Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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Covering space of compact surface with free fundamental group

Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ? I know that if we ...
Serge the Toaster's user avatar
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Hatcher Theorem 1B.8 / Proposition 1B.9 concluding that a map inducing the identity is homotopic to the identity?

In introducing the concept of $K(G, 1)$, Hatcher's Algebraic Topology proves the theorem that the homotopy type of a CW complex $K(G, 1)$ is uniquely determined by $G$, by citing a proposition and ...
I Eat Groups's user avatar
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Double cover vs universal cover for ${\rm SO}^+(p,q)$

I am confused about the double covers and universal cover of ${\rm SO}^+(p,q)$, where by this notation I mean the connected component with the identity. Previously I thought that ${\rm SO}^+(p,q)$ had ...
Gold's user avatar
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Covering spaces induced from a group homomorphism

I am reading a book and there is the following statement: Let $X$ be a nice enough space (e.g. a CW-complex) such that covering theory is applicable. Let $G$ be any group. Then one can identify the ...
The_Rookie's user avatar
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Bijection from (proper) Lorentz group to PSL(2,C)

It is well known that $SL_2(\mathbb{C})$ is the universal cover of $SO^+(1,3)$, see for example the Wikipedia page on the Lorentz group 1. The map goes like this (some of this is not standard ...
TheEmptyFunction's user avatar
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For $(G,X)$ - manifolds can we assume $X$ is simply connected?

This is based on my rough understanding, so let me know which part if any is wrong. Suppose $M$ is a $(G,X)$-manifold, where $X$ is a homogeneous $G$-space. Pullback a $(G,X)$ structure from $X$ to ...
subrosar's user avatar
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covering of a topological space that has coverings with $n$ sheets.

Let $X$ be locally path-connected, path-connected, and semilocally simply connected by paths. Suppose that $X$ admits a covering with $n$ sheets for every $n \in \mathbb{N}$. Show then that $X$ admits ...
Andreadel1988's user avatar
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covering of subspaces of $S^1 \vee S^1$

Let $Y=\mathbb{R}^2$ be the infinite square grid, i.e., the graph with vertices $\mathbb{Z}^2$ and edges representing all segments connecting points at distance $1$, which is the universal covering of ...
Andreadel1988's user avatar
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Construction of Covering space by Galois correspondence

We know, If $X$ is path connected,locally pathconnected,semi locally simply connected then $X$ has a universal cover $\overline X$.Now, By Galois Correspondence for any subgroup $H$ of $π_{1}(X)$ we ...
Nope's user avatar
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Requirements in addition to path lifting between manifolds that imply covering map

I am interested in what conditions can I add to path lifting that imply covering. Let $X$ and $Y$ be two topological manifolds, and $p:X\to Y$ be a continuous map for which path lifting holds. Theorem ...
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Is there path lifting from the circle to the long line?

I understand that the long line isn't a covering space for $S^1$, as has been discussed in other places, such as here. However, does the projection map to $S^1$ satisfy path lifting? Intuitively it ...
SarthakC's user avatar
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When does a covering map have a section?

This question is motivated by Show that the map $s : X \longrightarrow \widetilde X$ is well-defined. In the sequel let the following data be given: A covering map $p : \tilde X \to X$ be with path-...
Paul Frost's user avatar
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Computing the Fiber of This Map

I am reading the answer here, but I don't understand why "the fibres look like cosets $H/gHg^{-1}$." Here are my thoughts. Fix some $H$-orbit of $\tilde{X}$ and pick a point $x$ in the orbit....
Vasting's user avatar
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Constructing a covering space of the wedge of two circles correponding to a subgroup of the fundamental group

I am learning covering spaces for the first time and came across the following problem: Describe the covering space of the wedge of two circles that corresponds to the subgroup of $\pi_1(S^1\vee S^1) ...
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Covering the sphere by a torus, branched over four points with ramification indices 2,2,3,3?

I'm hoping to find a covering map $\Gamma: \Sigma_{g}\rightarrow CP^1$ satisfying following conditions $\Gamma(z_i)=x_i$, where $i=1,2,3,4$ $\Gamma(z)=x_i +a_i (z-z_i)^{w_i}+\cdots$, where $w_1=2,w_2=...
Vayne's user avatar
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Map between a connected covering space

Let $p : Y \to X$ be a covering map and $Y$ is connected. There exist another covering map $q : Y \to X$ and a map $r : Y \to Y$ such that $p \circ r = q$ Is it true that $r$ is a covering map of $Y$? ...
Eloon_Mask_P's user avatar
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If $p : Y \to X$ is covering and $Y$ is Hausdorff then $X$ is so? [duplicate]

I want to know if $p : Y \to X$ is covering and $Y$ is Hausdorff then is $X$ hausdorff (I dont know if this is true). My attempt is this : let $a \ne b \in X.$ There exist nbd $U_a$ of $a$ and $U_b$ ...
Eloon_Mask_P's user avatar
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How simple are simple spaces in regards to twisted coefficients?

