# Tagged Questions

**5**

votes

**0**answers

214 views

### Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where ...

**0**

votes

**1**answer

246 views

### Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...

**5**

votes

**1**answer

658 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...

**4**

votes

**3**answers

522 views

### Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...

**6**

votes

**2**answers

469 views

### Homotopy problem for infinite dimensional topological space II

This post here is a specification of this post.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of intrinsic metric spaces verifying :
$X_{n}$ have topological dimension $n$.
$X_{n+1}$ is ...

**0**

votes

**2**answers

282 views

### Homotopy problem for infinite dimensional topological space

Let $X$ be an infinite dimensional topological space such that :
$ \forall n \in \mathbb{N}$, $ \exists X_{n} \subset X$, $n$-dimensional subspaces verifying :
$\forall r<n$, the homotopy ...

**10**

votes

**0**answers

249 views

### When is the one-point compactification well-pointed?

This is a follow up to my previous
question.
Question:
Is there a reasonably natural set of conditions which guarantee that the one-point
compactification $X^+$ of a locally compact Hausdorff ...

**4**

votes

**1**answer

354 views

### “monotone” homotopy?

This is a question about a concept that I call "monotone homotopy" which arises in a natural way in some topological situations.
Let $X$ be a (bounded) metric space, $Y$ be a topological space and ...

**19**

votes

**4**answers

937 views

### The role of ANR in modern topology

Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...

**3**

votes

**2**answers

214 views

### Is there a good concept of a measurable fibration?

In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability ...

**0**

votes

**2**answers

259 views

### (Homotopy) Y ENR and contractible subset implies Y is a retract

I'm trying to solve the following question:
Suppose $Y \subset R^n$ is a Euclidean neighborhood retract. I want to prove that if $Y$ is contractible, then it is a retract of $R^n$.

**11**

votes

**1**answer

1k views

### Example of fiber bundle that is not a fibration

Hi all,
It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird ...

**2**

votes

**2**answers

379 views

### Is the wedge sum of two cones over the hawaiian earring contractible?

Let $X_1$ and $X_2$ be two cones over the hawaiian earring and let $X$ be the wedge sum of $X_1$ and $X_2$ (of course you join them in the special point of the hawaiian earring). How do you prove that ...

**6**

votes

**2**answers

369 views

### Proper maps and transversality

I'll begin with the question, which is intrinsically interesting:
Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map ...

**0**

votes

**0**answers

206 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 ...

**3**

votes

**1**answer

343 views

### Bitopological spaces and algebraic topology

Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy ...

**9**

votes

**2**answers

1k views

### Well-pointed space which is not locally contractible

I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the ...

**5**

votes

**2**answers

203 views

### Simultaneously minimizing intersections

This may be a standard problem in homotopy theory, but I don't know a good reference.
Let $\Sigma$ be a smooth, oriented surface, and let $X_1,X_2$ and $X_3$ be three smoothly embedded curves in ...

**1**

vote

**1**answer

145 views

### Shrinkable maps and universal weak equivalences

Recall that a morphism $f:X \to B$ is called shrinkable is there exists a section $s:B \to X$ together with a homotopy $$H:I \times X \to X$$ from $sf$ to $id_X$ over $B,$ i.e. for all $t$, the map ...

**3**

votes

**0**answers

141 views

### Mapping into Hurewicz cofibrations.

In Strøm's paper "The Homotopy Category is a Homotopy Category" he proves
(Lemma 4) that if $Y$ is compact and if $i:A\to X$ is a cofibration, then the induced map
$$
i_*: A^Y \to X^Y
$$
is also a ...

**27**

votes

**4**answers

945 views

### Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?

In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the catgory Top topological ...

**6**

votes

**0**answers

294 views

### The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes.
Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...

**6**

votes

**0**answers

393 views

### The Mapping Cylinder of a Pullback Square

Suppose I have a pullback square, which I think of as a map from the fibration
$q:X\to A$ to the fibration $p:Y\to B$. Then there is an induced map $m: M \to N$
from the mapping cylinder $M$ of ...

**5**

votes

**1**answer

510 views

### How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?

How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface ...

**2**

votes

**1**answer

363 views

### An example of a space which is locally relatively contractible but not contractible?

A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...

**5**

votes

**2**answers

470 views

### What is an example of a non-regular, totally path-disconnected Hausdorff space?

I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...

**11**

votes

**2**answers

443 views

### Noncontractible connected topological rings ?

Are there any non-contractible connected topological rings?
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by ...

**10**

votes

**3**answers

711 views

### Which properties of finite simplicial sets can be computed?

A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I ...

**2**

votes

**0**answers

243 views

### Homotopy equivalences and cores

Hi all,
Before asking my question, I need to fix some terms and notation.
Let $M$, $M'$ be locally compact, Hausdorff spaces, and $f:M\rightarrow M'$ a homotopy equivalence with homotopy inverse ...

**29**

votes

**3**answers

3k views

### Why do finite homotopy groups imply finite homology groups?

Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. ...

**2**

votes

**1**answer

411 views

### Are the C(S^n, S^n)'s homeomorphic ?

Let m, n > 1. Is it true that C(S^m, S^m), and C(S^n, S^n) are homeomorphic ?
[both endowed with the sup metric (or equivalently the compact-open topology)]
Generally, C(S^n, S^n), with n >= 1, is a ...

**7**

votes

**1**answer

308 views

### Stable presentable categories as module categories

There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I ...

**7**

votes

**3**answers

364 views

### relationship between universal coefficient theorem and [K(Z,n),K(G,n)]?

In short, I'm wondering whether the universal coefficient theorem can be understood/reinterpreted by using maps of Eilenberg-MacLane spaces. This is a wishy-washy idea and I don't have evidence to ...

**3**

votes

**1**answer

587 views

### properly interpreting Pi_0 in the homotopy exact sequence

Define the lens space L(m,n) as the quotient of S2m+1 by the action of the cyclic group ℤn⊂S1⊂ℂ*. We can create the infinite lens space L(∞,n) by a telescoping construction ...

**1**

vote

**2**answers

135 views

### Something like Yoneda's lemma

This is inspired by The Whitehead for maps question.
Consider two maps f, g: X\to Y which happen to induce the same maps (of discrete spaces) ...

**5**

votes

**1**answer

161 views

### Homotopy type of stabilizers

Let X be a contractible metric space and G a topological group acting transitively on X (i.e. given any two points x,y \in X, there exists g \in G such that gx=y).
My question is the following: is it ...

**7**

votes

**2**answers

406 views

### Base change for category objects in topological spaces

I was prompted by this question, but the motivation is different.
Suppose we have an internal category object in topological spaces, i.e. an object space X and a morphism space Y, together with ...

**14**

votes

**8**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...