# 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 "...
• 26.5k

### 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 ...
• 60.6k

### 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 ...
• 9,785

### $4$-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 $\mathbb{Z}/3\mathbb{Z}$ ...
• 96.8k
Accepted

• 33.6k

### 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 ...
• 6,662

### 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 ...
• 115k
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 ...
• 1,176
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 ...
• 6,215
Accepted

• 157k

### Intuition behind Khinchin's constant

If you want a geometric explanation of the result, the existence of Khinchin's constant is a consequence of the erodicity property of the geodesic flow on the modular surface (more precisely, on the ...
• 92.2k

### Existence of a vector field with one singularity on a surface

The statement actually holds in any dimension! There are references to this appearing in Hopf and Alexandroff's Topologie (1935), but I'm sure it was known earlier. Claim. Every compact, connected, ...
• 5,963

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