# Questions tagged [low-dimensional-topology]

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

566 questions
Filter by
Sorted by
Tagged with
9 views

### Is johnson-lindenstrauss lemma valid for a set of high dimensional complex vectors? [closed]

I apologize in advance if this is a simple question, but is it valid in the complex domain?
57 views

### The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
27 views

### Compressible torus in irreducible 3-manifold bounds a solid torus

In his 3-manifolds notes (page 11, item $(4)$), Hatcher shows that a 2-sided compressible torus $T$ in an irreducible 3-manifold $M$ either bounds a solid torus $S^1 \times D^2$ or is contained in a ...
23 views

1 vote
40 views

### Heegaard Floer homology of a genus two diagram of $S^3$

I am reading the introductory paper "Heegaard diagrams and holomorphic disks" by Ozsváth and Szabó (https://arxiv.org/abs/math/0403029v1, Section 2.2), and I do not understand one of the ...
1 vote
31 views

1 vote
19 views

### $s$-isotopy of braids is an equivalence relaiton

So, I have been reading Chapter 6 of "A Study of Braids" by Murasugi and Kurpita. The main goal of the chapter is to prove that many notions of equivalence of braids actually coincide. I ...
67 views

### Equivalence of triangulations and piecewise-linear triangulations in dimension $d\leq 4$

There are two different notions of triangulations for a manifold $\mathcal{M}$: A (simplicial) triangulation is an abstract simplicial complex $\Delta$ such that $\vert\Delta\vert\cong \mathcal{M}$. ...
52 views

### criterion to show manifold only admits trivial principal bundle

Suppose $M$ a $n-$ dimension manifold and $N$ a $n-$ dimension manifold with boundary. If there exist an embedding $N\to M$ and $M$ only admits trivial principal $G$ bundle. Can we show that $N$ only ...
94 views

### Quick proof clarification: Show 1 dimensional manifold is homeomorphic to $\mathbb{R}$ or a circle

I am trying to see it in the case where we just have to open sets $U,V$ that cover the connected manifold $M$, each of which is homeomorphic to the real line. I know that the answer depends on the ... 50 views

### 3-manifolds with one spherical boundary component

My question is the following: Can all compact, connected and orientable 3-dimensional topological manifolds $\mathcal{M}$ with connected boundary $\partial\mathcal{M}\cong S^{2}$ be obtained by ...
53 views

### Is the Alexander polynomial of the 2-cable of a knot zero?

I am trying to find a proof (or counterexample) of the following claim: the Alexander polynomial of the 2-cable of a knot (where the blackboard framing is considered) is zero. That is, the Alexander ...
66 views

### Intersection form of a homology manifold

Recall that for a closed orientable 4-manifold $M$ with fundamental class $M$, there is a intersection form $H^2(M)\times H^2(M)\to \Bbb Z$ defined by $(a,b)\mapsto \langle a\cup b, [M]\rangle$, where ...
54 views

### Prove that $\frac{H_1(\Sigma)}{[\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]} \cong H_1(Y)$

I am looking at Definition 2.11 in this paper: https://arxiv.org/pdf/math/0101206.pdf. In particular, given a Heegaard diagram $(\Sigma, \alpha, \beta)$ for a 3-manifold $Y$, how to prove the ...
61 views

### Homology of 4-manifold minus a small ball

Let $X$ be a simply connected compact 4-manifold without boundary. We know that $H_4(X)=\mathbb Z$. Intuitively, I can see why removing a small (open) 4-ball $D^4$ would kill the top homology, but I'm ...
72 views

### Explicit calculus of Hilbert-Smith conjecture

According to the answer of this related question, isometry actions on any manifold form a locally compact group and are closely related to Hilbert-Smith conjecture. As the $3$-dimensional case is ...
88 views

### How to recognize torus knots

I have a list of 185 knots with less than 15 crossings and I want to check which of them are torus knots. I know there are many invariants being able to exclude some of them being torus, but I was ...
1 vote
I am reading about essential surfaces and I know that $F\times \{pt\}$ is essential within the mapping torus of $F$. I also know there are essential tori in $F\times S^1$. Are there other broad ...