manifold

Let $S^2$ be an embedded sphere in a $3$-manifold $M^3$ such that $[S^2]$ is trivial in $\pi_2(M)$. Can we find an embedded disk $D^3$ in $M$ such that $\partial D^3=S^2$?
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Knot Group and the Unknot

Hi I am stuck in trying to show that given a knot $K$ such that the knot group $\pi_1(K)=\mathbb Z$ then $K\simeq U$. I tried to use the fact that the infinite cyclic cover is the universal cover but ...
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The Developing Map

I am looking for resources to learn about the developing map, but the only source I know is Thurston's book. Could anyone direct me to other sources? Lecture notes, articles, books, etc? Thanks in ...
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Existence of a vector field with one singularity on a surface

In a paper I'm reading, it is stated that "It is known that on a compact, connected, oriented two dimensional manifold, there exists a vector field with only one singularity". Where can I find a ...
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286 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
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Euler Integral of a self-overlapping tube with a cusp singularity

I am studying in depth the following paper on Euler calculus applied to target enumeration: https://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf Within this paper there is an ...
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Fundamental group of Antoine's necklace

Let $A \subset \mathbb{R}^3$ denote Antoine's necklace. It is well-known that $A$ is a Cantor space and that $\mathbb{R}^3 \backslash A$ is not simply connected. Futhermore, $\pi_1(\mathbb{R}^3 ...
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Covering maps between Seifert fibered manifolds

Let $M$ and $\widetilde{M}$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: \widetilde{M} \to M$ preserving the Seifert structure. What is the relation ...
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Casson handles neighborhoods are representable by $D^2$-bundles over $S^2$.

On 250 page of Scorpan's book Wild world of 4-manifolds. there is a construction of an exotic $\mathbb{R}^4$. It starts from taking manifold $M = \mathbb{C}P^2 \# 9 \overline{\mathbb{C}P}^2$ and ...
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on branched covers

Consider a branched cover $f:M\to N$ with branch set $A\subset M$ and $B\subset N$. In Rolfsen's book Knots and Links, it is assumed as part of the definition of a branched cover that the dimension ...
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Embedding 3-manifolds in Euclidean space

By the Whitney Embedding theorem, every 3-manifold can be embedded in $\mathbb{R}^6$. It's my understand that it's an interesting problem to see which 3-manifolds embed in $\mathbb{R}^4$; some do and ...
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Space of $G$-connections; respecting a spin structure

If I want to have a space of $G$-connections on a Riemann surface, I can take the fundamental group on the surface, represent its generators on $G$ and take (up to conjugation) those representation ...
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theorem relating the genus of a surface to the Euler characteristic

One of the most basic facts of the theory of surfaces is $$ \chi = 2-2g $$ where $$ \chi \triangleq V - E + F $$ for orientables surfaces. Does this result have a name? I am also after a proof or a ...
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p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball.

in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...
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Compute the fundamental and homology groups of $S^3 \setminus K$, where $K$ is two linked copies of $S^1$ in $\mathbb R^3$

Compute the homology groups of $S^3 \setminus K$, where $K$ is two linked copies of circles in $\mathbb R^3$. How about the homology group of $S^3 \setminus K'$ where $K'$ is just one copies ...
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Gluing 3 dimensional tetrahedra with orientation reversing edge

I am not sure how to proceed on exercise 3.2.3 in Thurston's book "Three Dimensional Geometry and Topology". The wording is as follows: "In a gluing of three dimensional simplices, each edge enters ...
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The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move ...
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Does pushing point along a loop on a surface induce a homotopy from identity to a homeomorphism of the surface?

Suppose $S$ is a closed surface (assume $\chi(S)\leq 0$ if necessary). Let $p$ be a fixed point on the surface $S$, and $\beta$ be an oriented loop based at $p$. Intuitively, I can put finger on $p$ ...
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Alexander polynomial of sums

A standard argument of knot theory reveals that the Alexander polynomial of the sum of two knots $K= K_1 \# K_2$ is equal to the product of the Alexander polynomial of the two summands $K_1$ and ...
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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 ...
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Looking for a “Guide for the Perplexed by Low-dimensional Topology”

The following excerpt is from pp. 56-57 of Loring Tu's (so far very enjoyable) textbook An Introduction to Manifolds (2nd ed.): One of the most surprising achievements in topology was John ...
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Presentation of Fundamental Group for a Seifert Fibred Space

I am trying to understand how to make sense of the presentation for a seifert fibred space geometrically. I understand that for each exceptional fibre (with index a/b) you get a generator with torsion ...
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Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
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1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
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What is the underlying geometry of the space in this advertisement

There is this really intriguing new ad by a large Japanase car manufacturer: Here. The universe in the ad looks to be an orbifold with a cone-axis of order 6, but this is not the math I am comfortable ...
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Legendrian isotopy

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ...