# Questions tagged [covering-spaces]

The covering-spaces tag has no usage guidance.

**3**

votes

**0**answers

43 views

### Galois Covering induces new Cover $Ind_H ^G(Y)$

I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84):
We consider a group $G$ which contains a ...

**3**

votes

**1**answer

150 views

### English literature close to “Algébre et Théories Galoisiennes” by Régine and Adrien Douady

I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...

**4**

votes

**1**answer

134 views

### Invariant lifts of a closed curve on a surface of genus > 1

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...

**5**

votes

**2**answers

307 views

### Galois categories for topological spaces?

Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?
...

**3**

votes

**1**answer

211 views

### If $X, Y$ are topological spaces, with $Y$ being a k-space, and $f : X \to Y$ is a proper covering map, is $X$ necessarily a k-space?

A k-space is a compactly generated Hausdorff topological space. (I used the terminology "k-space" in the question, in order keep the question within the limit of 150 characters.)
Note that under the ...

**1**

vote

**0**answers

86 views

### Idea behind definition of classifying space over an orbifold

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.
Definition : Let $X$ be a locally compact ...

**4**

votes

**1**answer

175 views

### Path-lifting property: function space interpretation

I asked this question on math.SE, but even with a bounty, there were no answers/comments. I hope this is not too low-level for this site.
Suppose I have a covering map $\pi:E\rightarrow B$, and a ...

**2**

votes

**2**answers

259 views

### covering theory with compact open topology

In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected.
Under ...

**2**

votes

**1**answer

70 views

### How to detect covering graphs

Let's say $G$ is a graph. How can we detect if $G$ is (nontrivially) a covering graph?
$G$ is nontrivial covering graph if there is a covering map $f : G \to C $ (for some graph $C$) such that $f$ is ...

**3**

votes

**1**answer

269 views

### Nonpathological nonnormal covering space

A topological covering $p : \tilde{X} \to X$ is normal when the group of deck transformations acts transitively on the fibers of $p$. This is equivalent to the fact that $p_* (\pi (\tilde{X}, \tilde{x}...

**3**

votes

**2**answers

344 views

### Coverings of a space and coverings of a groupoid

In algebraic topology, there is the well-known notion of coverings of a space. It is very nice, it has many properties, but I find it frustrating that:
1) some hypotheses are needed for them to work ...

**12**

votes

**0**answers

232 views

### Alexander modules and weight filtrations

$\def\Ext{\mathrm{Ext}}$I have the following situation: I have a complex algebraic variety $X$. I want to compute the weight filtration on various abelian covers $Y \to X$. As a concrete example, take ...

**4**

votes

**1**answer

184 views

### The Classification of all spaces for which $X$ is a covering space

A well-known problem is to classify all covering spaces of a topological space $X$. For example, if $X$ is a semi-locally simply connected space, then each equivalent class of a covering space of $X$ ...

**6**

votes

**0**answers

217 views

### Obstructions for existence of a fiber wise covering space structure( A bundle of covering spaces)

Let $S^n \times \mathbb{T}^n$ be the trivial Torus bundle over $S^n$.
Assume that we have a continuous fiber preserving map $\phi :TS^n \to S^n \times \mathbb{T}^n$ which restriction to each fiber ...

**1**

vote

**0**answers

189 views

### Is there a compatible metric on covering space of a metric space? [closed]

Let $p:\tilde{X}\longrightarrow X$ be a covering map and $(X,d)$ a metric space. Is there any metric $\tilde{d}$ on $\tilde{X}$ so that compatible with the metric $d$? i.e. a metric $\tilde{d}$ on $\...

**1**

vote

**0**answers

121 views

### Connectedness of symmetric subgroup of simply connected Lie group

Let $G$ be a connected real simple Lie group, and $\tau$ be an involutive automorphism of $G$. Then $\tau$ defines a symmetric subgroup $G^\tau$ of $G$. In general, $G^\tau$ is not necessarily ...

**12**

votes

**4**answers

881 views

### Self-covering spaces

Let $M$ be a connected Hausdorff second countable topological space. I will call $M$ self-covering if it is its own $n$-fold cover for some $n>1$. For instance, the circle is its own double cover ...

**11**

votes

**1**answer

443 views

### Finite covers of hyperbolic surfaces and the `second systole´

We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...

**10**

votes

**2**answers

314 views

### Spaces that are finitely covered by manifolds

Suppose $X \to Y$ is a finite-sheeted covering of CW-complexes. Moreover, assume that the total space is homotopy equivalent to a (closed, connected, smooth) manifold $M$. I am interested in ...

**4**

votes

**1**answer

317 views

### Descent theory, fibrations, and bundles

In the very last page of Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms, the authors relate the theory of fiber bundles (and covering spaces in particular) to descent ...

