**9**

votes

**1**answer

195 views

### Alexander duality for non-manifolds

Let $X$ be a CW complex and $A$ a subcomplex. I will assume that both are compact, and that $X$ is $n$-dimensional. Furthermore, assume that the local homology of $X$ is that of a manifold in a ...

**0**

votes

**0**answers

29 views

### Is every $C^1$-domain which is homeomorphic to the unit ball in $\mathbb{R}^d$ Lipschitz equivalent to the unit ball?

Suppose we have a domain $\Omega\subset \mathbb{R}^n$ which is homeomorrphic to the unit ball $B(0,1)\subset \mathbb{R}^n$ and such that $\partial \Omega$ is of class $C^1$ (technically, this means ...

**5**

votes

**1**answer

216 views

### Simply connected CW-complex with only finitely many nontrivial homotopy and homology groups

Let $X$ be a simply connected CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily contractible?
I ...

**3**

votes

**2**answers

290 views

### How to show whether a given knot and its mirror image are the same or not?

The title says it all:
How can I show that a knot $K$ is distinct from its mirror image?
May be I have to try different knot invariants. Not sure, I am new in this area.

**3**

votes

**1**answer

124 views

### Which automorphisms on $H_{1}(M^{3})$ are induced by homotopy equivalences?

Let $M^{3}$ be a closed orientable 3-manifold, and $\phi:H_{1}(M;\mathbb{Z})\to H_{1}(M;\mathbb{Z})$ be an automorphism of abelian groups.
My question is: Is there any characterization of $\phi$ ...

**17**

votes

**3**answers

686 views

### Degrees of self-maps of aspherical manifolds

In "Infinitesimal computations in Topology", Publ IHES, page 318, Dennis Sullivan writes "Recall any self-mapping
of a Riemann surface of genus $g>1$ either has
degree $0$ or degree $\pm 1$." ...

**0**

votes

**1**answer

75 views

### Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all:
Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...

**2**

votes

**1**answer

97 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 ...

**15**

votes

**1**answer

494 views

### Is every degree 1 self-map a homotopy equivalence?

In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this ...

**-1**

votes

**0**answers

35 views

### Which spaces admit bump functions? [migrated]

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets.
Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...

**3**

votes

**1**answer

138 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, ...

**9**

votes

**2**answers

190 views

### Question about the Weeks Manifold

I am wondering about the existence of incompressible surfaces in the Weeks manifold. Is this space a Haken manifold?

**11**

votes

**2**answers

225 views

### When is it possible to “shrink” a polyhedron?

Let $P$ be a (not necessarily convex or simply-connected) polyhedron, and $\gamma(t)$ a homotopy of $P$, i.e. a continuous displacement of the vertices of $P$ that keeps its faces planar.
I call ...

**8**

votes

**0**answers

97 views

### Doubly slice knots and an embedding of 4-manifold

It is well-known that the existence of topologically slice knot which is not smoothly slice implies the existence of exotic $\mathbb{R}^4$. For example, see the answer of K. Davis.
To prove the above ...

**40**

votes

**1**answer

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 ...

**-1**

votes

**1**answer

83 views

### unordered configuration space of pointed space

Let $(X,*)$ be a pointed topological space.
Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid x_i\neq x_j, i\neq j\}$.
Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.
Is there an inclusion ...

**6**

votes

**3**answers

2k views

### How are fiber bundles, transition functions and principal bundles related?

Please read the edit below.
Is my understanding of this correct? Fix a sufficiently nice and connected topological space $B$ and a topological group $G$. A principal bundle $E\to B$ with structure ...

**11**

votes

**1**answer

1k views

### How well can we localize the “exoticness” in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, Exotic Structures on smooth 4-manifolds by Selman Akbulut and Localized Exotic Smoothness by ...

**5**

votes

**1**answer

139 views

### Triangulations of submanifolds of smooth manifolds

Every smooth manifold $M$ has a PL structure, and therefore a triangulation. Given a submanifold $N$ of $M$, does anyone know some nice conditions for $N$ to be the subcomplex of some triangulation of ...

**2**

votes

**1**answer

88 views

### Thickening graphs to get honest actions

Let $X$ be a finite graph. Its fundamental group is the free group $F_n$ on (say) $n$ generators. Let further an automorphism $\phi$ of $F_n$ be given.
It is not true in general that this ...

**8**

votes

**0**answers

71 views

### Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$
induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes.
Given a ...

