**21**

votes

**1**answer

955 views

### Homeomorphism historically: When did it reach its modern formulation?

Q. When did the notion of homeomorphism reach its
modern formulation as a bicontinuous bijection, i.e., a
continuous bijection
between topological spaces whose inverse is also continuous?
...

**3**

votes

**1**answer

134 views

### Chern-Simons invariants of 2-bridge knots

2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:
Looking at ...

**10**

votes

**1**answer

366 views

### Pseudomanifolds and Poincaré duality

1) A $n$-dimensional homology manifold is a topological space $X$ such that for any $x\in X$, the homology groups
$$H_p(X,X-x,\mathbb{Z})$$
are trivial unless $p=n$ where
$$H_n(X,X-x,\mathbb{Z})\cong ...

**2**

votes

**1**answer

162 views

### cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...

**8**

votes

**2**answers

669 views

### Morgan Shalen compactification of $\mathbb C^2$

I'm reading the Otal's survey on the compactification of Morgan Shalen.
(available here)
He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...

**12**

votes

**1**answer

348 views

### Geometric intersection with incompressible surfaces

Let $M$ be a oriented compact $3$-manifold, closed or with boundary.
For any incompressible surface $F$, define a function $i_F$ on the set of homotopy classes of closed curves in $M$ by $$i_F ...

**7**

votes

**2**answers

190 views

### What are known examples of a 3-manifold $Y$ embedded into $Y'\times I$ where $Y'$ is another 3-manifold?

The question I have is the following:
Let $Y,Y'$ be two integer homology 3-spheres. Can we embed $Y'$ into $Y\times I$ such that $Y'$ separates the two boundary components apart?
Do we know any ...

**4**

votes

**1**answer

201 views

### Quotient of principal congruence subgroups

This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?

**0**

votes

**2**answers

133 views

### Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?

Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?

**5**

votes

**2**answers

176 views

### 2-bridge knots in the Rolfsen's table

2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a_1,a_2,\ldots,a_k\right]$.
Has somebody worked out a ...

**3**

votes

**0**answers

112 views

### Dimension of Birman-Murakami-Wenzl Algebra

I was reading the paper Braids, Link Polynomials and A New Algebra by J. S. Birman and H. Wenzl, and I was wondering is there a combinatorial way to compute the dimension of the algebras ...

**2**

votes

**1**answer

105 views

### Why does the Gluck twist on a spun knot give the standard $S^4$?

Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) ...

**1**

vote

**0**answers

150 views

### Geometric representatives of homology classes of manifolds

Is it true that for even dimensional differentiable manifold $M^{2n}$ all singular homology classes in dimension less than $n$ can be represented by a submanifold?

**2**

votes

**0**answers

124 views

### Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...

**1**

vote

**1**answer

112 views

### Boundary components of a subsurface

Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...

**0**

votes

**0**answers

91 views

### permutation action on cohomology of configuration space

Let $F(M,n)$ be the $n$-th configuration (ordered) of manifold $M$.
In the paper The Integral Cohomology Algebras of Ordered Configuration Spaces of Spheres, ...

**1**

vote

**1**answer

173 views

### permutation action on cohomology of Stiefel manifolds

Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.
In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ...

**7**

votes

**2**answers

124 views

### Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.
Suppose that we have closed surface of genus $g\geq 2$, and ...

**6**

votes

**2**answers

326 views

### contractible configuration spaces

Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$.
My question: is $F(S^\infty,k)$ ...

**6**

votes

**0**answers

183 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**2**

votes

**2**answers

182 views

### boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$

Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...

**0**

votes

**1**answer

87 views

### How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...

**1**

vote

**1**answer

113 views

### configuration spaces of real projective space

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring
$$
H^*(F(\mathbb{R}P^n,k);R)$$
is obtained for any ...

**2**

votes

**2**answers

208 views

### Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...

**1**

vote

**0**answers

116 views

### Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...

**0**

votes

**1**answer

105 views

### Maps of balls with fixed value along boundary

Suppose I wish to find the homotopy classes of maps of $B^3 \rightarrow M$ which along the boundary are fixed by a (particular) map $f: S^2 \rightarrow M$. Take $M$ to be a closed orientable ...

**1**

vote

**0**answers

128 views

### The fundamental group of the complement of codimension 2 submanifold

Suppose that $M^n$, $V^{n+2}$ are connected, compact smooth manifolds.
Let $f\colon M^n\to V^{n+2}$ is a smooth embedding.
Let $K_f$ be the kernel of the inclusion induced homomorphism ...

