Questions tagged [covering-spaces]

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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 ...
Michael Albanese's user avatar
6 votes
2 answers
638 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 $k$...
Federico Fallucca's user avatar
4 votes
0 answers
180 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 ...
Hugo's user avatar
  • 384
3 votes
0 answers
150 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, ...
Jesse Wolfson's user avatar
16 votes
4 answers
3k 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 ...
GSM's user avatar
  • 343
4 votes
0 answers
362 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 ...
Martin Tancer's user avatar
-2 votes
1 answer
445 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 ...
Eduardo Longa's user avatar
23 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 ...
Kim's user avatar
  • 4,034
3 votes
0 answers
228 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 ...
Kim's user avatar
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3 votes
0 answers
246 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 ...
Arrow's user avatar
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1 vote
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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 ...
Eduardo Longa's user avatar
10 votes
2 answers
1k 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 ...
GSM's user avatar
  • 343
2 votes
2 answers
287 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 ...
user470881's user avatar
0 votes
1 answer
192 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-...
user138375's user avatar
7 votes
2 answers
2k 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 ...
user267839's user avatar
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5 votes
2 answers
402 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 ...
Daniel Loughran's user avatar
4 votes
0 answers
186 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 $\mathcal{O}$ ...
FKranhold's user avatar
  • 1,623
3 votes
1 answer
305 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 ...
FKranhold's user avatar
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3 votes
1 answer
79 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 ...
M.Ramana's user avatar
  • 1,172
2 votes
0 answers
157 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 ...
FKranhold's user avatar
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2 votes
0 answers
65 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 ...
user267839's user avatar
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3 votes
2 answers
329 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 ...
Natalio's user avatar
  • 133
4 votes
1 answer
253 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 ...
TheSilverDoe's user avatar
8 votes
2 answers
666 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)? ...
jlk's user avatar
  • 3,234
3 votes
1 answer
341 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 ...
Ian Iscoe's user avatar
  • 301
1 vote
0 answers
204 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 ...
Praphulla Koushik's user avatar
4 votes
1 answer
552 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 ...
Hempelicious's user avatar
2 votes
2 answers
483 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 ...
Paul Pfeiffer's user avatar
2 votes
1 answer
78 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 ...
Christopher King's user avatar
3 votes
2 answers
849 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}...
Jean Van Schaftingen's user avatar
3 votes
2 answers
806 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 ...
Jeremy's user avatar
  • 121
13 votes
0 answers
258 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 ...
David E Speyer's user avatar
5 votes
1 answer
490 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 $X$ ...
M.Ramana's user avatar
  • 1,172
7 votes
0 answers
359 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 ...
Ali Taghavi's user avatar
1 vote
0 answers
224 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 ...
Hebe's user avatar
  • 821
16 votes
4 answers
2k 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 ...
Bedovlat's user avatar
  • 1,939
12 votes
1 answer
774 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 ...
rpotrie's user avatar
  • 3,878
10 votes
2 answers
347 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 ...
Jens Reinhold's user avatar
4 votes
1 answer
500 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 ...
Arrow's user avatar
  • 10.3k
-1 votes
1 answer
124 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. ...
Arrow's user avatar
  • 10.3k
13 votes
0 answers
282 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 ...
evgeny's user avatar
  • 1,980
3 votes
0 answers
152 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 ...
მამუკა ჯიბლაძე's user avatar
6 votes
1 answer
993 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 ...
Arrow's user avatar
  • 10.3k
2 votes
1 answer
253 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 $...
Álvaro Krüger Ramos's user avatar
0 votes
1 answer
223 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 ...
Tensor_Product's user avatar
2 votes
2 answers
413 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....
Arrow's user avatar
  • 10.3k
2 votes
1 answer
442 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. ...
QSR's user avatar
  • 2,213
1 vote
1 answer
108 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\...
Shiquan Ren's user avatar
  • 1,970
6 votes
1 answer
182 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_{\...
Shiquan Ren's user avatar
  • 1,970
8 votes
2 answers
825 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 $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ ...
Shiquan Ren's user avatar
  • 1,970