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

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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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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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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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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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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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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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$3$-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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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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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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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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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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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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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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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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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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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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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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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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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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Visualization of the fact that the integers defining lens spaces must be coprime

This is related to this question I asked: Visualization of Lens Spaces and is also related to this question by @EarthliƋ: Why are the integers appearing in lens spaces coprime? I understand the ...
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Visualization of Lens Spaces

I am trying to visualize lens spaces geometrically. While I am aware of the fact that most manifolds which cannot be embedded in $\mathbb{R}^3$ are hard to visualize because of the obvious ...
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Incompressible tori in 3-manifolds obtained by Dehn surgery on knots

Let $K \subset S^3$ be a knot. Given $r \in \mathbb{Q} \cup \{ \infty\}$, denote by $S^3_{r}(K)$ the 3-manifold obtained by Dehn surgery on $K$ with coefficient $r$. Is it true that: $S^3_r(K)$ ...
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A(nother ignorant) question on phantom maps

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were ...
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Using Alexander's Theorem to show that the sphere $S^3$ is a prime manifold

I'm completely aware of the triviality of this question, but for some reason, I can't visualize the argument. In Hatcher's 3-manifold notes, the form of Alexander's theorem stating that Every ...
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A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
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Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
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Introduction to Euler structures

I am looking for a basic text on Euler structures, in particular smooth Euler structures, and the relation to combinatorial Euler structures; It is known that given a combinatorial Euler structure on ...
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existence of a 1-form

Suppose $L$ is a fibered link in $S^3$ and consider the fibration $f:S^3-L\to S^1$. Is it possible to write down the 1-form $df$ in the form $df=md\mu+ld\lambda$ near each component of $L$, for which ...