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

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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 ...
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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 ...
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Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed (or just compact) 4-manifolds. Are there any closed (or compact) ...
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Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
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What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
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Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
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How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
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High-Dimensional Topology vs. Low-Dimensional Topology: What are the hard questions in the former?

This is a somewhat vague/non-technical question. I've heard a lot about how the topology of manifolds (smooth or otherwise) is simpler in dimension at least 5, due to the applicability of surgery ...
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fibered knots in $ S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2 $, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
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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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References about 3-manifolds

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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For any diagram of a surface-knot, what are the possible connections of edges?

The singular set of a surface-knot diagram is a disjoint union of i) a graph, which has 1- and 6-valent vertices corresponding to branch points and triple points respectively. ii) circles which has no ...
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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 ...
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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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The Geometrical Mean of $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$

This is my homework: $T^2$#$RP^2=RP^2$#$RP^2$#$RP^2$ Below is what I have think: From the book of M.A.Armstrong, I find $RP^2$#$RP^2$#$RP^2$ is a sphere which is attached by three Mobius ...
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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 ...