# Tagged Questions

**2**

votes

**0**answers

98 views

### when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can prevent $\rho$ from being ...

**0**

votes

**0**answers

67 views

### moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc.
Fix a non-negative integer $g$ and consider
the space
...

**4**

votes

**1**answer

172 views

### Relation between Milnor ring and middle dimensional homology of hypersurface

I have suspected that the following is well-known:
If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ...

**2**

votes

**2**answers

277 views

### Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.
The first paper I am aware of that uses the ...

**9**

votes

**6**answers

1k views

### Is there a good notion of morphism between orbifolds?

Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable ...

**-2**

votes

**2**answers

221 views

### topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology.
I am not sure if the following two results are true, and where can I find the references:
(1) ...

**2**

votes

**1**answer

320 views

### looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e,
($J^1M=T^*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...

**9**

votes

**1**answer

497 views

### moduli spaces are kahler?

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see ...

**0**

votes

**0**answers

166 views

### Fractional degree of a map?

Is there some natural notion of a fractional degree of a map?
The degree of a map is a generalization of the winding number,
and fractional winding numbers appear in the (mathematical physics)
...

**3**

votes

**1**answer

266 views

### On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $.
What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...

**2**

votes

**1**answer

216 views

### Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?

**6**

votes

**3**answers

557 views

### Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$.
The ...

**7**

votes

**1**answer

253 views

### Generators for exact representations of 3-manifold groups

Does anyone have a list of matrix generators for a bunch of hyperbolic 3-manifold groups? I am testing an algorithm and am looking for a collection of test cases. I am looking for exact values of ...

**4**

votes

**0**answers

202 views

### Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...

**1**

vote

**1**answer

391 views

### Is surface $x^2+z^2=2\cdot y^2$ something of a Möbius strip?

This question is naive. My association with Möbius strip comes from not being able to smoothly extract positive solutions of the diophantine equation
$$x^2+z^2=2\cdot y^2$$
I got a parametrization ...

**0**

votes

**0**answers

151 views

### Casson's invariant and intersection homology

EDIT: Immediately after I wrote this question, I remembered the elegant paper "An intersection homology invariant for knots in a rational homology 3-sphere" by Frohman and Nicas, which I believe does ...

**1**

vote

**1**answer

90 views

### On the simply connectedness of Symmetric products and Hilbert schemes of points

My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$.
The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where ...

**10**

votes

**2**answers

1k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

**9**

votes

**2**answers

308 views

### Algebraic Stratifications of $G$-varieties

My question is simple:
Given an algebraic group $G$ acting on a variety $X$ algebraically. If the orbits are of finite number then they form what is called an algebraic stratification of $X$.
Now my ...

**1**

vote

**0**answers

65 views

### Complements to reducible plane projective curves

Hello,
Suppose that $C_1,C_2\subset\mathbb P^2$ are projective curves (over $\mathbb C$); $C_1$ and $C_2$ may be reducible but they must not have a common component. Let $L\subset \mathbb P^2$ be a ...

**11**

votes

**1**answer

487 views

### When are Brieskorn Manifolds Homeomorphic?

Let $a_0, \dots, a_n, b_0, \dots, b_n \in \mathbb{N}$ and consider two polynomials $f = \sum_{i = 0}^{n} z_{i}^{a_i}$, $g = \sum_{i = 0}^{n} z_{i}^{b_i}$. Given two Brieskorn manifolds $\Sigma(a_0, ...

**3**

votes

**1**answer

186 views

### Vanishing of !-restriction of constructible sheaves

If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about ...

**16**

votes

**2**answers

371 views

### A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface?

In a 1986 paper, Harer and Zagier proved the recursion:
$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$
where e(g,n) is the number of ways of grouping sides $S_1...S_{2n}$ of a 2n-gon ...

**1**

vote

**1**answer

166 views

### Relation between the Character variety of a knot $K\subset M$ and that of $M$

Suppose there is a knot $K\subset M$,$M$ is a closed 3-manifold. What's the relation between $\chi(\pi_{1}(M-K))$ and $\chi(\pi_{1}(M))$?
($\chi(G)$ means the $\text{SL}_{2}(C)$ representation of $G$ ...

**15**

votes

**1**answer

451 views

### Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$:
$$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$
...

**4**

votes

**1**answer

206 views

### Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action?

My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod_i S^{n_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I ...

**5**

votes

**2**answers

266 views

### Given the vertices of a convex polytope, How can we construct its Half-Space representation

HI,
I have a question regarding convex polytopes. Let us say I have the vertices of a polytope which I name as $ V = \{v_1,\cdots,v_k\}$. Each of the $v_k$ are n-dimensional vectors, i.e. ...

