Questions tagged [covering-spaces]
The covering-spaces tag has no usage guidance.
3
votes
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
4answers
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
$$
f:M\to M/\Sigma_k.
...
1
vote
1answer
82 views
vector bundles induced by an action of a finite subgroup of $O(n)$
Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow M\...
6
votes
1answer
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
$$
\xi:\mathbb{R}^k\longrightarrow M\times_{\...
6
votes
2answers
645 views
quotient space of Eilenberg-MacLane space
Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
...
1
vote
0answers
129 views
Acyclicity of covering space
Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...
2
votes
2answers
316 views
Is the following 3-manifold irreducible?
We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...
1
vote
1answer
130 views
triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle
Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \...
6
votes
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
2answers
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
2answers
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
3answers
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
0answers
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
1answer
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 ...
