# Tag Info

Accepted

### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

I don't have a precise answer, but the genus of $S$ has to grow like $n!$. To see this, note that when $n$ is large enough, $S$ cannot be the sphere or torus. So $S$ admits hyperbolic metrics. By ...
• 27.3k
Accepted

### Connectedness of Milnor fiber

I am just posting my comment as one answer. For every complex polynomial $f(z_1,\dots,z_n)$, denote by $f^*$ the polynomial map, f^*:\mathbb{C}^n\setminus \text{Zero}(f) \to \mathbb{C}\setminus \{0\...

### Is it possible to fill a boundary component of an irreducible 3-manifold using a handlebody so that the resulting manifold is still irreducible?

EDIT: Here is a substantial rewrite of my previous (incomplete) answer. I think that this proof is a bit "heavy", but I haven't yet thought of a better approach. The answer is "yes&...
• 27.3k
Accepted

### What are the covering spaces of $\mathbb{Q}$?

Yes. For a cardinal $c$ let $V_c\subset X$ be the set of all $x$ such that $c$ is the cardinality of $p^{-1}(x)$. So $X$ is the union of some set of disjoint nonempty open sets $V_c$ such that when ...
• 55.2k
Accepted

### When is the action of a mapping class group on the set of punctures realized by a finite subgroup of mapping classes?

I don't think what you're asking for is true. Namely, I think the sequence does split for $S_3$ when $g\equiv 2$ mod $3$. For example, if $g=2$ then you can make a surface that looks like three tubes ...
• 14.5k

### Equivalence of knotted spheres in $S^4$

$\DeclareMathOperator{\Diff}{Diff}$ The answer is "yes modulo some small potatoes". There is one case where the answer is a simple no: if $K$ and $K'$ are mirror images of each other you can ...
• 43.6k