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

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branched cover over a symplectic surface

Consider a branched cover $X$ of $(B^4,\omega_{st})$ over a symplectic surface $F$. Is it true that there is a naturally induced symplectic structure on $X$? Of course, there is a natural symplectic ...
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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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86 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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37 views

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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29 views

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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36 views

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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41 views

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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38 views

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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188 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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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 @Earthliŋ: Why are the integers appearing in lens spaces coprime? I understand the ...
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161 views

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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43 views

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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59 views

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 ...
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60 views

Isotopic tori in $\mathbb{R}^4$

Intuitively it seems to me that two tori in $\mathbb{R}^4$ are isotopic to each other. By isotopic, I mean a smooth family of deformations beginning in one and ending in the other, and each member of ...
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Does a closed surface in the 3-sphere bound a handlebody? [closed]

If a closed surface is embedded in the 3-sphere, then does it bound a handlebody?
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The notion of a right-angled hexagon in hyperbolic geometry

I was hoping someone would help me understand better what a "right-angled hexagon" is in hyperbolic geometry. I know these are glued together somehow to produce hyperbolic pairs-of-pants. The only ...
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Surface groups as 3-manifold groups

From studiosus' answer to A 3-manifold with fundamental group isomorphic to a surface group. a closed 3-manifold cannot have a fundamental group isomorphic to that of a closed surface of genus $\geq ...
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Reeb orbit and open books

Weinstein conjecture is about existence of a closed orbit of the Reeb vector field on every contact manifold. On the other hand, we know every contact 3-manifold admits a compatible open book, which ...
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Lens Space Orientation Reversing Homeomorphism

I am thinking of an example where the connected sum of two three Manifolds depends on the chosen orientation. Hempel gives in his book "3-Manifolds" an example, namely lens spaces. He shows that two ...
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Heegaard splitting and mapping class group

I would like to ask questions about the definition of the Heegaard splitting. The following are the facts I know. A Heegaard splitting says that any 3-manifold is built up from two ...
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1answer
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Discontinuity of the identity function in topology

According to a theorem I was taught, the identity function $id(x)=x$ from $(\mathbb{R}, \tau_1)$ to $(\mathbb{R}, \tau_2)$ is continuous if $\tau_1 = \tau_2$. Are there any examples of topologies ...
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Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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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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Plane models from the “word”

I have a "word" for a plane model $abacdc^{-1}db^{-1}$. From what I reckon, it's a torus. But I am not too sure of it. I sketched it up and did some "adjustments". Could it be a projective plane ...
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Classification of compact 3-delta-complexes made of a single simplex

With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see ...
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Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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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!
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characteristic foliation on a page of an open book

I cannot figure out the mistake in the following argument: The characteristic foliation on a page of a compatible open book on a three manifold (intersection of the contact planes with the page), is ...
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Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
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Is a Möbius Strip in > 4 dimensions impossible?

I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra ...
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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 ...
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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 ...
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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 ...
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Questions about the Nature of Chirality (with some focus on dimensionality)

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in ...