For questions about or involving covering spaces in algebraic topology.

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Covering space covers simply connected set correctly

I found this proposition in a paper stated as a well known result from topology, but I can neither find this result in my textbooks nor proof it by myself: Let $p:E \rightarrow B$ be a covering ...
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54 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over ...
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25 views

Is it possible to “lower” a $\pi_1$-invariant differential function defined over the universal covering manifold to the base one?

Consider a differentiable manifold $M$ and its smooth universal covering $\pi:\tilde{M}\rightarrow M$. There is a canonical action of the fundamental group $\pi_1(M)$ on the covering manifold ...
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82 views

find a necessery and enough condition just using $\pi_{1}$.

suppose $p:\widetilde{X} \rightarrow X$ will be a covering space and $X$ is path connected and locally path connected, also $\widetilde{X}$ is connected, then find a necessary and enough condition ...
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1answer
39 views

“the standard two-fold branched cover of $CP^2$”

What could the following sentence mean: $\iota : S^2\times S^2 \rightarrow \mathbb{C}P^2$ is the standard two-fold branched cover, branched along the diagonal. What I can think of is to ...
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100 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
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32 views

Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
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2answers
62 views

Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?
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52 views

Morphism between covering spaces.

Let $p:Y\to X$ and $q:Z\to X$ be covering maps (of course $X,Y,Z$ are all Hausdorff, arcwise connected and locally arcwise connected) and $g:Y\to Z$ a morphism such that $q\circ g=p$. Then, $g$ is a ...
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37 views

Each open covering of F has a finite subcovering, how can F not complete be a problem?

I understand that if for each open covering, $\mathcal{O}$, of a set $F$, which is subset of a metric space $(X,d)$, there is a finite subcovering $\implies F$ is compact $\implies F$ is complete ...
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304 views

Universal Cover of a Surface (with Boundary)

I'm trying to see if there is a "nice-enough" way of describing/constructing the universal cover for a compact surface with n boundary components. Clearly, if $n=0$ , the classification theorem for ...
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1answer
81 views

Covering spaces and homotopical equivalence

I have this simple question: if $X$ and $Y$ are two topological spaces homotopically equivalents, have they the "same" covering spaces? (and if yes, in which sense?) This question derive from an ...
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1answer
290 views

Covering space(s) of $\mathbb{R}\text{P}^2$ minus one point

I know that the covering space of $\mathbb{R}P^2$ is $S^2$, and it is unique unless than isomorphism of covering spaces. Now, $S^2$ minus one point is homeomorphic to $\mathbb{R}^2$ (by stereographic ...
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169 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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1answer
90 views

Is a covering space of a completely regular space also completely regular

I'm trying to solve a problem in Munkres' Topology book. Let $p: E \rightarrow B$ be a covering map and suppose that $B$ is completely regular (for any closed subset $A$ and disjoint point $a$ there ...
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1answer
71 views

existence of double covering [duplicate]

Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?
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27 views

Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
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1answer
124 views

Constructing explicit lift of a circle homeomorphism

Studying a book by Luis Barreira in the Universitext Collection -- Dynamical Systems: an Introduction -- I'm told that given $f: S^{1} \to S^{1}$ homeomorphism, it's always possible to construct a ...
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1answer
46 views

Finite coverings are closed.

I'm working on solving as many of the exercises in Lenstra's Galois Theory for Schemes as possible, but there is one problem I'm partially stuck on. The statement of the problem is: ...
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58 views

Cover polygon with rectangles

I need to cover some polygon with rectangles here's an example : The black figure in a black square is the polygon that i need to cover with those green rectangles but i need to do it more ...
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0answers
64 views

Covering Spaces and Fundamental Groups

Can somebody tell me if what I did is right? I need to Draw the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B},v)$ corresponds to the subgroup $\langle a^{3}, a^{2}b\rangle$ ...
2
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1answer
146 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
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41 views

Degree and picture of a Covering map

I need to know if I am right: I need to know the degree of this covering map $R \rightarrow S$: $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ I have that genus of $R$, $g_{R} ...
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1answer
105 views

Genus of a surface

Let $S$ be a genus 3 closed orientable surface. Let $R\rightarrow S$ be a degree 2 covering map. What is the genus of $S$ ? Do I have to use the Euler characteristic of a surface presentation which is ...
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2answers
159 views

Covering map of a Torus

How would I draw (describe) a covering map given by $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ $T^{2}\#T^{2}\#T^{2}\#T^{2}\rightarrow T^{2}\#T^{2}$ and what would be the ...
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1answer
94 views

Find The Automorphism Group

I am taking my first course in Geometry and Topology and we are seeing the automorphism group of a covering. In class, my teacher gave some graphs and their automorphism groups, but he did not explain ...
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1answer
95 views

Automorphism groups of Graphs

What are the automorphism groups of the following regular covering spaces? I think the first picture is an 8-degree cover of the figure 8, whereas the second one is an infinite degree cover. I ...
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25 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
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1answer
76 views