**5**

votes

**1**answer

121 views

### Embedded spheres in the K3 surface

Using the Seiberg-Witten theory, we know that every (smoothly) embedded $S^{2}$ in $K3$ with trivial normal bundle is null-homologous. We know we have a lot of interesting knotted $S^{2}$ inside ...

**4**

votes

**1**answer

150 views

### A question about simple closed curves in finite dimensional Euclidean spaces

Let n be a positive integer not less than 2. Does anyone know of a theorem stating that- for each n- there exists a simple closed curve c(n), which (1) is a subset of n-dimensional Euclidean space ...

**2**

votes

**2**answers

170 views

### A Jordan Separation Theorem for Polyhedral Surfaces

Let me begin by defining what a polyhedral surface is.
A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...

**6**

votes

**2**answers

563 views

### fundamental groups of curves

I saw the following statement made without proof in a paper of Bogomolov and Tschinkel:
If $X$ is an algebraic surface, and $C$ is an ample smooth curve in $X,$ then the fundamental group of $C$ ...

**4**

votes

**1**answer

236 views

### When are principal bundles preserved by colimits?

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...

**25**

votes

**1**answer

993 views

### Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...

**2**

votes

**1**answer

116 views

### Is there a straightforward way to define a differentiable structure on a localic manifold?

I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.

**2**

votes

**1**answer

134 views

### A basic question of Khovanov-Rozansky Homology

thank you for spending time on the following question.
In [1] Khovanov and Rozansky categorifized $sl_n$ version of HOMFLY Polynomial, in page 11, they mention that what they defined in [1] is ...

**6**

votes

**1**answer

237 views

### 3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...

**1**

vote

**1**answer

218 views

### Limit Group decomposition

I would need a clarification about a statement in the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel.
First recall that a limit group is a finitely ...

**8**

votes

**0**answers

487 views

### Commutator subgroup of a surface group

Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset ...

**1**

vote

**0**answers

142 views

### Surgery to unlink $S^p$ and $S^q$ in $S^d$

We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres.
I am looking for a surgery to unlink $S^p$ and ...

**2**

votes

**2**answers

207 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

84 views

### Canonical metric of toric Kahler manifolds

Let $X$ be non-compact toric Kahler manifold associated to a Delzant polygon $P$
and $g$ be the canonical Kahler metric constructed by Guillemin. Is it true that
the real part of $g$, as a ...

**8**

votes

**2**answers

205 views

### Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...

**2**

votes

**0**answers

94 views

### McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...

**20**

votes

**0**answers

577 views

### Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...

**9**

votes

**1**answer

121 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 ...

**14**

votes

**1**answer

164 views

### $(n-1)$-dimensional sphere in $S^n$ such that the closure of a component of complement is not contractible

Let $f:S^{n-1} \rightarrow S^n$ be a topological embedding and let $A_f$ and $B_f$ be the components of $S^n \setminus f(S^{n-1})$. If $\overline{A}_f$ and $\overline{B}_f$ are manifolds with ...

**3**

votes

**1**answer

119 views

### Hopf link from analytic geometry

I am a condensed matter physicist, and want to understand the Hopf link from analytic point of view. My question is as follows.
We have two sets of equations, and each set of equations describes a ...

**4**

votes

**2**answers

928 views

### A question about Marsden-Weinstein reduction theory

Let $G$ be a compact 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 $J \colon M\to \frak g^*$ be its moment map then the ...

**1**

vote

**1**answer

124 views

### Chern classes of three (two) dimensional complex vector bundles

Let $M$ be a manifold.
Let $F(M,3)=\{(m_1,m_2,m_3)\mid m_1, m_2, m_3\in M, m_i\neq m_j, \text{ for any } i\neq j\}$.
Let $S_3$ be the symmetric group of order $3$.
Let $S_3$ act on $F(M,3)$ by ...

**0**

votes

**1**answer

256 views

### Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space

This is a follow-up to this question.
Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$).
I'm interested in the topological ...

**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 ...

**4**

votes

**2**answers

373 views

### How many metrics of constant curvature exist on a Riemannian surface?

I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...

**3**

votes

**2**answers

121 views

### Teichmuller distance between isospectral riemann surfaces

Let $S$ be a surface of negative Euler characteristic (for simplicity let's assume $S$ to be closed), and let $\mathcal{M}(S)$ denote the moduli space of hyperbolic surfaces homeomorphic to $S$.
...

**5**

votes

**2**answers

309 views

### Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes

I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type ...

**2**

votes

**1**answer

190 views

### Ehresmann fibration theorem for manifolds with boundary

All manifolds in consideration may have nonempty boundary and may be disconnected.
Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...

**20**

votes

**0**answers

311 views

### Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...