Questions tagged [covering-spaces]

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Classification of bundles with fixed total space

I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
Matthew Kvalheim's user avatar
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1 answer
260 views

what is this simple topological space?

Take $M_p$ the mapping cylinder (MC) of the $p=3$-fold cover of $S^1$, $M_q$ the MC of the $q=2$-fold cover of $S^1$, where for both, the identification of the MC is done on the side $\{0\}$ of $[0,1]$...
Virgile Guemard's user avatar
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0 answers
65 views

Existence of covering space with trivial pullback map on $H^1$

I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
F. Müller's user avatar
7 votes
0 answers
293 views
+50

Is there a purely topological definition of $\text{Spin}(p,q)$?

I'm cross-posting this question from Math.SE, as it didn't get much attention there (even after a bounty). A common way to define the group $\text{Spin}(p,q)$ is via Clifford algebras. However, $\text{...
WillG's user avatar
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3 votes
2 answers
328 views

Determine monodromy representation from local system

Let $X$ be a path-connected manifold nice enough such it's universal covering space $p:\widetilde{X} \to X$ exists, $k$ a field. Then there exist a wellknown correspondence $$ \{\textit{linear}\text{ ...
JackYo's user avatar
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2 votes
0 answers
247 views

Homological algebra generalization of covering map

I would like to know if there exists an operation in homological algebra that generalizes the notion of covering maps for abstract chain complexes (over any field or ring, or maybe just certains where ...
Virgile Guemard's user avatar
2 votes
0 answers
128 views

Symmetric group-cocycle descends to symmetric product

Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
KuSi's user avatar
  • 53
0 votes
1 answer
242 views

Universal covering of symmetric product

Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
KuSi's user avatar
  • 53
9 votes
2 answers
481 views

For which spaces $S^n$ ($n\geq 2$) is a universal covering space?

I know that $S^n$ $(n\geq 2)$ is a universal covering space for itself and $\mathbb{RP}^n$. But my question is, for which spaces (up to homotopy equivalence) is $S^n$ ($n\geq 2$) a universal covering ...
M.Ramana's user avatar
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1 vote
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92 views

Relation of branched covers and groups

I am self-studying covering spaces of topological spaces. The following question comes to my mind. In the case of topological covering spaces, we have a nice relation between the fundamental group of ...
KAK's user avatar
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64 views

Confusion about Turaev's description of G-bundles on the cylinder and pairs of pants

In Homotopy Field Theory in dimension 2 and group algebras, section 4.6, page 24, Turaev considers an annulus $C = S^1 \times [0,1]$ (thought of as a cobordism from $C_0 = S^1 \times \{ 0\}$ to $C_1 = ...
Tanny Sieben's user avatar
5 votes
2 answers
221 views

References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?

Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category? I know that one can define "categorical ...
Tanny Sieben's user avatar
7 votes
1 answer
376 views

When do covering spaces correspond to characteristic subgroups?

Given a covering space $p \colon X \to Y$, we get an injection $p^* \colon \pi_1(X) \to \pi_1(Y)$, and we know that the image $p^*(\pi_1(X))$ is normal in $\pi_1(Y)$ if an only if $p$ is regular, that ...
Anschel Schaffer-Cohen's user avatar
7 votes
2 answers
506 views

Covering of a knot complement

Let $B=S^3\setminus K$ for some (tame) knot $K$. Suppose we have a covering $E\to B$ with a finite fiber. Question: is $E$ homeomorphic to a knot/link complement? On this question I found only the ...
Andrey Ryabichev's user avatar
3 votes
0 answers
81 views

(When) can you embed a closed map with finite discrete fibers into a (branched) cover?

Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers. Questions. Given closed map $X\to S$ ...
Arrow's user avatar
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2 votes
1 answer
65 views

Sets with a good lift under a covering

Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V_d \subset X$ ...
Ville Salo's user avatar
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6 votes
1 answer
400 views

Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?

I essentially am asking for an explanation of the comment under this post by Tom Goodwillie. In the "Kerodon", Lurie defines a simplicial covering map as follows: A map $p:E\to X$ of ...
FShrike's user avatar
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Relation between projective representation and the representation of the universal cover of a Lie Group

I am reading this paper, in what says exactly: "Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...
Gabriel Palau's user avatar
3 votes
1 answer
80 views

Can a compact good orbifold be realized as a global quotient of a compact manifold?

Let $\mathcal{O}$ be a compact good orbifold, where we understand a good orbifold to be an orbifold obtained as a global quotient $M/G$, where $M$ is a manifold and $G$ is a discrete group. Are there ...
gpr1's user avatar
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0 answers
33 views

Arbitrarily high degree planar covers?

All the graphs I want to discuss are finite, simple, and connected. A graph $G_1$ covers another graph $G_2$ if there is a surjective map $\pi : V(G_1) \to V(G_2)$ that sends edges to edges and such ...
Sprotte's user avatar
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5 votes
0 answers
158 views

Length metrics on covering spaces

This is a question (Exercise 3.30(2)) in the book `Metric spaces of non-positive curvature' written by Bridson and Haefliger. In the book, there is the following proposition (Proposition 3.28) Let $p:\...
Sangrok Oh's user avatar
1 vote
1 answer
161 views

Does fiber bundles admits good properties of covering spaces?

