# Tag Info

Accepted

### Why can't Antoine's necklace fall apart?

if we remove a segment of one of the tori the object still cannot be separated by a sphere, and yet can fall apart macroscopically. Well, sure it can. The key observation is that "sphere" means "...

### Intuition behind Khinchin's constant

Here is what I think of as the standard explanation. I learned this from The Art of Computer Programming, Section 4.5.3. Start with some number $z$ whose continued fraction we want to find. I'll ...

### What are some applications of these specific pure math areas?

In most fields of pure mathematics, there aren't a lot of industry options that rely directly on your expertise or involve continuing with your research program. Instead there are jobs which use the ...

### $4$-manifold with fundamental group $\Bbb Z/4\Bbb Z$

As pointed out, every finitely presented group arises as the fundamental group of a closed smooth orientable four-manifold. The same is not true of non-orientable manifolds as $\mathbb{Z}/3\mathbb{Z}$ ...
Accepted

### What's true in $\mathbb{R}^4$, false in $\mathbb{R}^3$ and uninteresting in $\mathbb{R}^5$?

Lets bump knot theory up a dimension. In general, an $n$-sphere can be non-trivially knotted in $\mathbb{R}^{n+2}$. Obviously, an $n$-sphere can be embedded in $\mathbb{R}^{n+1}$, where it is ...

### Fundamental group of the total space of an oriented $S^1$ fiber bundle over $T^2$

Let me use $T$ to denote the torus. Oriented circle bundles over $T$ (indeed over any closed, connected surface) are classified up to orientation preserving bundle isomorphism by an integer known as ...
Accepted

### Fundamental group of the total space of an oriented $S^1$ fiber bundle over $T^2$

Here's a geometric view point which computes (a presentation of) $\pi_1(M)$. As a byproduct, we establish The group $\pi_1(M)$ is nilpotent of at most $2$-steps. That is, $[\pi_1(M),\pi_1(M)]$ need ...

### Prove by elementary methods: the plane cannot be covered by countably many copies of the letter "Y"

An idea that occured to me: Let there be some countable covering $G$ of the plane through "Y letters". Pick some circle $S$ in the plane. Clearly $G$ covers $S$. But each "Y letter" has at most 6 ...
Accepted

### What closed 3-manifolds have fundamental group $\Bbb Z$?

The first argument of your proof sounded a little familiar to me, and I realized I saw it in Hempel. So for a maybe more citeable reference and also some other infinite non-$\mathbb Z$ examples, let ...
Accepted