**14**

votes

**0**answers

597 views

### Horrible sets and blowups in Hubbard's Teichmuller theory

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...

**0**

votes

**1**answer

298 views

### K3 surfaces as double covers of P2

Consider a K3 surface given as the double branched cover over P^2, branched along a smooth sextic. What is the ramification divisor in the K3 surface?

**0**

votes

**1**answer

356 views

### A Question about SO(n)

My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...

**10**

votes

**4**answers

2k views

### Geometric invariant theory for geometers

I am trying to learn "Geometric invariant theory" like it was introduced by Mumford. But I do not have a strong background in algebraic geometry since I work in geometric topology and geometry.
So ...

**4**

votes

**1**answer

238 views

### braids and dynamics of roots of a polynomial

The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time ...

**7**

votes

**1**answer

403 views

### Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...

**5**

votes

**1**answer

183 views

### Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense.
Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...

**18**

votes

**2**answers

1k views

### Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?

I stumbled upon the fact that the Bolza surface can be obtained as the locus of the equation,
$y^2 = x^5-x$
Its automorphism group has the highest order for genus $2$, namely $48$. I recognized ...

**2**

votes

**2**answers

611 views

### One tetrahedron inside another tetrahedron

Consider two tetrahedrons one placed inside the other.
Prove that the sum of all 6 sides of inner tetrahedron is at most the sum of the 6 sides of exterior tetrahedron.

**15**

votes

**2**answers

646 views

### When does the blow-up of $CP^2$ at N points embed in $CP^4$?

Write $X_N$ for this blow up. Place the N points in 'general position' as needed. Then $X_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other ...

**4**

votes

**2**answers

311 views

### Twisted cohomology of the mapping class group

Let $M_{g,1}$ be the mapping class group of surfaces of genus g $\geq 1$ with one boundary component. By $S_g$ we denote a closed surface of genus $g$.
In the paper "Families of jacobian manifolds ...

**8**

votes

**3**answers

824 views

### Two rectangular parallelepiped

Prove that if we have two rectangular parallelepiped (cuboids) such that one of them is placed inside the other then the sum of the three lengths of the inner parallelepiped is at most the sum of the ...

**8**

votes

**2**answers

368 views

### Cohomology of representation varieties

Perhaps this question is too general then I am sorry about this.
My question is the following.
Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...

**7**

votes

**1**answer

561 views

### Manifolds and Polynomials

Given a compact smooth manifold $M \subset R^k$ there is a Polynom $f\in R[x_1,..x_n]$ such that the zero set of $f$ is diffeomorphic to $M$. Can the coefficients of $f$ be pertubated slightly to a ...

**13**

votes

**3**answers

1k views

### When are (finite) simplicial complexes (smooth) manifolds?

Hi,
is there an algorithm that determines if a given simplicial complex is
a.) a manifold
b.) a smooth manifold
c.) homotopy equivalent to a manifold
d.) a real algebraic variety
?

**14**

votes

**0**answers

414 views

### Do mapping classes have gonality?

(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)
The hyperelliptic mapping class group is (by ...

**12**

votes

**1**answer

548 views

### Are surface bundles over a surface with non-zero signature necessarily complex (or algebraic)?

By "surface bundle over a surface" I mean a compact, oriented 4-manifold $X$ which is the total space of an oriented fiber bundle $X\to B $ over an oriented 2-manifold $B$. Assume that the signature ...

**7**

votes

**1**answer

764 views

### Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?

The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...

**3**

votes

**0**answers

292 views

### What is the behaviour of a smooth 3-manifold acting by a circle?

As Mumford pointed out in his paper 'Topology of Normal Singularities and a Criterion for Simplicity'(1961), every point $p$ on a normal complex surface $V$ has an associated 3-manifold $M$ which is ...

**8**

votes

**3**answers

642 views

### orbifold covering

Given two compact surfaces $S_1$ and $S_2$ of genus at least $2,$ it is easy to tell when $S_1$ covers $S_2:$ whenever $\chi(S_2)$ divides $\chi(S_1).$ Now, suppose I have two orbifolds of negative ...

**4**

votes

**1**answer

446 views

### Orbifold Beauville-Siu

The Beauville-Siu theorem states that for a compact Kahler manifold the following two statements are equivalent:
$M$ admits a surjective holomorphic map with connected fibers to a compact Riemann ...

**1**

vote

**0**answers

138 views

### Generalized vector bundles with singularities on Riemann surfaces

Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve.
Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex ...

**5**

votes

**0**answers

493 views

### Computing the Chern class for a flat line bundle using the holonomy group?

Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From ...