Skip to main content

Questions tagged [low-dimensional-topology]

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

Filter by
Sorted by
Tagged with
4 votes
1 answer
75 views

Equivalence of two strengthenings of open neighborhood

I'm currently pursuing a project (about Markov conditions in Minkowski spacetime) in which a certain equivalence between two strengthenings of open neighborhood plays an important role. Say that set $...
Jens's user avatar
  • 119
1 vote
1 answer
76 views

Approximating an arbitrary set in $\mathbb{R}^n$ with countable unions of cubes?

Let a neighborhood of a set $R\subseteq \mathbb{R}^n$ be any $U\subseteq \mathbb{R}^n$ such that $\bar{R}\subseteq \langle U\rangle$, i.e. such that the closure of $R$ is a subset of the interior of $...
Jens's user avatar
  • 119
0 votes
0 answers
23 views

Braid form of general link embedded in a genus $g$ surface?

It is known that a general braid word for a $T(m, n)$ torus link is $(\sigma_1\sigma_2\cdots\sigma_{n−1})^m$; for example, see here. Does anyone know if there is a braid form for a general link ...
Zuriel's user avatar
  • 5,451
0 votes
0 answers
16 views

How to consider M\\S as a sutured manifold?

I'm reading guts in sutured decompositions and the thurston norm by Agol and Zhang. Now $M$ is a compact 3-manifold and $S$ is a properly norm-minimizing surface in M. $M\backslash\backslash S$ means $...
lena11's user avatar
  • 1
0 votes
0 answers
30 views

Equivalence between two definitions of a Seifert Fibered Homology Sphere

I am reading Savaliev's Invariants for Homology 3-spheres. Here, he defines the Seifert Fibered Homolgy Sphere as $\Sigma(a_1,...,a_n) = V_B(a_1,...,a_n) \cap S^{2n-1}$ where $V_B(a_1,...,a_n) = \{b_{...
user13121312's user avatar
0 votes
0 answers
60 views

Why can we choose lifts $\tilde{\alpha}$ and $\tilde{\beta}$ of $\alpha$ and $\beta$ that have the same endpoints in $\partial\mathbb{H}^2$?

My question refers to the proof given in http://euclid.nmu.edu/~joshthom/Teaching/MA589/farbmarg.pdf, page 34. Proposition 1.10 states the following: Let $\alpha$ and $\beta$ be two essential simple ...
olchew's user avatar
  • 3
2 votes
0 answers
57 views

Open neighborhood of a geometric knot in an orientable 3-dimensional manifold.

I was reading "Braid Groups" by Christian Kassel and Vladimir Turaev where I found the following question: Prove that an arbitrary geometric knot $L$ in an orientable 3-dimensional manifold ...
ripan sharma's user avatar
0 votes
0 answers
53 views

Slicing and gluing topological objects : a Klein bottle circle dance

Consider this Klein bottle model Slicing the model with a horizontal plane from top to bottom gives this beautiful circle dance (also reminiscent of this MRI scan). Hurray! As far as I know, this is ...
vallev's user avatar
  • 286
0 votes
0 answers
26 views

Generalizing Reidemeister-Singer Theorem to manifolds with boundary

I was studying some Heegaard splittings theory for a course; we saw the Reidemeister-Singer theorem, which states that for a closed, connected 3-manifold $M$, any two Heegaard splittings share a same ...
Nennee's user avatar
  • 158
3 votes
1 answer
74 views

Lie group structure on exotic $\mathbb{R}^4$

Are there Lie group structures on exotic $\mathbb{R}^4$s? By the theorem that every continuous group homomorphism of two Lie groups is smooth, we can conclude that if $G$ and $H$ are two Lie groups ...
Strichcoder's user avatar
  • 2,005
1 vote
0 answers
29 views

What are some books recomended for learning low dimensional topology as a begginner? [duplicate]

What are some books recomended for learning low dimensional topology as a begginner? I would like to learn low dimensional topology, as I am good at general topology. What are some advice could you ...
Ahmed's user avatar
  • 21
1 vote
2 answers
55 views

Do increasing the number of variables in a function leads us to new dimensions in space.

