**2**

votes

**1**answer

76 views

### Quasiconformal deformation

Given a finite set $A$ on the Riemann sphere and a homeomorphism $f$, may I say there exists a quasiconformal homeomorfism isotopic to $f$ relative to the set $A$?

**1**

vote

**1**answer

88 views

### Sutured Manifolds and minimal genus

Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share.
In other matters, do we know how to ...

**3**

votes

**1**answer

135 views

### Parity of knot signatures

I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally ...

**5**

votes

**0**answers

89 views

### Distribution of random Projections on the Stiefel Manifold

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is:
$\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar ...

**12**

votes

**2**answers

915 views

### Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...

**20**

votes

**1**answer

498 views

### good covers of manifolds

It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the ...

**2**

votes

**1**answer

150 views

### homology of punctured manifolds

Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have
$$H_k(M - x, \mathbb{Z}) \cong H_k(M, \mathbb{Z})$$
for $k<n-1$. For ...

**3**

votes

**2**answers

150 views

### Circle Bundles of surfaces

Let S be a surface with a metric of constant curvature and finite area. Is there a classification of the circle bundles over S?

**4**

votes

**2**answers

311 views

### a question on rank of fundamental group

Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let
$G_{1} = \langle a_{1},...,a_{k} \rangle$ be a free subgroup of $G$, and let $G_{2}=\langle a_{k+1} \rangle$ be a ...

**2**

votes

**0**answers

40 views

### Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state:
Any reduced diagram of an alternating link has the fewest possible crossings.
Any two reduced ...

**10**

votes

**1**answer

317 views

### Finitely presented group in which every element is conjugate to its square

Does there exist a nontrivial finitely presented group in which every element is conjugate to its square? Is this an open problem?
Motivation: Jahren proved in [Geom Dedicata (2010)] that if $M$ is a ...

**2**

votes

**2**answers

109 views

### linking number and covering

Recently I read Dale Rolfsen's paper –A surgical view of Alexander’s polynomial. This is a good paper. But there is a lemma which I don’t know how to prove.The lemma is following:
Lemma:In the ...

**6**

votes

**0**answers

153 views

### Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...

**2**

votes

**2**answers

209 views

### manifolds with unusual rational cohomology rings

I'm looking for examples of 3-manifolds with unusual rational cohomology rings. I'm curious about what the cup product structure can actually look like, and I'd like some examples to play with. Does ...

**9**

votes

**2**answers

518 views

### The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group

My question is about the Alexander polynomial of a slice knot.
For a slice knot $K$,
Fox-Millnor and Terasaka proved that
$$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$
for some polynomial $f(t) \in ...

**7**

votes

**1**answer

129 views

### Relation between the Alexander module of a link and intermediate free abelian covers

I'm working through McMullen's paper "The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology" and have a question concerning the following setup:
Given a link complement $(X, ...

**4**

votes

**2**answers

590 views

### A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...

**2**

votes

**1**answer

194 views

### The behaviour of holomorphic mapping of curves

Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ ...

**1**

vote

**0**answers

49 views

### Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometry

Let $M$ be an open Riemannian surface of bounded geometry. Let $\Gamma$ be a group of diffeomorphisms of $M$. Suppose that $\Gamma$ is equi-quasi-isometric; i.e., its elements are (differentiable) ...

**1**

vote

**0**answers

100 views

### Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...

**4**

votes

**1**answer

114 views

### Rationality of translation lengths in hyperbolic groups

Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...

**7**

votes

**1**answer

256 views

### Casson invariant

Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does ...

**7**

votes

**2**answers

221 views

### Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that
there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
His main technique of ...

**9**

votes

**2**answers

380 views

### Vector bundle for prescribed Stiefel-Whitney classes

I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...

**0**

votes

**0**answers

116 views

### Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there.
For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...

**2**

votes

**1**answer

221 views

### Meaning of “ Open Book cannot be Stabilized Further”?

I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...

**1**

vote

**0**answers

123 views

### The relations between some 3-components links and trefoil knots [closed]

It is intuitive to see that the 3-components links (under Alexander–Briggs notations) $6^3_1, 6^3_2, 6^3_3$ are closely related to each other; in a sense by doing a cut-gluing or sew-gluing surgery, ...

**9**

votes

**0**answers

129 views

### When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...

**3**

votes

**0**answers

102 views

### Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...

**9**

votes

**1**answer

245 views

### Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$

Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties:
P1: $M$ can ...

**4**

votes

**1**answer

294 views

### What are the low-degree group cohomology of the mapping class group of a surface

Let $MCG_g$ be the mapping class group of genus $g$ closed surface.
(Say $MCG_1=SL(2,Z)$).
I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as ...

**5**

votes

**1**answer

281 views

### Finite group acting on sphere

Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?

**3**

votes

**1**answer

126 views

### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...

**6**

votes

**2**answers

125 views

### On trivalent spines of surfaces

Let $\Sigma$ be a surface with non-empty boundary labelled by finitely many points $x_i$.
For our purposes a spine $s$ for $\Sigma$ is a uni-trivalent graph embedded in $\Sigma$ whose 1-vertices are ...

**1**

vote

**0**answers

163 views

### Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...

**8**

votes

**2**answers

454 views

### Covering the space by disjoint unit circles

Sierpinski has proved the following two interesting theorems.
Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate disjoint circles.
Theorem 2. The Euclidean space ...

**11**

votes

**1**answer

292 views

### Are the mapping class groups of manifolds finitely presentable?

The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving ...

**14**

votes

**2**answers

586 views

### Low dimensional topological manifolds [duplicate]

There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: ...

**3**

votes

**1**answer

156 views

### Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures

Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...

**8**

votes

**1**answer

149 views

### Hakenness of Heegard splitting

This is somewhat related to this previous quesiton. Suppose I give you a Heegard splitting of $M^3$ of genus $g$ with a gluing map $\phi.$ Is there some condition on $\phi$ which would guarantee that ...

**6**

votes

**0**answers

167 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**8**

votes

**1**answer

303 views

### Must a closed totally path-disconnected subset of the sphere have connected complement?

This question (which is more a curiosity than a research problem) originates from these two:
...

**3**

votes

**0**answers

212 views

### Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...

**6**

votes

**1**answer

597 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

**2**

votes

**1**answer

183 views

### Surface curves equidistant from a simple closed geodesic

Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is ...

**0**

votes

**0**answers

60 views

### Suitable references for Zappa-Szep products of groups

Can anyone provide references for the following: isoperimetric (Dehn) functions of semi-direct products of groups and, more generally, of Zappa-Szep products of groups?

**0**

votes

**0**answers

29 views

### definition of an avatar of a tree-automorphism group

Would anyone be able to show me where I can read about what the avatar of a tree-automorphism group is? This is so that I can understand Pierre-Emmanuel Caprace's article "Simple locally compact ...

**5**

votes

**1**answer

155 views

### Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...

**2**

votes

**0**answers

145 views

### Canonical trivialization of Stiefel-Whitney classes on the frame bundle

Let $X$ be a smooth manifold, perhaps oriented if necessary. The frame bundle $\pi:P \to X$ carries a canonical trivialization of the pullback of the tangent bundle of $X$ and thus a canonical ...

**2**

votes

**0**answers

120 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 ...