Let $X$ and $Y$ be non compact complex manifolds and $f:X\to Y$ be a holomorphic fiber bundle with fibers $F$ such that $f^*:\pi_1(X)\to\pi_1(Y)$ is injective and let for any $f_1,f_2\in F$ there ...
tota's user avatar
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6 votes
0 answers
178 views

G-sheaves on spaces with a free G-action

Let $X$ be a topological space equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined "$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space, ...
Misha Verbitsky's user avatar
0 votes
2 answers
246 views

Finite sheeted covering of the complement of a finite set in $\mathbb{C}$

For figure "eight" there is a list of finite sheeted covering discussed in Hatcher's book "Algebraic topology". I was thinking about the following question: Let $S$ be a finite ...
piper1967's user avatar
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0 votes
1 answer
724 views

What is definition of branched covering?

What is definition of branched covering in the page 10 of following paper ? In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...
Usa's user avatar
  • 119
2 votes
1 answer
251 views

Explicit universal covering map for higher genus algebraic curves

Suppose I have a projective plane curve $C = V(F)$ defined over $\mathbb{C}$, where $F$ is some homogeneous polynomial in three variables. For the sake of simplicity, let's assume that $C$ is ...
Danny Stoll's user avatar
3 votes
1 answer
206 views

The monodromy in the proof of Little Picard via Klein's $J$

First of all, my question is in the context of formalising proofs in the proof assistant Isabelle/HOL, so my questions are a bit guided by what material we have in the complex analysis library there. ...
Manuel Eberl's user avatar
  • 1,181
5 votes
1 answer
235 views

Nondegeneracy of kernel of map on homology induced by covering of surfaces

Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f_\ast: H_1(X) \rightarrow H_1(Y)$. Here homology has rational ...
Nicole's user avatar
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11 votes
1 answer
529 views

Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$?

There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with ...
Michael Albanese's user avatar
2 votes
0 answers
151 views

Semisimple covers of varieties

Let $X$ be an algebraic variety. The finite étale covers of $X$ are measured by the étale fundamental group $\pi_1^{\rm et}(X)$. On the other hand, the Cox ring ${\rm Cox}(X)$ of $X$ (finitely ...
Joaquín Moraga's user avatar
5 votes
0 answers
309 views

To what extent are geometric methods being used to attack the inverse Galois problem?

My limited knowledge so far is that some groups have been constructed geometrically using the theory of covering spaces, then applying Hilbert irreducibility. Is there a deeper way in which inverse ...
Nicolas Banks's user avatar
0 votes
0 answers
87 views

Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]

Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
wonderich's user avatar
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2 votes
0 answers
155 views

Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?

Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
wonderich's user avatar
  • 10.3k
16 votes
2 answers
725 views

What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...
Thom's user avatar
  • 169
5 votes
1 answer
367 views

Minimum number of generators for quotients of congruence subgroups of SL(2, Z)

For a given positive integer $N$ let $L(N)$ denote the principal congruence subgroup of $\operatorname{SL}(2, \mathbb{Z})$ of level $N$. It is known that $L(N)$ is a finitely generated free group. Let ...
sugata mondal's user avatar
1 vote
1 answer
479 views

Local diffeomorphisms, covering maps and smooth path lifting

Let $f: M\to N$ be a surjective local diffeomorphism of noncompact smooth manifolds. Suppose that every smooth path is liftable, that is, for any smooth path $\gamma: [0,1]\to N$ and any point $p\in f^...
Dmitrii Korshunov's user avatar
5 votes
1 answer
725 views

Construction of the universal covering space of the etale homotopy type $Et(X)$

Let $X$ be a nice scheme (additional assumptions could be added), and let $Et(X)$ be its (Artin-Mazur) etale homotopy type. I am looking for a/the scheme $Y$ over $X$ whose etale homotopy type $Et(Y)$...
Moutand Mohammed's user avatar
3 votes
0 answers
222 views

Universal cover of finetely connected surface with boundary

Let $M$ be a finetely connected orientable surface with compact boundary. This means $M$ is homeomorphic to a compact orientable surface $\Sigma$ of genus $g \geq 0$ minus $r \geq 1$ points and minus $...
Eduardo Longa's user avatar
4 votes
0 answers
58 views

Fundamental group of the complement of some quadric cones

cross-posting from MathSE Problem Consider the domain $$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$ and the map $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...
Samuele's user avatar
  • 1,185
12 votes
1 answer
417 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 ...
Johnny El Curvas's user avatar
4 votes
1 answer
430 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 $\...
cellular's user avatar
  • 913
2 votes
1 answer
200 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/...
André Gomes's user avatar
2 votes
1 answer
556 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 ...
Nikhil Sahoo's user avatar
  • 1,175
2 votes
1 answer
251 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 ...
Someone's user avatar
  • 265
13 votes
0 answers
670 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 ...
Sumanta's user avatar
  • 622
3 votes
0 answers
204 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\...
Chetan Vuppulury's user avatar
3 votes
2 answers
1k 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 ...
user avatar
1 vote
1 answer
215 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 $X'$. ...
Shiquan Ren's user avatar
  • 1,970
2 votes
0 answers
124 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 ...
Odylo Abdalla Costa's user avatar
22 votes
2 answers
1k 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 ...
Michael Albanese's user avatar