# Tagged Questions

**12**

votes

**2**answers

496 views

### Low dimensional topological manifolds

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: ...

**8**

votes

**1**answer

249 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:
...

**16**

votes

**1**answer

589 views

### A function composed with itself produces the identity

Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...

**1**

vote

**0**answers

132 views

### The image of homomorphism of fundamental group of closed surface [closed]

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to S$ ...

**7**

votes

**3**answers

776 views

### Is the list of “known” 3D compact manifolds complete?

"it is an open question if the known compact manifolds in 3-D are complete."
This is a quote from Eric Weisstein's
CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480.
(Google ...

**1**

vote

**0**answers

131 views

### Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...

**1**

vote

**0**answers

163 views

### Topological razors (ball-like spaces)

Introduction
Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...

**5**

votes

**0**answers

107 views

### Is the dimension given by Klee trick ever sharp?

The Klee Trick allows one to find an $\mathbb{R}^m$ where two embeddings of same compact metric space have homeomorphic complements. More precisely, given two embeddings of a compact metric space $K$ ...

**11**

votes

**1**answer

272 views

### Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?

Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2?
Thanks

**6**

votes

**0**answers

188 views

### Spaces that never separate the Hilbert cube

I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...

**2**

votes

**1**answer

143 views

### Extend Homeomorphism to Uniformly Continuous Function

I have a space $A$ which is homeomorphic to the open $n$-ball $B_n$.
I'm trying to build a CW-complex with it, so
I want a continuous function from the closed ball $\overline{B}_n$
to the closure ...

**9**

votes

**0**answers

277 views

### 3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?

**5**

votes

**1**answer

265 views

### Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in
when mapping cone is contractible
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...

**6**

votes

**1**answer

309 views

### How unique is a conformal compactification?

I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, ...

**3**

votes

**1**answer

295 views

### Homotopy equvalence from contractibility of fiber.

Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that ...

**5**

votes

**3**answers

360 views

### Probing a manifold with closed curves

Since fedja's excellent comment on Joseph's question on probing a manifold with geodesics remained uncommented (especially by topologists), I'd like to make a question out of it:
Conjecture: Given ...

**1**

vote

**0**answers

127 views

### Reference needed: Does pseudo laminated compact subsets of the plane separate the plane?

Hi,
doing my research I found the following problem and I´ll be glad if someone could give a reference.
We say that a compact connected subset $K$ of the plane is psuedo laminated if the following ...

**2**

votes

**1**answer

242 views

### Homotopy groups of K3

Let X be a K3 surface and $Y=X/\mathbb{Z}_2$, an Enrique surface.
Long exact sequence of homotopy groups corresponding to fiberaion $\pi:X\to Y$, says that $\pi_2(X)=\pi_2(Y)$, while we know $H_2(X)$ ...

**3**

votes

**2**answers

226 views

### continuous R^2xR^2xR^2/E^+(2) -> R^3 injection?

This is a question that comes from my (biological) research. I'm very weak in topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two ...

**1**

vote

**0**answers

212 views

### Closed irreducible subset

A subet, $A$ of a topological space $X$ is called irreducible if $A\subset A_{1}\cup A_{2}$ and $A_{1}, A_{2}$ are closed subset of $X$, then $A\subset A_{1}$ or $A\subset A_{2}$. We know that in a ...

**5**

votes

**2**answers

266 views

### Given the vertices of a convex polytope, How can we construct its Half-Space representation

HI,
I have a question regarding convex polytopes. Let us say I have the vertices of a polytope which I name as $ V = \{v_1,\cdots,v_k\}$. Each of the $v_k$ are n-dimensional vectors, i.e. ...

**1**

vote

**1**answer

120 views

### Nonhomeomorphic CW-complexes that are “stably” homeomorphic

Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but $X \times I$ and $Y \times I$ are homeomorphic? Here $I$ denotes the unit interval $[0, 1]$.

**7**

votes

**3**answers

719 views

### Associativity of topological join and join of spheres

This must be an elementary question, as I couldn't find any proofs on the Internet, but I still can't do it. And yes, Hatcher says that the join is not actually associative for general topological ...

**6**

votes

**3**answers

615 views

### Classification of Platonic solids

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula ...

**12**

votes

**4**answers

428 views

### Characterization of cocompact group action

Wikipedia claims the following:
In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space or, equivalently, if there is a compact ...

**-1**

votes

**1**answer

204 views

### the space of maximal ideals in C(X) and C*(X) [closed]

Let $C(X)$ be the continous function ring and $C*(X)$ be the bounded continous function ring.$Max C(X)$ consisting of all maximal ideals in $C(X)$.
Question:why $Max C(X)$ and $Max C*(X)$ are compact ...

