# Questions tagged [low-dimensional-topology]

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

674 questions
Filter by
Sorted by
Tagged with
75 views

• 119
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 ...
• 5,451
16 views

• 173
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 ...
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 ...
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 ...
• 286
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 ...
• 158
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 ...
• 2,005
1 vote
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 ...
• 21
1 vote
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 ...
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 ...
• 309
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....
• 327
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 ...
• 1,097
74 views

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, ...
• 141
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 ...
• 1,540
1 vote
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 ...
• 1,540
50 views

• 343
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 ...
• 139
1 vote
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 ...
• 11
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 ...
218 views

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. ...
1 vote
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 ...
• 55
1 vote
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 ...
1 vote
64 views

160 views

1 vote
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, ...
• 11
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 ...
• 1,206
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 ...
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)...