# Tagged Questions

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### Complexity of a function [closed]

I am looking for a natural definition of the complexity a function. If the image is discrete, i was thinking it could be: consider the preimage of an element of the image, count the number of ...
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### Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
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### Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded？

I have the following question: Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...
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### 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 ...
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### Problem on infinite dimensional metric space, with rigidity assumption

By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : ...
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### How many ways do we have to prove that a mapping is open?

Given a continuous mapping $f$ between Euclidean domains (or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? ...
478 views

### Which spaces have the (weak) homotopy type of compact Hausdorff spaces?

Inspired by the discussion in the comments of this question, I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak ...
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### How many ways we have to prove that a topologically (or analytically) nice mapping is injective?

I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and ...
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### Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
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### Simply connectedness of minimal resolution of Kleinian singularities

Is the minimal resolution of Kleinian singularities of type $D_k$ (i.e. the minimal resolution of singularities of the action of the binary dihedral group of order $4(k-2)$ on $C^2$ simply connected? ...
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### Subdividing Simplicial complexes

Barycentric subdivision is an important tool in simplicial homology theory, where it is used as a means of obtaining finer simplicial complexes from a given one. Are there other common subdivision ...
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### CW complex and group action

This is a general question and any reference or related result will be extremely helpful. Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...
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### 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 ...
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### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a regular CW complex of constant local dimension $n$. $X_{n}$ is of finite type, ...
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### Topological degree of homogeneous function of degree k [closed]

Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k? This is true if F is ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### Topology of Asymmetric Symmetric Products

Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting ...
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### 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 ...
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### Topological dimension of quotient group determined by the inverse limit of discrete free monoids

Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero? The question might well be open, but I would be grateful for news ...
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### Topological question about right-lifting property and the evaluation map

Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ ...
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### Only finitely many fundamental groups in $M(n,k,v,D)$?

Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
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### Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here. Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$. Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...
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### When does the sheaf cohomology of a topological space vanish?

The question is in the title. A more precise formulation is: Let $X$ be a topological space. When does $H^i(X,F) = 0$ for all $i > 0$ and all abelian sheaves $F$ on $X$? The obvious example is a ...
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### 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 ...
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### Finitely cocomplete categories of compact Hausdorff spaces

Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 ...
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### 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 ...
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### 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 $Y$ ...
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### 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 $H_2(X)$ ...
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### (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 $R^n$.
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### 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 ...
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### Making CW-complexes metrizable

It is a basic topological fact that CW-complexes aren't typically metrizable (they must satisfy a certain local finiteness condition) and the quotient topology is to blame. Suppose $X$ is a finite ...
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### Manifolds covered by an n-dimensional torus

It is well-known that classification of manifolds up to homemorphism is, in general, out of question. However, this task is sometimes tractable under some additional assumptions on manifolds one would ...
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### 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 ...
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### Free and cellular G-action implies free G-complex?

Recall that a CW-complex $X$ with an action of a group $G$ which permutes the cells (i.e., for any $g \in G$ and any cell $\sigma \subseteq X$, $g\sigma$ is a cell) is called a $G$-complex. If the ...
566 views

### Universal covering space for non-semilocally simply connected spaces

Consider a topological space $X$. Let us consider a universal covering space to be a covering $p : \tilde{X} \rightarrow X$ which is a covering of all other covering spaces. (Perhaps I should call ...
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### Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint ...
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### 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 ...
298 views

### Geometry Realization of Homology Class

Hello! My question is about the realization of homology class. The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a ...
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### A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer. So I hope people here can help me. In the book, Lectures on tensor categories and modular functors by Bakalov ...
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### H-spaces without rational homology

Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space, and whose rational homology groups vanish in positive degrees? My space $M$ is in fact homotopy equivalent ...
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### De Rham representatives of the cohomology classes in $H^*(SU(3))$

I would like to know differential forms representing the cohomology classes of $SU(3)$. I know that there exist a unique bi-invariant form in each class, but I'm not highly motivated by simply putting ...
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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 ...
### Quotient rings of $C(X)$
Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
Let $G$ be a discrete group, acting on a space $X$ (by homeomorphisms). I will say that the action is properly discontinuous if for any $x, y \in X$, there are neighborhoods $U_x$ and $U_y$ such that ...