# Questions tagged [low-dimensional-topology]

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

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### Confused about the quotient goup of fundamental group of hyperbolic manifolds by boundary group.

I'm reading a paper (Arithmetic of Hyperbolic Manifolds). I'm confused about a statement of the proof of corollay 2.3 at page 277. It says "P is the subgroup of $\Gamma$ generated by parabolic ...
1 vote
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### Kirby Diagram of Enriques Surface

I would like a Kirby/handle diagram of the Enriques surface, and I whilst I haven't been able to find one in the literature, there is this diagram (originally due to Kondo I believe, though taken here ...
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### Question on a stable geodesic lamination on a closed hyperbolic surface

Let me first state a theorem in Casson-Bleiler Automorphisms of Surfaces after Nielsen and Thurston. Theorem 5.5: Let $h:F\to F$ be a non-periodic irreducible automorphism of a closed orientable ...
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### Classification of good foliations of a pair of pants

The following is a proposition from FLP (Thurston's work on surfaces). Proposition 6.7 (Classification of good foliations of a pair of pants) The function $\mathcal{MF}_0(P^2)\to\Bbb R^3_+$, which to ...
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As is discussed in this answer, the connected sum of two copies of $\mathbb{RP}^3$ can be realised as the compact affine grassmannian of $2$-planes in $\mathbb{R}^3$, denoted by $\operatorname{cGraff}... 1 vote 0 answers 37 views ###$C^\infty$equivalence and complex analytic equivalence between elliptic surface [closed] In Friedman and Morgan's book about elliptic surface, the authors claimed that$C^\infty$isomorphism between elliptic surfaces$A$and$B$doesn't imply that$A$and$B$are deformation equivalent, ... 4 votes 1 answer 83 views ### Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory. As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ... 3 votes 1 answer 59 views ### Isotopic if and only if homotopic for essential simple closed curves The following is Proposition 1.10 of A primer on mapping class group. Let$\alpha$and$\beta$be two essential simple closed curves in a surface$S$. Then$\alpha$is isotopic to$\beta$if and only ... 1 vote 0 answers 34 views ### Not primitive and not a multiple closed curve The following definitions are written in A primer on mapping class group.$(1)$An element$g$of a group$G$is primitive if there does not exist any$h\in G$so that$g = h^k$where$|k|>1$.$(2)...
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I am not very familiar with low-dimensional topology and I was wondering if we know the classification of links (in $\mathbb R^3$) that can be isotoped into a position where every link is a rigid (...
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### How to discribe the complement of a Seifert surface

We can construct a covering space of a knot complement by cutting along a Seifert surface and glue several copies together. So I want to know how the complement of a Seifert surface looks like. What I ...
1 vote
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### Simple left earthquakes are dense

i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows: Definition. If $\lambda$ is a geodesic ...
1 vote
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### Example of boundary parallel surfaces with boundary [closed]

"boundary parallel annulus" always appears in books/papers on 3-manifold theory. I know a surface is called boundary parallel if there is an isotopy sending it onto a boundary component. But ...
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### $\Gamma_{\tilde{P}}$ is a free group generated by two hyperbolic transformations

Book: An introduction to Teichmuller spaces by Imayoshi & Taniguchi. Let $R$ be a Riemann surface whose universal cover is $\Bbb D^2$ (i.e. Hyperbolic surface). Consider cutting $R$ by a family of ...
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### Joining the edges of a disc in a way that prevents mirroring

Imagine an object moving across a square and when it reaches an edge, it is transported to the opposite side of the square. Now imagine the same thing, but with a disc. When you reach the edge, you ...
1 vote
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### Simple Closed Geodesics in Mapping Torus Via Pseudo Anasove Homeomorphism?

Recently I have been reading the book A Primer on Mapping Class Groups by Benson Farb and Dan Margalit. Let $S_g=$ Closed surface of genus $g$, and $Mod(S_g)$ denoted the corresponding mapping class ...
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### If the second homotopy group is trivial, surface integral is independent with surface

Let $\Omega$ be a domain in $\mathbb R^3$ and let $\pi_2(\Omega)$ be the second homotopy group of $\Omega$. Let $\mathbf{F}=(P,Q,R)$ be a $C^1$ vector field. If $\pi_2(\Omega)=0$, can we deduce that ...
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### Lefschetz fibration on product of surfaces

If I understand the literature correctly, then every symplectic 4 manifold (potentially up to connected sum with complex projective space) admits a lefschetz fibration with codomain a two sphere. In ...
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### Orbit Space of Moduli Space in Morse Homology