Let $f(x) = x^2$ then $y=x^2$ here the values of $x$ gives the solution of $y$ in $Y$-axis means the graph so forming will be in second dimension. Similarly if $f(x,y) = x^2+y^2$ then $z=x^2+y^2$ here ...
Rigg Verma's user avatar
0 votes
0 answers
28 views

Intersection number and linking number of arcs

Part 1: Let S be a surface with boundary (we assume everything is oriented and compact if necessary). It's classical that we can define the intersection number of any two closed curves. Now I want to ...
Chard's user avatar
  • 309
0 votes
1 answer
49 views

Understanding the proof of Fenn-Rourke Theorem

Fenn-Rourke Theorem states that Framed links can be transformed into each other by Kirby moves if and only if they can be done by Fenn- Rourke moves. I'm trying to understand the proof of it in V.V....
user540663's user avatar
3 votes
2 answers
152 views

Open sets on a surface with locally connected boundary

Let $\Sigma$ be a surface and $\Omega$ be an open subset of $\Sigma$. Suppose that $\Omega$ is homeomorphic to the open unit disk $\mathbb{D}$ and is relatively compact in $\Sigma$. I'm interested in ...
Dilemian's user avatar
  • 1,097
5 votes
0 answers
74 views

Intuition about r-spin structures

I should preface this by saying that I am a physicist. My question pertains to the paper https://arxiv.org/abs/1802.09978, where $r$-spin structures are defined (Def. 2.1). To orient the discussion, ...
johnny's user avatar
  • 141
2 votes
1 answer
67 views

Different statements about the peripheral system as a complete knot invariant

I am somewhat confused about the different flavours in which the statement "the peripheral (group) system is a complete knot invariant" usually comes, and I believe not all of them have ...
Minkowski's user avatar
  • 1,540
1 vote
0 answers
27 views

Peripheral subgroup determined up to conjugation

Let $K$ be an oriented knot in $S^3$, let $X_K := S^3 - \mathrm{int} \ N(K)$ be its knot exterior and let $i: \partial X_K \hookrightarrow X_K$ be the subspace inclusion. The peripheral subgroup is ...
Minkowski's user avatar
  • 1,540
2 votes
1 answer
50 views

Does a torus knot give a Seifert fibering of the 3-sphere?

Let $K$ be a $(p,q)$ torus knot on the torus $T_1$. Via the map \begin{equation*} H=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} \end{equation*} $K$ becomes a $(q,p)$ torus knot on $...
Hempelicious's user avatar
0 votes
0 answers
50 views

Show that the two-fold cyclic branched cover of a knot $K $ with trivial Alexander polynomial is a homology sphere.

I meet this when reading a paper about cyclic branched covering spaces of knots. There is no explanation nearby, can anyone tell me how to understand this?
Tsoshamry's user avatar
  • 316
0 votes
0 answers
18 views

Visualising the interior gluings of a 3D shape in 2D

I have a small triangulation of a 3-ball that I'm trying to form a nice 'visualisation' of for a paper/talk. The best I've got so far is a few rough sketches like the one below, where I've tried to ...
Finn T's user avatar
  • 83
1 vote
0 answers
82 views

Can an isotopy between diffeomorphisms be through diffeomorphisms?

Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth (say $C^{\infty}$) isotopy between two diffeomorphisms $f_0$ and $f_1$ of M in the sense that $\...
Skyskie's user avatar
  • 343
5 votes
2 answers
159 views

Understanding $\mathbb{C}P^2$

I am trying to understand $\mathbb{C}P^2$. Since I understand the Hopf fibration quite well, I like the following construction: Attach a $\mathbb{D}^2$ (2-cell) to a point $\mathbb{D}^0$ (0-cell) to ...
Thomas's user avatar
  • 139
1 vote
0 answers
43 views

$L(p,q)$ diffeomorphic to $L(p,q+np)$

I'm wanting to show the two lens spaces are diffeomorphic using a Rolfsen twist. I know under this Kirby move, the resulting framing co-efficient of $\frac{p}{q}$ surgery is $\frac{p}{q+np}$, so I ...
Bugzy's user avatar
  • 11
2 votes
2 answers
113 views

Why do $2g+1$ distinct closed curves separate a compact orientable surface of genus $g$?

The genus of a surface is the maximum number of pairwise disjoint simple closed curves that do not separate a surface. Why do $2g+1$ distinct closed curves separate a compact orientable surface of ...
Fernando Oliveira's user avatar
2 votes
1 answer
218 views

Lifting maps of Sphere to the Torus

If on the square $S=\left[-\frac{1}{2},\frac{1}{2}\right]\times\left[-\frac{1}{2},\frac{1}{2}\right]$ we identify the points $\left(-\frac{1}{2},y\right)\sim\left(\frac{1}{2},y\right)$ for every $y\...
Bernardo Carvalho's user avatar
2 votes
1 answer
61 views

Deducing inequality from exact triangle in Heegaard Floer homology

In Hom's lectures on Heegaard Floer homology, pages 8 and 9 contain a proof that $rk \widehat{HF}(Y) \geq |H_1(Y, \mathbb{Z})|$ for rational homology spheres. The proof involves using an exact ...
horned-sphere's user avatar
1 vote
0 answers
54 views

Etymology of the term "stabilised" for Heegaard splittings

A Heegaard splitting of a 3-manifold $ M = H \cup H' $ is stabilised if there exist essential discs in $ H $ and $ H' $ whose boundary curves intersect transversely at a single point. Equivalently, a ...
Alex Elzenaar's user avatar
2 votes
1 answer
56 views

2-bridge knot with straightened strand

Apparently, every 2-bridge knot can be drawn such that of the four strands in the braid word, one strand remains straightened and is not crossing any of the other strands. Is there a general algorithm ...
Philippe Knecht's user avatar
0 votes
1 answer
59 views

Is this a sound technique for visualizing four dimensions?