Connected Covering Space of a Bouquet of 3 Circles

Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. What is $\pi_{1}(\hat{X})$ (It is a free group of what rank?). Any hints?
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1answer
85 views

Universal covering Spaces Drawings

I just have trouble drawing universal covers, how can I draw the universal covers of the following spaces: $X$ is the union of a circle with a projective plane $\mathbb{P}^2$ identified along a ...
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1answer
119 views

Homotopy groups of a covering space

This is a question related to the exercise 2218 from the book "Problems and Solutions in Mathematics" by Ta-Tsien, $2^{nd}$ Ed. Let $Z$ denote the figure 8 space, $Z = X \vee Y$, $X$ and $Y$ circles. ...
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80 views

For any element $e$ of an open set $V$ of a covering space, does there exist a sheet $S$ such that $e\in S\subseteq V$

Let $p:E\rightarrow X$ be a covering map. Let $V$ be any open subset of $E$ and $e$ be any element of $V$. I feel that the following statement must be true: There exists an evenly covered open subset ...
2
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1answer
226 views

On double covering of projective plane and map preserving antipodal points

Let $p:S^2\to P^2$ be the double covering of the real projective plane. Let $g:P^2 \to P^2$ be a map such that its induced homomorphism on fundamental group is not trivial. I'd like to show that ...
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1answer
83 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
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1answer
691 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
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2answers
453 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
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1answer
85 views

What's the colimit of the n-sheet covering spaces over the circle?

I was thinking in computing the filtered colimit of the n-sheet covering spaces $f_n: \mathbb{S}^1 \longrightarrow \mathbb{S}^1$ ($f(z) = z^n$) in the comma category of topological spaces with the ...
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42 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
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2answers
155 views

How to compute the fundamental group of a necklace of $\mathbb{S}^1$' s?

I was trying to compute $\pi_1 (X)$ where $X =$ "necklace of $n$ $\mathbb{S}^1$'s". At first, I tried using Van Kampen theorem however I could not find open sets $U$ and $V$ such that $U \cap V$ is ...
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86 views

finite covering space of non-orientable surfaces

Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers. A ...
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1answer
62 views

Monomorphisms and epimorphisms in the category of finite coverings of a topological space

I'm working my way through Lenstra's Galois Theory for Schemes, and I've run into a bit of a conundrum with Exercise 3.14(b). In this exercise, we consider the category $\textbf{FC}_X$ of finite ...
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1answer
77 views

Any homeomorphism is a covering map

Prove that any homeomorphism is a covering map. My thought: Let $p:X\to Y$ be a homeomorphism. Choose $y\in Y$. Then $Y$ is a open neighbourhood of $y$. Since $p$ is a homeomorphism, $p^{-1}(Y)=X$ ...
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3answers
130 views

Covering Space of the Pearl Necklace

Let $S^{2}_{1},\dots, S^{2}_{n}$ be disjoint copies of the unit sphere, and, for each $i\in\{1,\dots, n\}$, let $p_i,q_i\in S^{2}_{i}$ be distinct points. Define an equivalence relation $E\subseteq ...
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2answers
620 views

Function doesn't have a lift in a space related to Topologist's sine curve

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology: Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the ...
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0answers
112 views

Connected Components of Covering Space

Let $\pi : \tilde M\to M$ be a covering map, $K_1, K_2\subset\tilde M$ are two different connected components and there exists such points $x\in K_1, y\in K_2$ that $\pi(x)=\pi(y)$. In other words, ...
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1answer
89 views

Lifting elements of $SO(3)$ to $SU(2)$.

Let $A$ an element of ortogonal group $SO(3)$ such that the orders of $A$ is $>2$. We have that $SU(2)$ is a $2$-fold cover of $SO(3)$: $$ \mathbb{Z}_2 \to SU(2) \to SO(3) .$$ So how can I build a ...
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1answer
117 views

Does every map $\mathbb{R}P^n\rightarrow\mathbb{R}P^n$ lift to a pair of maps $S^n\rightarrow S^n$?

Question: Given a continuous map $f:\mathbb{R}P^n\rightarrow\mathbb{R}P^n$, is there automatically a continuous map $g:S^n\rightarrow S^n$ such that $f,g$ commute with the covering map ...
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1answer
91 views

Lifting problems existence

Let $g:\mathbb{R}^m \longrightarrow \mathbb{R}^m,g\in C^1(\mathbb{R}^m)$ such that: $\|g'(x)(v)\|\geq\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$ show that any rectilinear path ...
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3answers
294 views

Existence of a Minimal Cover

I'm well aware that for the sequence $x_n=\frac{1}{n}$, $\text{inf }x_n=0$ but $0 \notin (x_n)$. This made me think about something similar but when we are no longer thinking about existence of a ...
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0answers
95 views

Fundamental group and path-connected

Let $p:E \rightarrow B$ be a covering space, $E$ and $B$ are path-connected. Let $A$ be a path-connected subset of $B$ . How to use fundamental group to give a sufficient and necessary condition to ...