Questions tagged [covering-spaces]

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4
votes
0answers
123 views

Necessary and sufficient condition for the lifting of a proper map to be proper

For $i=1,2$ let $(X_i,v_i)$ be two connected topological manifolds, two subgroups $H_i\leq \pi_1(X_i,v_i)$, two coverings $q_{H_i}:\big(\overline{X_i}(H_i),\overline{v_i}\big)\to (X_i,v_i)$ ...
9
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1answer
213 views

Construction of the universal covering space via compact-open topology

This is a re-post of a question I asked a month ago on MSE, but unfortunately didn't receive any answers. I'm hoping someone could help me with it. Here it goes: Recently I've been self-studying the ...
4
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1answer
185 views

Cellular homology of the universal cover

Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$. Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...
2
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1answer
175 views

Automorphisms of $G/Z(G)$ with $G$ simply connected

Let $G$ be a simply connected (if necessary, compact Lie) group with finite center $Z$ and $p:G/\to G/Z$ be the canonical projection. Is there any way to know if every element in $\operatorname{Out}(G/...
1
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1answer
141 views

Principal G-bundles over the circle

To edify my understanding of fiber bundles with structure groups, I was currently trying to reconcile two classifications (in a particular case). For simplicity, I'm taking the base to be $S^1$ and ...
2
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1answer
139 views

Lifting of a proper map in the cover is a proper map

Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...
12
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0answers
248 views

Covering image of a connected CW-complex need not be a CW-complex

This question is already asked here MSE, and there is an answer based on some conjecture (probably still open). I am posting the same question for a counterexample (if any, not based on such unsolved ...
3
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0answers
89 views

Is the category of covering spaces always a topos?

It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...
1
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2answers
640 views

Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
1
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1answer
89 views

can we take skeletons of covering maps to give new covering maps?

Let $X$ be an $n$-dimensional cell complex. We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$. Take the universal cover (or a general covering space) $\tilde X'$ of $X'$. ...
2
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0answers
81 views

Rotation set vs existence of rotation number

Let $f\colon \mathbb{S}^{1}\to\mathbb{S}^{1}$ be a continuous function of degree 1 and $F\colon \mathbb{R}\to \mathbb{R}$ a lift of $f.$ One can define, for each $x\in \mathbb{R}$, the rotation number ...
18
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2answers
891 views

If the universal cover of a manifold is spin, must it admit a finite cover which is spin?

If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover). If $M$ is non-spin, then it does not necessarily have a finite cover which is ...
6
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2answers
338 views

The variety induced by an extension of a field

If $K$ is a finitely generated field extension of $k$, then there exists an irreducible affine $k$-variety with function field $K$. The idea is that if $x_1, \dots, x_n$ are generators of $K$ under $k$...
3
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0answers
82 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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0answers
23 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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0answers
113 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, ...
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4answers
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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0answers
213 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 ...
-2
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1answer
228 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 ...
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5answers
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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0answers
213 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 ...
2
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0answers
131 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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0answers
48 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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2answers
929 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
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2answers
233 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
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1answer
162 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-...
4
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2answers
1k 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
2answers
303 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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0answers
168 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
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1answer
267 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
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1answer
67 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 ...
2
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0answers
102 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 ...
2
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0answers
53 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
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2answers
269 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
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1answer
186 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
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2answers
446 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
1answer
248 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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0answers
142 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
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1answer
297 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
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2answers
382 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
1answer
75 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
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1answer
510 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
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2answers
582 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 ...
13
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0answers
247 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
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1answer
254 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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0answers
277 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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0answers
313 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 $\...
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0answers
155 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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4answers
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
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1answer
624 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 ...