A four dimensional space, as I understand it, is just a set of points each defined by four numbers. It's easy to visualize the first three, so then I just imagine the last one as the color of the ...
Rincewind's user avatar
1 vote
0 answers
35 views

Torsion spin-c structures on $S^1\times S^2$

I have been reading the paper https://arxiv.org/pdf/1902.04050.pdf by Zemke and at some point we have the following: My question is, how can one make sense of torsion $\mathrm{Spin}^c$ structures on $...
horned-sphere's user avatar
0 votes
0 answers
42 views

References for Kleinian Groups

I am keen in learning hyperbolic manifolds and Kleinian groups. Can anyone please suggest me references for Kleinian groups? I know (I maybe wrong) that there are four faces of Kleinian groups: 1. ...
user1180312's user avatar
1 vote
1 answer
113 views

Fundamental group of $4$ spheres pairwise tangent

I intuitively got that the fundamental group of the topological space given by considering $4$ spheres pairwise tangent in $ \mathbb{R}^3$ is $\langle a,b,c\rangle$. My approach was that if you ...
ccnptr's user avatar
  • 55
1 vote
1 answer
99 views

proving $\mathbb{Q}^2$ is regular space in separately open topology.

I am studying the proof of theorem 5.5, regularity of the topology of separate continuity We are familiar with Euclidean topology on $\mathbb{R}^2$, in which open sets are defined with respect to the ...
Ashutosh Shinde's user avatar
1 vote
1 answer
64 views

Three-sphere as a sphere bundle over circle

When we view $S^3$ as a subset of $\mathbb{C}^2$ through the equation $|z_1|^2+|z_2|^2=1$, the Hopf fibration $S^1\to S^3\xrightarrow{p} S^2$ is usually given by (see, e.g., Wikipedia) $$ p(z_1,z_2)=(...
sam's user avatar
  • 53
1 vote
0 answers
57 views

$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$

How does one prove the following statement : $$E(n)\# \mathbb{CP}^2=2n\mathbb{CP}^2\# (10n-1)\overline{\mathbb{CP}^2}$$ Here $E(n)$ denotes the $n$-th elliptic surface formed by fiber summing $E(1)$ $...
Chanel Rose's user avatar
3 votes
1 answer
179 views

What is the algorithmic complexity of knot equivalence?

Question. Given two (tame) knots by their link diagrams, what is the algorithmic complexity (e.g. time in the size needed to store the link diagrams) to decide if the represented links are isotopic? ...
Thomas Preu's user avatar
  • 2,002
6 votes
1 answer
156 views

Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus? Attempting to google for this I found this example of non-isotopic ...
Sidharth Ghoshal's user avatar
0 votes
0 answers
18 views

After cutting along a maximal collection of essential annuli and disks, does there exist such surfaces?

Let $M$ be an irreducible oriented compact 3-manfold. Take a maximal collection of disjoint non-parallel essential disks and essential annuli (in fact this collection is finite). Then cut $M$ along ...
Chard's user avatar
  • 309
0 votes
0 answers
29 views

Hyperbolization theorem for closed, reducible manifolds

Ashenbrenner et al. state the Hyperbolisation theorem as follows Let $N$ be a compact, orientable, irreducible $3$-manifold with empty or toroidal boundary. If $N$ is atoroidal and $π_1(N)$ is ...
Dinisaur's user avatar
  • 1,085
3 votes
1 answer
71 views

Understanding 3-manifold retraction to graph

I am reading On Fibering Certain 3-Manifolds by Stallings. It's a short paper, and I think I understand at a high level what's happening, but there are a few technical details I don't quite get. This ...
Hempelicious's user avatar
1 vote
1 answer
100 views

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 ...
one potato two potato's user avatar
2 votes
0 answers
77 views

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\...
hitoriboku's user avatar
0 votes
0 answers
160 views

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 $...
Chard's user avatar
  • 309
0 votes
1 answer
58 views

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 ...
one potato two potato's user avatar
2 votes
1 answer
221 views

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}...
Dimpi Paul's user avatar
1 vote
0 answers
44 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, ...
Yi Du's user avatar
  • 11
5 votes
1 answer
122 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 ...
ZSMJ's user avatar
  • 1,206
3 votes
1 answer
125 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 ...
one potato two potato's user avatar
1 vote
0 answers
71 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)...
one potato two potato's user avatar

1
2 3 4 5
14