For context, I am working on my bachelor thesis on algebraic K-theory. I have "finished" the basics of algebraic topology, by which I mean the main content of Hatcher's book. For my project ...
DevVorb's user avatar
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$f$ continuous over $S^{1}$ has a fixed point

Let $f:S^{1}\rightarrow S^{1}$ be a continous map such that $f$ is homotopic to a constant function $c_{x_{0}}$. Then $f$ has a fixed point. I already know how to prove this using fixed point ...
Dungessio's user avatar
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Obstruction to construction of $n$-sheeted covering space of simply connected Riemann surface

When considering the covering spaces of $S^1$ (I use as base point $0$), usually in textbooks or courses, you see the spiral with some winding number $n$ to represent a $n$-sheeted covering space. To ...
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restriction on number of branch points of holomorphic functions

The image of the function $z^n$ is an $n$-fold covering of the Riemann sphere with two branch points at 0 and $\infty$. Is there a holomorphic function that covers the Riemann sphere with exactly one ...
node196884's user avatar
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A complete exercise of Algebraic topology [closed]

My teacher of Algebraic topology has shared exams from past years and this exercise appears on them: Consider the following sets on the Euclidean plane $\Bbb{R}^2$: $$X_1 = \{(x,y) \in \Bbb{R}^2 : x^...
Superdivinidad's user avatar
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Universal cover $S^1\vee S^2\vee S^3$

Im trying to understand the Universal Cover of the space $S^1\vee S^2\vee S^3$. I've been told it has the form of attaching a copy of $S^2\vee S^3$ at every integer of the real numbers (we call this $...
NoIdea's user avatar
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Computing the covering space of a cylinder

Denote for every $n \in \Bbb{Z}, n\geq 1$ the Euclidean topology of $\Bbb{R}^n$ as $\mathcal{T}_E^n$. From my Algebraic topology lesssons, I have the following excercise: Consider $C:=\{(x,y,z) \in \...
Daniel García's user avatar
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Non-normal coating of the Klein bottle

(a) Provide an example of a non-normal connected covering map $p:(\tilde X, \tilde x_0) \to (K, x_0)$ where $K$ is the Klein bottle. (b) Choose $x_0 \in K$ and $\tilde x_0 \in p^{-1}(x_0)$. State what ...
Andreadel1988's user avatar
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Image of paths closed in a covering

Let $p:(\tilde X, \tilde x_0)\rightarrow(X, x_0)$ be a covering map that is path-connected and locally path-connected. Is it true that given $\gamma$ and $\gamma'$, two continuous paths in $\tilde X$ ...
Andreadel1988's user avatar
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H-spaces and universal covering space

Let $X$ be an $H$ space and $\tilde{H}$ it's universal cover. It is well known that the action of $\pi_1(H)$ on $\pi_n(H)$ is trivial. I am seeking an alternative proof of this fact using the ...
DevVorb's user avatar
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Covering space of the unit circle

I am reading algebraic topology on my own. I am facing some problem in notation. Consider one of the covering space of circle, that is $\mathbb R$. And consider the covering map $p: \mathbb R \...
user1180312's user avatar
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1 answer
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The universal cover $\Phi: H \longrightarrow G$ of a connected Lie Group $G$ induces a homeomorphism $H/Ker(\Phi) \cong G$

Let $G$ be a connected Lie Group. Let $H$ be a simply connected Lie group and $\Phi:H \longrightarrow G$ a Lie group homomorphism, such that the associated Lie Algebra homomorphism $\phi: Lie(H) \...
Drops's user avatar
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Why does the covering map $\pi: \tilde{G} \longrightarrow G$ of a connected Lie group $G$ induces a homeomorphism $\tilde{G}/\ker \cong G$?

Let $G$ be a Lie group, $\tilde{G}$ its simply connected universal cover and $\pi: \tilde{G} \longrightarrow G$ the associated covering map. Then $\tilde{G}$ is also a Lie group and $\pi$ is a Lie ...
Drops's user avatar
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What Precisely is an "n-fold covering space"?

The problem I am working on asks the following: Explicitely construct a 3-fold covering of $S^1\vee S^1$. The question I have, however is more general than this one. I have seen questions similar to ...
Kenneth Winters's user avatar
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What is the relationship between the kernel of a homomorphism and its degree as a covering map?

Let $\phi: A \rightarrow B$ be a continuous homomorphism between Lie groups (or any appropriate generalization). Is there any relationship between the degree of $\phi$ as a covering map of and the ...
CBBAM's user avatar
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2 votes
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Reference request for the fact that if the base space is orientable then the total space is orientable too (for surfaces)?

I need a reference for the fact that if you have a covering $f:\tilde{\Sigma}\longrightarrow\Sigma$ and $\Sigma$ is orientable then $\tilde{\Sigma}$ is also orientable. $\tilde{\Sigma}$ and $\Sigma$ ...
MySelf's user avatar
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Pushfoward of a CW complex structure by a covering map

Let $p:Y\to X$ be a covering map. If $X$ has a CW complex structure, then we can give a CW complex structure on $Y$ so that $p$ becomes a cellular map, by lifting the characteristic maps (cf. Euler ...
user302934's user avatar
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A problem about covering space in Munkres. Why the elements in the neighborhood can only has exactly $k$ preimages.