If I have a group $G$ acting on a topological space $X$, and this action is free (no fixed points), what can I say about the orbit space $X/G$? I am aware that freeness plus proper discontinuity leads ...
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### Isometric embedding of Teichmüller spaces

Let $S_{g,k}$ be a genus $g$ surface with $k$ punctures. Let $\mathrm{Mod}(S)$ be the extended mapping class group of a surface, defined as the isotopy classes of the self-homeomorphisms of surface $S$...
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### Specific isomorphism between presentations of fundamental group of figure eight knot complement

I'd like to find a specific isomorphism between the following finitely presented groups, (hopefully up to conjugation) preserving the meridian and longitude. Capital letters denote inverses. ...
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### Exterior of Alexander horned sphere is not simply-connected

By a compactness argument, in order to prove that the exterior of the Alexander horned sphere is not simply-connected, it suffices to show that any finite stage of the construction is not simply-...
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### Is every Seifert fiber space a fiber bundle?

Recently I've been reading the Jankins/Neumann notes Lectures on Seifert Manifolds (PDF). In the notes it is claimed that Seifert spaces are almost fiber bundles (with $S^1$ fiber), except around ...
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### Can framings on plumbed manifolds be taken to be even?

The definition of "plumbed manifold" that I'm using in this context is the following - given a weighted tree $\Gamma$, build up a framed link $L(\Gamma)$ by chaining together two copies of ...
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### Heegaard diagram on a surface

Textbook: Lectures on the topology of 3-manifolds by Nikolai Saveliev In this book, there is a discussion about the Heegaard diagram of a closed orientable 3-manifold $M$. During the discussion it ...
1 vote
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### Standard fact in knot theory

In "Knots and Links in Spatial Graphs" (Journal of Graph Theory, vol.7 1983, 445-453), Conway and Gordon write : "Now it is a standard fact in knot theory, not hard to prove, that any ...
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### Constructing a pseudo-Anosov homeomorphism

Let S be a genus 2 closed surface. Consider the two non-separating red curves (say $\alpha$ & $\beta$) in the picture. I want to construct a pseudo-Anosov homeomorphism $\phi: S \rightarrow S$ ...
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### Genus 1 fibered knots in an integer homology sphere.

It is well known that genus 1 fibered knot in $S^3$ consist of trefoil and figure-eight knot. My classmate tells me that genus 1 fibered knots in an integer homology sphere must be trefoil or figure-...
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### Reference request: Every self-diffeomorphism of $S^3$ is isotopic to the identity

I am searching for a reference with a proof of the statement: Every orientation preserving self-diffeomorphism of $S^3$ is isotopic to the identity. I searched in many books aswell as online, but did ...
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### Continuous Map on Compact Oriented Manifold of Higher Degree

Let $M$ be a compact oriented manifold of dimension $\ge 3$. Is there any known obstructions of $M$ for it to admit a continuous self-map of degree $>1$? Note: the case when $\dim(M) = 2$ is known (...
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### Let $X$ be a topological space such that $X \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^2$. Must $X$ be homeomorphic to $\mathbb{R}$?

This question was posted on twitter here as a quiz but the author never gave an answer, so I thought I'd try here. I don't have much experience with topology so I'm stumped. From searching online it ...
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### Exotic manifolds in three dimensions [closed]

Suppose, we are working in three dimensional setting. Are there any exotic manifolds?(manifolds which are homeomorphic but not diffeomorphic)
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### Let $X$ and $Y$ are copies of $S^2$ taking points $x_1,x_2 \in X$ and $y_1, y_2 \in Y$ then $\pi_1(X \bigsqcup Y/x_1\sim y_1,y_2\sim x_2 )?$

Recently I have been reading Hatcher's algebraic topology book; in chapter zero, there is a problem that if take the sphere of dimension $2$ (symbolically $S^2$)and identify two points on $S^2$ then ...
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### Using Laudenbach-Poénaru to justify Kirby diagrams

It is always referred as the "classical argument" that one only needs to specify the 1 and 2-handles in the Kirby diagram of a 4-manifold $X$ since by Laudenbach-Poénaru there is a unique ...
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### Small and large extra dimension(s) of the physical space

Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example. Consider a two dimensional manifold like $\mathbb{R}^2$ and we are trying to add a small and ...
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### Homeomorphism of open disks on a sphere

Are open disks defined on the unit sphere $S^2$ homeomorphic to open disks on $\mathbb{R}^2$? I know that the unit sphere is a 2D manifold, but that tells me the (seemingly weaker?) point that any ...
Suppose I have Heegard splitting $M^3 = S_g \bigcup_f S_g$, where $S_g$ are solib bodies of genus g. I know how to construct Morse function on $S_g$ , so my question is:How can i fing a Morse function ...