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 ...
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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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Compute the cohomology of the n dimensional torus with disjoint circles removed
I'm particularly interested in the case of $n=3$ but the most general formulation of the problem I'm trying to tackle is the following:
Let $\mathbf{T}^n$ be the n-dimensional torus and $W=L_1\sqcup\...
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Visualize the lens space $L(1,1)\cong S^3$
I know $L(p,q)\cong L(p,q-np)$ for all $n\in \mathbb{Z}$, so in particular $L(1,1)\cong L(1,0)\cong S^3$. It is easy to visualize $L(1,0)\cong S^3$ since the complement of an unknotted solid torus in $...
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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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Can the connected sum of three copies of $\mathbb{RP}^3$ be the total space of a fiber bundle?
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 $\operatorname{cGraff}...
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$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, ...
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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 ...
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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 ...
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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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Classification of links made of rigid circles
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 ...
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Is $K_1\# K_2$ isotopic to $K_1\#\overline{K_2}$?
Let $K_1$ and $K_2$ be two oriented knots (with fixed embeddings in $S^3$). Then we have a well-defined oriented connect sum $K_1\#K_2$. We can also take $K_2$ with the opposite orientation to form $...
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The uniqueness of minimal genus Seifert surfaces for knots in $S^3$.
I was reading some materials about knots, some procedures inspired me to ask this question. Given a knot $K$ in $S^3$, one can use Seifert's algorithm to obtain a surface in $S^3$ whose boundary is $K$...
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Is self homotopy equivalence close to self homeomorphism?
It is known that self homotopy equivalence of closed surfaces is homotopic to a self homeomorphism. I wonder whether this statement holds for all closed manifolds. Note that no smoothness of the self ...
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Reference request for books that cover topics similar to Rolfsen's Knots and Links but are more detailed.
I have been told to study the mapping class group of the torus for a summer project from Dale Rolfsen's Knots and Links (Chapter 2-D) in preparation of studying 3-manifolds. However Rolfsen proof of ...
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Existence of normal crossing of knot diagram via Mather's stable mapping theory
Using Mather's theorem on stable mappings/using multijet transversality, we know that the set of immersion with normal crossing $\mathcal M\subseteq C^\infty(S^1,\mathbb R^2)$ is dense and open in the ...
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Can't remember about homotopy equivalences of surface with boundary.
Suppose that $F$ and $G$ are compact surfaces and $h:F\rightarrow G$ is a homotopy equivalence. I believe that there is a theorem that gives a standard form for homotopy equivalences, but I don't ...
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is this set already closed?
Let the set
$D=\{(x, y)\in\mathbb{R}^2|0 < x^2 + y^2 \leq 4\}$
is this set closed?
I know that this set is closed if $\mathbb{R}^2\backslash D$ is open.
The set $\mathbb{R}^2\backslash D=\{0\}\cup \...
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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 ...
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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 ...
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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 ...
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Stabilization of Heegaard splitting
So I'm trying to understand the stabilization operation which is to obtain a Heegaard splitting of closed orientable $3$-manifold $M$ of genus $g+1$ from genus $g$.
Given a Heegaard splitting $M = H_g\...
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Any closed orientable $3$-manifold admits a Heegaard splitting
Textbook: Lectures on the topology of 3-manifolds by Nikolai Saveliev
Theorem. Any closed orientable 3-manifold admits a Heegaard splitting.
Proof. Let $T$ be a triangulation of a closed orientable $...
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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 ...
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Morse function "induced" by Heegard splitting
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 ...
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How to find the described circle?
Given a linearly connected domain $\Omega$ homeomorphic to a $2D$ disc $D^2$ with a simple regular ($C^1$ and nowhere vanishing derivative) boundary $\partial\Omega$, find such a circle $\omega(\rho, ...
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Invariance of oriented intersections on surface with boundary
Suppose that we have two smooth, simple closed curves $\alpha, \beta : S^{1} \rightarrow \Sigma$ in a closed, connected, oriented surface $\Sigma$, and suppose that $\alpha \cap \beta$ consists of ...
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