**-1**

votes

**1**answer

119 views

### General description of transition arrows of covering morphisms in family fibrations

For sets and functions, I think the following data are equivalent:
A function $g:A\times B\to B$ such that $(\pi_1,g):A\times B\to A\times B$ is a bijection;
a function $A\to \mathrm{Aut}B$.
Proof. ...

**13**

votes

**0**answers

257 views

### Actions of $\mathbb Z/2\mathbb Z$ on algebraically closed fields and even-dimensional spheres and parallel between Galois theory and covering theory

It is well known that there is a parallel between Galois theory and covering theory. So I wonder whether there is a deep similarity between the following two facts:
Artin-Schreier theorem. The only ...

**3**

votes

**0**answers

121 views

### Making extensions $L/K$ aware of the Galois group coming from $K/k$

Although inspired by my question on math.SE https://math.stackexchange.com/q/1902190/214353 this is not a crosspost. What happened is that after an answer and some comments I realized more clearly ...

**5**

votes

**1**answer

478 views

### Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...

**2**

votes

**1**answer

194 views

### How to increase the injectivity radius function of a hyperbolic 3 manifold of finite volume?

Let $N$ be an oriented hyperbolic 3-manifold of finite volume and let $\Delta \subset N$ be a smooth connected compact subdomain such that the restriction of the injectivity radius function of $N$ to $...

**0**

votes

**1**answer

153 views

### Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$

I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...

**2**

votes

**2**answers

345 views

### For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?

Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....

**2**

votes

**1**answer

328 views

### induced group actions and covering maps on Eilenberg-Maclane space

Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma_k.
...

**1**

vote

**1**answer

82 views

### vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow M\...

**6**

votes

**1**answer

155 views

### non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle
$$
\xi:\mathbb{R}^k\longrightarrow M\times_{\...

**6**

votes

**2**answers

645 views

### quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
...

**1**

vote

**0**answers

129 views

### Acyclicity of covering space

Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...

**2**

votes

**2**answers

316 views

### Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...

**1**

vote

**1**answer

130 views

### triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle

Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \...

**6**

votes

**1**answer

336 views

### vector bundles associated to a covering space

Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...

**4**

votes

**0**answers

174 views

### characteristic classes of a covering space with symmetric group action

Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...

**6**

votes

**0**answers

161 views

### Overview and/or reference of theory of pro-universal covers?

This question will contain very little in the way of concrete information, because I don't have much to go on. I've heard whispers of something called a "pro-universal cover," which is the inverse ...

**2**

votes

**1**answer

282 views

### Does every connected component of a covering space over a connected base intersect all the fibers of the covering space?

This statement is used without explanation in the proof of a Corollary to Proposition 4.3.5, pages 210-211 of the book "Algèbre et théories galoisiennes" by R. and A. Douady, second edition, Cassini, ...

**4**

votes

**1**answer

254 views

### Universal covering and double cover functors

Initially posted on MSE
Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to ...

**0**

votes

**1**answer

121 views

### Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...

**1**

vote

**1**answer

563 views

### cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field.
Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...

**0**

votes

**1**answer

114 views

### Genus of Covering Space of 3-Manifold

Let $M_g$ and $M_h$ be closed orientable 3-manifolds of genus $g$ and $h$ respectively and suppose that $M_g$ is an $n$-sheeted cover of $M_h$. Is there a formula that would allow us to compute $g$ if ...

**1**

vote

**2**answers

260 views

### Graph lifts and representation theory

Is there any connection known between the two?
One can naturally define lifts of graphs by groups like $\mathbb{Z}_k$ and hence I wonder if representation theoretic properties can be used to say ...

**0**

votes

**1**answer

207 views

### When is a $2$-lift of a graph connected? [closed]

Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ ...

**1**

vote

**0**answers

306 views

### Ramification: Riemann surfaces vs Number fields

I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...

**2**

votes

**2**answers

302 views

### A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another
metric $d_{f}$ on $Y$ With $$ d_{f}(y_{1},y_{2})=Hd(f^{-1}(y_{1}), f^{-1}(...

**22**

votes

**2**answers

2k views

### Teaching the fundamental group via everyday examples

This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...

**10**

votes

**3**answers

527 views

### Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...

**3**

votes

**0**answers

149 views

### Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors.
Have the possible universal covering spaces of $U$ been classified?
Do we know when the ...

**6**

votes

**1**answer

793 views

### A generalization of covering spaces to fiber bundles with totally path-disconnected fibers

There is a classical theorem about covering spaces and the actions of the fundamental group.
Theorem 1: Let $B$ be a non-empty locally path-connected and path-connected space. The category of ...