When I tried to deal with this problem, I defined 2 sets $$\begin{align*} C= \left\{ b \in B: |p^{-1}(b)| =k \right\} \\ D= \left\{ b \in B : |p^{-1}(b)| \neq k \right\} \end{align*}$$ Since $b_{...
M_k's user avatar
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3 votes
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Group objects in the category of covers

Let $X$ be a topological space. Consider the category of (finite) covers of $X$, which we denote by $\mathrm{Cov}(X)$. What are the group objects in $\mathrm{Cov}(X)$? Assume that $X$ admits an ...
Ben's user avatar
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Finding covering spaces of $\mathbb RP^2\vee \mathbb RP^2$

My question is connected (pun intended) to the Exercise 1.3.14 of Allen Hatcher's Algebraic Topology. The question is to find all connected covering spaces of $\mathbb RP^2\vee \mathbb RP^2.$ I ...
Susan Bradely's user avatar
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Prove $X$ has the weak topology where $p:X\to Y$ is a covering map.

This question has been answered here, but the method used is different than the one I learned in my algebraic topology course, which I believe I should use. I showed that given that $Y$ is a CW ...
user13121312's user avatar
8 votes
1 answer
205 views

Does every finitely presented group have a finite index subgroup with free abelianisation?

Let $G$ be a finitely presented group. Does there exist a finite index subgroup $H$ such that its abelianisation $H^{\text{ab}} = H/[H, H]$ is free abelian? Note, if $G^{\text{ab}}$ is not already ...
Michael Albanese's user avatar
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Can you explain to me the relationship between those two definitions of branched covering on surfaces?

I'm studying branched (or ramified) coverings between surfaces and the definition that was given in the book I'm reading is the following: "Lets consider two closed and connected surfaces $M$ and ...
MySelf's user avatar
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1 answer
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Can one prove the covering space of a manifold is second countable without using the fact that the fundamental group of a manifold is countable?

I'm trying to prove that the covering space X of a manifold Y is a manifold of the same dimension. I was stuck on proving X is second countable, more specifically, on proving the fiber of any element ...
minukesis's user avatar
1 vote
1 answer
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Reference for universal cover of $\mathrm{SL}_2(\mathbb{R})$

I want to know about books, expository papers o lecture notes on the universal covering group $\widetilde{\mathrm{SL}_2(\mathbb{R})}$ of the Lie group $\mathrm{SL}_2(\mathbb{R})$. The Wikipedia page ...
Albert's user avatar
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2 votes
1 answer
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Manifold of $SU(2)$ and it's relation to physical rotations

I'm a physicist trying to make sense of some topics of group theory and their applications to quantum mechanics. Let me state the things I think I understand Special Orthogonal Group $SO(3)$ This is ...
P. C. Spaniel's user avatar
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0 answers
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Covers of $S^1 \vee S^1$ are directed 2-colored graphs

I understand that every directed 2-colored graph defines a cover for $S^1 \vee S^1$. But how would one go about showing the other direction, that $\textbf{every}$ cover of $S^1 \vee S^1$ is a directed ...
Minerva's user avatar
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Fundamental group of one of the covering spaces of $S^1 \vee S^1$ from Hatcher's book [closed]

I have a question about the fundamental group of the covering space of $S^1 \vee S^1$ from part (6) of the following table from Hatcher's book enter image description here Hatcher claims that all the ...
John Do's user avatar
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1 answer
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Proof of injective correspondence on covering space

Let $ \Gamma $ $:\hat {X} \rightarrow X$ be a covering space and $ x \in \hat {X}.$$\quad$$ \Gamma (x) = x \in X$. Let $ F = \Gamma ^{-1} (x)$. We define the function $\phi : \pi_{1}(X,x) \rightarrow ...
User2018's user avatar
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2 votes
1 answer
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Characterization of connected space via fully faithfulness of trivial covering functor

Let $X$ be a topological space, and $F: \text{Set} \to \text{Cov}_X$ be the functor sending a set $S$ to the trivial covering $S \times X \to X$ of $X$ with fiber $S$. I want to prove that $X$ is ...
Alice in Wonderland's user avatar
1 vote
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How do we generate the loop $ba$ from the loops $a^2,b^2$ and $ab\ $?

In the second diagram a $2$-sheeted connected covering of the figure eight has been described. The image of the fundamental group of the covering space has the generators $a^2, b^2$ and $ab$ as ...
Akiro Kurosawa's user avatar
2 votes
1 answer
279 views

How to determine the fundamental group of $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.

Consider a manifold $ N $ defined as follows $$ N=\{n\otimes n-m\otimes m:n,m\in S^2,\,\,(n,m)=0\}\subset M^{3\times 3}, $$ where $ S^2 $ denotes the two dimensional sphere, $ (\cdot,\cdot) $ ...
Luis Yanka Annalisc's user avatar
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1 answer
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Visualizing the last two paragraphs of 3.2.3(b) solution.

Here is the question I am trying to understand its solution: Finish the proof of Borsuk-Ulam theorem (Hatcher) I did not understand this part of the solution: Let $\gamma:[0,1]\to \Bbb RP^n$ be a ...
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