# Questions tagged [covering-spaces]

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78
questions

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43 views

### When is the quotient of a manifold by a discrete group of diffeomorphisms a diffeological covering space?

I was reading An Introduction to Diffeology by Patrick Iglesias-Zemmour and he defines a diffeological covering space as a diffeological fiber bundle with discrete fiber.
My question: Consider a ...

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votes

**0**answers

20 views

### Upper bound for eigenvalue of symmetric kernel

Let $V \in L^2(D \times D)$ be symmetric kernel defining the compact and nonnegative integral operator
\begin{equation}\mathcal{V}: L^{2}(D) \rightarrow L^{2}(D), \quad(\mathcal{V} u)(x)=\int_{D} V\...

**3**

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87 views

### Mixed Hodge structures on (infinite) covers of complex varieties?

Let $X$ be a complex variety, and let $\tilde{X}\to X$ be a covering map. Does the singular cohomology $H_\ast(\tilde{X};\mathbb{Z})$ carry a natural mixed Hodge structure?
If the cover is finite, ...

**14**

votes

**4**answers

1k views

### The homology of the universal covering space, why so difficult to compute

Let suppose that we are given a connected CW-complex $X$, such that we know
All its homology groups.
All its homotopy groups, in particular we know $\pi_{1}(X)$.
As far as I know there is no ...

**4**

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**0**answers

159 views

### Contractibility and orientation double cover

Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...

**-3**

votes

**1**answer

107 views

### Local isometry implies covering map: nonempty boundary case [closed]

The following theorem is well known in the literature:
Let $M$ and $N$ be riemannian manifolds and let $f : M \to N$ be a local isometry. If $M$ is complete and $N$ is connected, then $f$ is a ...

**22**

votes

**5**answers

2k views

### Does anyone know a basepoint-free construction of universal covers?

Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...

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207 views

### Is there a reasonable notion of universal cover for schemes over arbitrary fields?

Let $X$ be smooth projective variety, such as an elliptic curve. We know that $X$ does not have a universal cover in the category of schemes.
However, if $k = \mathbf{C}$, then $\tilde{X}$ exists as ...

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108 views

### Pushforward of covering maps

Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid.
Given a covering map of $A$ and a ...

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vote

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40 views

### Existence of holomorphic coverings having small degree

Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal ...

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votes

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741 views

### Homology of the universal cover

$k$ is a field. Let $X$ be a connected pointed $CW$-complex such that the homology
$H_{n}(X;k)$ is a finite dimensional $k$-vector space for any $n\in \mathbb{N}$. Suppose that we have continuous ...

**2**

votes

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217 views

### $PSL_2(\mathbb{R})$ representations of free groups

Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...

**0**

votes

**1**answer

159 views

### Same fiber of induced covering map [closed]

Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...

**5**

votes

**2**answers

687 views

### Monodromy groups from Galois's viewpoint

According to the Wikipedia article about monodromy, the monodromy group can be defined in terms of Galois theory in following way:
Let $F(x)$ denote the field of the rational functions in the ...

**5**

votes

**2**answers

281 views

### Finite etale covers of products of curves

Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.
Let $C_1, C_2 \subset \mathbb{P}^1$ be non-empty ...

**4**

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**0**answers

158 views

### Dyer–Lashof operations for more than 2 inputs

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over it. Let the base ring be $\mathbb{Z}_2$. If $C_*$ denotes the singular chain complex over $\mathbb{Z}_2$, the action of $\mathcal{O}$ ...

**3**

votes

**1**answer

252 views

### Covering with Deck group $\mathfrak{S}_3$

I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...

**3**

votes

**1**answer

63 views

### Concerning the Spanier group relative to an open cover

Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi_1 (X, x)$ which contains all homotopy classes having ...

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97 views

### Singular homology: Lifting simplices gives map in homology

Let $X$ be a space, $k=k_1+\dotsb+k_r$ and let $G:=\mathfrak{S}_{k_1}\times\dotsb\times \mathfrak{S}_{k_r}$ act freely on the right on $X$. Fix a commutative ring $R$ and another space $Y$.
Then the ...

**3**

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50 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

**2**answers

245 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 ...

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**1**answer

160 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

374 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

230 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 ...

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111 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

247 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

329 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

73 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

418 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}...

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**2**answers

475 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**

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238 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

241 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 $X$ ...

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258 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 ...

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262 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**

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**0**answers

138 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 ...

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**4**answers

1k 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 ...

**12**

votes

**1**answer

555 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 ...

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votes

**2**answers

320 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

371 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

120 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. ...

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264 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 ...

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votes

**0**answers

129 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 ...

**6**

votes

**1**answer

545 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

206 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

165 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 ...

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votes

**2**answers

374 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....

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votes

**1**answer

354 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

91 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

157 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

697 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
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
...