**2**

votes

**0**answers

195 views

### Fixed points of self maps

Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...

**31**

votes

**1**answer

854 views

### Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...

**9**

votes

**1**answer

198 views

### regular tiling of a surface of genus 2 by heptagons

Let $S$ be a Riemann Surface of genus 2. Is there a picture in the literature for a regular tiling of $S$ by 12 heptagons (where three heptagons meet at each vertex). Also, apart from the obvious ...

**5**

votes

**1**answer

287 views

### Geodesics on manifolds with boundary

Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...

**3**

votes

**2**answers

296 views

### Is this a $C^0$ foliation of $\mathbb{R}^2$?

Let $f(x)=\frac{1}{\sin(\pi x)}$ for $x\in (0, 1)$ and let
$\Gamma=\left\{(x,f(x)): x\in (0, 1)\subset \mathbb{R}^2\right\}$ be its graph.
For any set $X\subset \mathbb{R}^2$ and $\lambda>0$ and ...

**0**

votes

**0**answers

59 views

### discrete group action on Stiefel manifold

Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by
$$
(1,2)(u,v)=(-u,v-u),
$$
$$
...

**22**

votes

**2**answers

872 views

### fake $S^{2k}\times S^{2k}$

Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...

**5**

votes

**1**answer

180 views

### Symmetric L-groups of integral group ring of finite cyclic groups

Where can i find the results about $L^{\ast}(\mathbb{Z}\pi)$ for $\pi$ a finite cyclic group?

**11**

votes

**1**answer

258 views

### What is obstructing two stably-isomorphic vector bundles from being isomorphic?

The specific situation is the following:
Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...

**1**

vote

**0**answers

45 views

### Local section of Lie Groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...

**2**

votes

**2**answers

149 views

### Minimal genus of Seifert surface of torus knot

Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince ...

**2**

votes

**1**answer

147 views

### Oriented knot complement conjecture for fibered knots

Suppose I have two inequivalent fibered knots in a homology sphere. When I say 'inequivalent', I mean that there is no orientation-preserving homeomorphism of the space that takes one to the other. ...

**2**

votes

**0**answers

114 views

### Piecewise smooth boundary version of Schoenflies Theorem

It is known that given any smooth ($C^\infty$) simple closed curve $\gamma$ in the 2-sphere, there is a smooth diffeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator $x^2+y^2=1$.
...

**7**

votes

**1**answer

211 views

### Can two fibered knots have the same exterior?

Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors?
See Oriented knot complement conjecture for ...

**0**

votes

**0**answers

140 views

### Pontryagin class of quaternionic line bundle

Let $\xi^{\mathbb{C}}$ be a complex line bundle over a CW complex $B$. Then
$$
VB_{\mathbb{C}^1}(B)\cong [B,BU(1)]=[B,\mathbb{C}P^\infty]=[B,K(\mathbb{Z},2)]\cong H^2(B;\mathbb{Z}).
$$
Hence if ...

**9**

votes

**3**answers

325 views

### Which polygons have *simple* periodic billiard paths?

I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...

**4**

votes

**0**answers

133 views

### When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?

Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a ...

**0**

votes

**0**answers

132 views

### obstructions of Chern class and Pontryagin class

Let $\xi$ be a real $n$ dimensional vector bundle over a CW-complex $B$. Then the Stiefel-Whitney class (coefficient in $Z/2$)
$$
w_i(\xi)=0$$
if and only if $\xi|_{sk^i(B)}$ has $n-i+1$ linearly ...

**1**

vote

**0**answers

72 views

### Regularity of the taut foliation

In Eliashberg-Thurston's famous paper "Confoliations" Corollary 3.2.11, they proved that Irreducible three manifold with $b_{1}>0$ admits semi-fillable contact structure using Gabai's theorem in ...

**6**

votes

**2**answers

326 views

### Making spheres shellable

This is equivalent to my earlier question A question about something like "shelling" in a PL manifold, but maybe more comprehensible and to the point.
Given a triangulation of the PL sphere ...

**4**

votes

**2**answers

227 views

### Geometrisation of inclusion-like epimorphisms to free groups

Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$.
Call $F_g$ be the free group of rank $g$.
Denote by $i:S_g \to ...

**9**

votes

**1**answer

198 views

### Asymptotic dimension of $C'(1/6)$ small cancellation groups

Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?
To offer a little more motivation, Roe proves that all hyperbolic groups have finite ...

**0**

votes

**1**answer

136 views

### Ambient isotopy of the diagonal submanifold in product space

Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
...