10
votes

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 ...

8
votes

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\...

Community wiki

6
votes

### 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&...

6
votes

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 ...

6
votes

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 ...

5
votes

### 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 ...

5
votes

### $\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducible 3-manifold produces a reducible 3-manifold

Suppose that $S$, the relevant boundary component of $M$, is a torus. Suppose that $G$ is the given essential two-sphere in the filled manifold $N$. We isotope $G$ to have minimal intersection with $...

3
votes

Accepted

### Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?

This is also proved by Wilkes in Corollary 6.20.

2
votes

### A stable splitting of linear surjections

Greg Arone has a very general stable splitting result using Weiss calculus in PAMS, vol 129 (2000), pp 1207-1211. Take a look and see if your example fits.

2
votes

Accepted

### Proving the Cork Theorem

The proof starts with an $h$-cobordism $W:M_0\to M_1$ between two manifolds $M_0, M_1$ which are not diffeomorphic and produces a cork embedded in $M_0$ such that the cork twist yields $M_1$.
I'll ...

1
vote

Accepted

### Is there a Dehn-like presentation of a knot quandle?

Reposting an answer from a deleted account:
Yes, there is an analogous Dehn-like presentation for knot quandles where generators correspond to regions of a knot diagram instead of arcs. This ...

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