**17**

votes

**0**answers

801 views

### Fake versus Exotic

Without recourse to the Disc Theorem (or its progeny), is it true that all known examples of exotic differentiable structures on 4-manifolds would be fake rather than exotic?
Terminology (perhaps ...

**8**

votes

**7**answers

1k views

### Undergraduate Topology

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...

**2**

votes

**0**answers

141 views

### Are open convex PL subsets of R^n PL homeomorphic to R^n?

This is a basic issue of PL topology that I assume must be true, but I can't find a written reference: is a convex open PL subset of $\mathbb R^n$ PL homeomorphic to $\mathbb R^n$? I've scanned ...

**0**

votes

**1**answer

356 views

### A Question about SO(n)

My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that ...

**0**

votes

**0**answers

192 views

### Killing homotopy groups by removing subsets

Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from ...

**8**

votes

**1**answer

344 views

### Uniform Embedding into Euclidean Space

Given a locally compact, separable, metric space $X$.
When does $X$ uniformly embed into some Euclidean space?
This means, when does there exist some integer $n$ and a closed subset ...

**0**

votes

**1**answer

192 views

### Diffeomorphisms of a surface in terms of generators.

I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism ...

**2**

votes

**1**answer

162 views

### In which cases a fiber bundle over a circle is a graph-manifold?

A fiber bundle over a circle $M^{3} \longrightarrow S^{1}$ with fiber a surface $F_{g}$ is characterized via a homeomorphism $\varphi \colon F_{g} \to F_{g}$. It can be one of the following: periodic, ...

**8**

votes

**2**answers

825 views

### Connected components of the boundary of an open subset

Hi!
Let f be a (continuous, $C^\infty$... whatever) function from $\mathbb{R}^n$ ($n \geq 2$) to $\mathbb{R}$. Assume that each connected component of $f^{-1} (0; \infty)$ and $f^{-1} (-\infty; 0)$ ...

**7**

votes

**3**answers

392 views

### Surface Eversions: Generalizing from Sphere and Torus Eversions

In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to ...

**3**

votes

**1**answer

192 views

### When is the Freudenthal compactification an ANR?

Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many ends or is of finite dimension. My question is:
What ...

**3**

votes

**1**answer

289 views

### Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...

**6**

votes

**3**answers

840 views

### When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact ...

**5**

votes

**1**answer

239 views

### Characteristic Classes of a Fibered Sum

I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.
Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for ...

**27**

votes

**3**answers

2k views

### Independent evidence for the classification of topological 4-manifolds?

Is there any evidence for the classification of topological 4-manifolds, aside from Freedman's 1982 paper "The topology of four-dimensional manifolds", Journal of Differential Geometry 17(3) 357–453? ...

**3**

votes

**1**answer

386 views

### Klein Bottle exception to the Heawood Conjecture [duplicate]

Possible Duplicate:
The Klein bottle and the Heawood Conjecture
It is well known that the Heawood Conjecture states that the bound for the number of colours which are sufficient to colour a ...

**8**

votes

**0**answers

431 views

### Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as ...

**32**

votes

**3**answers

1k views

### Does Euclidean space have a compact factor?

Is $\mathbb{R}^n$ homeomorphic to a product $X \times Y$ with $X$ compact and not a point?
Bing's Dogbone space is a quotient of $\mathbb{R}^3$ with fibers points and arcs, and whose product with ...

**6**

votes

**2**answers

298 views

### Higher dimensional Heegaard splittings?

Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...

**1**

vote

**3**answers

394 views

### Does the manifold of the three dimensional group of rotations SO(3) cause a separation of space in the group of rigid motions SE(3)?

The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective ...

**0**

votes

**1**answer

358 views

### When is a bijective map between bundles a homeomorphism?

Let $F \rightarrow E_i \rightarrow X_i$ be a bundle with fibre $F$ for i=1,2.
Let $f:E_1 \rightarrow E_2$ be a bijective continuous map and $h: X_1 \rightarrow X_2$ a homeomorphism.
Is f then also ...

**0**

votes

**0**answers

218 views

### Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$

Hi All:
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves ci; i=1,2,..,n. embedded ...

**1**

vote

**2**answers

394 views

### Has this kind of question in topology a special name?

Consider a space $X$ and the group $Homeo(X)/\sim$ of homeomorphisms on $X$ modulo homotopies which are homeomorphisms in each step. One could also consider diffeomorphisms on $X$ or whatsoever.
Have ...

**4**

votes

**2**answers

401 views

### space of homotopy equivalences of $S^1$

Does the space of homotopy equivalences of $S^1$ deformation retract onto the space of homeomorphisms of $S^1$? If so, does anyone have a reference?
I found that Kneser proved that $Homeo(S^1)$ ...