**11**

votes

**1**answer

434 views

### Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$

I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi_1(M), \mathrm{SL}(n,{\mathbb C}))$$
It is known ...

**3**

votes

**1**answer

207 views

### When is an irreducible $\mathrm{SL}_2(\mathbb{C})$ representation of a cusped hyperbolic 3-manifold scheme reduced or smooth?

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in \mathrm{Hom}(\pi_1(M),\mathrm{SL}_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can ...

**0**

votes

**0**answers

69 views

### asymptotic behavior of minimum dilatations on punctured surfaces

Let $l_{g,n}$ be the logarithm of minimum dilatation for pseudo-Anosov homeomorphisms on surface of genus $g$ with $n$ punctures. Let $n$ be fixed and $g$ varies. Is the asymptotic behavior of ...

**5**

votes

**0**answers

69 views

### Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...

**1**

vote

**1**answer

90 views

### homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe:
$\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...

**125**

votes

**27**answers

54k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**2**

votes

**2**answers

244 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**7**

votes

**4**answers

824 views

### What is the “right” definition of the homology(cohomology) of an orbifold?

What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
...

**11**

votes

**1**answer

230 views

### Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure?

It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every ...

**19**

votes

**2**answers

2k views

### First Chern class of a flat line bundle

A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?
Let $X$ be a nice space ...

**5**

votes

**1**answer

324 views

### Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...

**9**

votes

**1**answer

171 views

### Complements of unknotted tori (higher dimensions)

It is well-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori.
It is also known that an unknotted 3-torus in ...

**-3**

votes

**0**answers

64 views

### Volume of a scaled ball [closed]

Let $B \subset \mathbb{R}^n$ be the unit ball with respect to an arbitrary norm $\|.\|$ (e.g. $B=\{x \in \mathbb{R}^n:\|x\| \le 1 \}$).
I read in a book that it is easy to show:
$vol_n(\epsilon ...

**14**

votes

**1**answer

294 views

### Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...

**28**

votes

**1**answer

721 views

### Open immersions of open manifolds

For concreteness, I will work in the category of smooth manifolds, but my question makes sense in topological and PL category as well. Recall that a manifold $M$ is called open if every connected ...

**3**

votes

**1**answer

82 views

### Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes

The conjecture is something like the following:
The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a ...

**6**

votes

**2**answers

532 views

### Geometric or topological results from group theory

Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:
...

**10**

votes

**1**answer

180 views

### Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$).
Let $x \in \Sigma$, and suppose you have the following: for every $r<1$,
the open ...

**8**

votes

**2**answers

424 views

### Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...

**1**

vote

**0**answers

43 views

### Change of length of curve when Fenchel-Nielsen length coordinate increase

Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...

**4**

votes

**2**answers

146 views

### Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise.
Does anyone know of an example of a large ...

**0**

votes

**1**answer

60 views

### Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...

**2**

votes

**1**answer

159 views

### Prove that a metric space is intrinsic

Let $(X,d)$ be a general locally compact metric space (in particular not a Riemannian manifold). Suppose we don't know if $(X,d)$ is complete. To prove $(X,d)$ is intrinsic. I have to compute the ...

**5**

votes

**1**answer

429 views

### TQFT and Mapping Class Groups

It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference ...

**4**

votes

**0**answers

65 views

### Points of failure in definition of X- and A-moduli spaces for arbitrary G

In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...

**1**

vote

**1**answer

205 views

### Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...

**3**

votes

**1**answer

219 views

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

Suppose there is a knot $K\subset M$, where $M$ is a closed 3-manifold. What's the relation between $\chi(\pi_{1}(M-K))$ and $\chi(\pi_{1}(M))$?
Note: $\chi(G)$ means the ...

**1**

vote

**0**answers

85 views

### Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...

**9**

votes

**1**answer

436 views

### What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...

**7**

votes

**2**answers

208 views

### Heegard genus of hyperbolic Haken 3-manifolds

Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?

**2**

votes

**1**answer

77 views

### Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...

**4**

votes

**1**answer

90 views

### Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...

**8**

votes

**0**answers

157 views

### Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...

**9**

votes

**2**answers

175 views

### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold.
It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ?
...

**6**

votes

**1**answer

242 views

### “structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration".
Does it make sense to talk about "structure ...

**15**

votes

**3**answers

921 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 ...

**21**

votes

**1**answer

303 views

### Is the normal bundle of a torus trivial?

Question:
Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial?
What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the ...

**30**

votes

**4**answers

1k views

### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...

**4**

votes

**2**answers

311 views

### Are homotopy braid groups residually nilpotent?

A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...

**0**

votes

**0**answers

20 views

### Extending a symplectomorphism of a surface with boundary, properly embedded in a 4-disk, to the ambient space

Let $S$ be a surface with boundary that is properly embedded, via a map $f$
say, inside a $4$-disk. When is the embedding isotopic to the identity? --
All embeddings and isotopes are in the symplectic ...

**3**

votes

**0**answers

99 views

### Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...

**5**

votes

**2**answers

117 views

### Immersed quasi-Fuchsian surfaces surviving Dehn fillings

In papers like, Cooper - Long - Some surface subgroups survive surgery or Li - Immersed essential surfaces in hyperbolic 3-manifolds the game is to find some quasi-Fuchsian immersed surface $Q ...

**20**

votes

**2**answers

541 views

### Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity?
Of course, such a manifold must not admit a diffeomorphism ...

**3**

votes

**1**answer

101 views

### What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?

There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology.
What is the (Heegaard) Floer homology of a ...

**8**

votes

**1**answer

184 views

### Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...

**3**

votes

**1**answer

126 views

### Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties:
1) $M_g$ is an ...

**4**

votes

**1**answer

156 views

### contact surgery diagram on Brieskorn manifolds

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...

**3**

votes

**1**answer

186 views

### Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...

**5**

votes

**1**answer

239 views

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

**5**

votes

**1**answer

104 views

### Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt.
At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is ...