Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube ...

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2
votes
0answers
41 views

cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...
25
votes
7answers
3k views

What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello, I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ : For example : 1) Are all ...
0
votes
1answer
21 views

Existenc conditions of single crossing [on hold]

There are two density function $f(x)$ and $g(x)$.They have the commnon support set $[\underline{x},\bar{x}]$. Condition (1)$f(\underline{x})<g(\underline{x})$ and $f(\bar{x})>g(\bar{x})$. ...
6
votes
1answer
183 views

Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation $$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$ In other words, for every finite simple nonabelian group $G$, do there exist ...
23
votes
3answers
1k 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 ...
7
votes
1answer
281 views

Relation between moduli spaces and classifying spaces

I hope this question is suitable to be posted here on MO. I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...
15
votes
1answer
545 views

Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on ...
1
vote
0answers
72 views

How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...
17
votes
0answers
143 views

Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...
4
votes
1answer
116 views

Counterexample to high dimensional Nielsen realization problem

Can someone give me an example of a closed smooth oriented manifold $M$ and an orientation preserving diffeomorphism $f:M\rightarrow M$ such that $f^k$ is isotopic to the identity for some $k\geq 1$, ...
-6
votes
0answers
72 views

I need to know all geometry source to study geometry [closed]

I'm not specialist in geometry and I try to start learn all ph.d student need to know for their works. could you please introduce me which book should I start and please mention all book which is ...
2
votes
1answer
171 views

Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange. Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...
5
votes
1answer
328 views

Homeomorphism of closed manifold

Suppose that we have two closed n-manifold $M$ and $N$ such that the topological group of homeomorphisms $Homeo(M)$ is homotopy equivalent to $Homeo(N)$ (maybe as topological groups if needed), can ...
6
votes
2answers
194 views

Equivariant smoothing of PL structures on $S^3$

Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms? I am not quite ...
2
votes
0answers
60 views

section spaces related to configuration spaces

In the paper Configuration spaces of positive and negative particles, Dusa McDuff, a section space $\Gamma(M)$ is constructed: And in the paper ON THE HOMOLOGY OF CONFIGURATION SPACES. C.-F. ...
3
votes
0answers
40 views

Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
3
votes
2answers
116 views

Brieskorn homology spheres

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from ...
2
votes
0answers
92 views

homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$ $$ ...
9
votes
0answers
127 views

Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...
104
votes
2answers
10k views

What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
3
votes
1answer
205 views

Rigidity of secondary characteristic classes

For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes $$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$ ...
41
votes
1answer
4k views

Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times \mathbb R^2$ such that the projection $D^4\to S^2$ is an open map? Here $D^n$ denotes closed $n$-ball. An open map D⁴ → S². It ...
11
votes
1answer
282 views

Homology spheres and fundamental group

I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic ...
9
votes
1answer
134 views

bi-Lipschitz gluing

Let $H$ be the Heisenberg group with left invariant sub-Riemannian metric and $\varepsilon>0$ is small. Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$. I have a bi-Lipschitz ...
1
vote
1answer
111 views

Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...
22
votes
6answers
2k views

Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
-1
votes
0answers
21 views

Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately. I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...
5
votes
1answer
176 views

Hyperbolic knot complement groups and relative dimension

Let $G$ be a the fundamental group of a hyperbolic knot complement. Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$. The knot complement has a $2$-dimensional spine ...
5
votes
0answers
65 views

Complements of unknotted tori (higher dimensions)

It is weil-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 ...
1
vote
6answers
754 views

Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
5
votes
0answers
170 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 ...
7
votes
2answers
194 views

Automorphism of genus 2 surface with 5 fixed points

Is there a self-homeomorphism of a genus 2 (closed, orientable) surface, which has finite order and exactly 5 fixed points? Of course, the same question can be asked replacing 2 by $g$ and $5$ by any ...
7
votes
2answers
168 views

What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot. Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$? Are there any ...
19
votes
3answers
2k views

Riemann mapping theorem for homeomorphisms

How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
6
votes
1answer
192 views

Can a 3-ball divide a standard 4-ball into two exotic 4-balls?

Let $B^n$ denote the unit ball in $\mathbb{R}^n$ (wrt the standard euclidean metric) and $\bar{B}^n$ denote the unit closed ball. Suppose that $\Sigma$ is a a smooth embedded hypersurface with ...
1
vote
0answers
196 views

On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix. Let $J$ be all $1$ matrix. Let $\bar{A}=J-A$. Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...
3
votes
1answer
88 views

Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution) $B\varphi ^*$ on ...
4
votes
1answer
176 views

Can a homology $n-1$-sphere divide $\mathbb{S}^{n}$ into non-contractible components?

This is a follow-up to my earlier question. Let $\Sigma\subset \mathbb{S}^{n}$ be a hypersurface -- here $\mathbb{S}^{n}$ is a smooth sphere (possibly exotic ... if this makes a difference). If ...
4
votes
1answer
162 views

Topology of hypersurface of sphere fixed by homeomorphic involution

I'm not an topologist, so I apologize in advance if this is a silly question. I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and ...
8
votes
0answers
170 views

Simplices and cubes

Question: What is the first appearance in the literature of one of the following statements: The result of intersecting a simplex with a cell of the dual subdivision is a cube There ...
5
votes
0answers
100 views

Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...
4
votes
2answers
179 views

Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology. The evaluation map $$ev\colon ...
7
votes
1answer
290 views

Z/p action on finite contractible complex

Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or ...
3
votes
1answer
215 views

Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems '': Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, ...
6
votes
1answer
91 views

Reference for a PL flat torus embedding in $\mathbb{R}^3$

A piecewise linear flat torus embedded in $\mathbb{R}^3$ is shown at http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml. It is flat in the sense that the angle defect at the vertices is zero. ...
37
votes
11answers
4k views

Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
3
votes
1answer
230 views

Simple proof for property R conjecture

Gabai's property R theorem is: If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot. Recently, 3-manifold topology has been developed rapidly by Agol, ...
7
votes
2answers
512 views

Embedded (framed) cobordisms

[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.] This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...
0
votes
0answers
80 views

Codimension one embeddings

For smooth knots in $S^3$ "isotopy" and "ambient isotopy" are equivalent (although this is not true in the topological category). I guess that therefore also for tori in $S^3$ "isotopy" and "ambient ...
13
votes
1answer
165 views

Three-manifolds having a Reebless foliation but not a taut one

A straightforward argument reveals that a taut foliation is Reebless, and of course there are many examples of Reebless foliations that are not taut. I guess that there are many examples of ...