For questions about or involving covering spaces in algebraic topology.

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14
votes
1answer
639 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$? [duplicate]

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...
29
votes
3answers
3k views

When is a local homeomorphism a covering map?

if $X$ and $Y$ are Hausdorff spaces, $f:X \to Y$ is a local homeomorphism, $X$ is compact, and $Y$ is connected, is $f$ a covering map? It seems to be, and I almost have a proof, but I'm stuck at the ...
5
votes
2answers
593 views

If a covering map has a section, is it a $1$-fold cover?

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
5
votes
4answers
2k views

Why this map is a covering map?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows ...
7
votes
0answers
2k views

The simply connected coverings of two homotopy equivalent spaces are homotopy equivalent [duplicate]

This is exercise 1.3.8 in Hatcher: Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then $\...
6
votes
1answer
848 views

composition of certain covering maps

This problem was posted before, but not the proof (because the asker knowed the answer), only a counterexample without the hypothesis of finite fibres. I want to know how to prove this proposition: ...
8
votes
2answers
289 views

If $\|\left(f'(x)\right)^{-1}\|\le 1 \Longrightarrow$ $f$ is an diffeomorphism

Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n,f\in C^1(\mathbb{R}^n)$ such that $\forall x \in \mathbb{R}^n\;,\;f'(x)$ is an isomorphism and: $$ \|\left(f'(x)\right)^{-1}\|\le 1\;,\forall x \in \...
8
votes
3answers
1k views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
4
votes
2answers
1k views

Covering space of a non-orientable surface

I have the following problem: Find the 2-sheeted (orientable) cover of the non-orientable surface of genus g. The cases $g=1,2$ are well-known, we have that the cover of $\mathbb{R}P^2$ is $\...
4
votes
2answers
617 views

The fiber of a covering space over a connected space has constant cardinality

Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for every $b \in B$. I know that $E$ has a ...
3
votes
3answers
626 views

Cayley complex as universal covering space

In Combinatorial Group Theory, Lyndon and Schupp construct a complex $K(X;R)$ from a presentation of group $G=(X;R)$, such that $G \simeq \pi_1(K,v)$ (proposition 2.3, p.117). Moreover, the Cayley ...
3
votes
1answer
965 views

covering map with finite fibres and preimage of a compact set

Let $f:X\to Y$ be a covering map (covering maps are surjective) , Y be compact set. And suppose that $f^{-1}(y) $ is finite for each $y\in Y$. Prove that $X$ is also compact. I think that this ...
2
votes
3answers
411 views

Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
2
votes
0answers
83 views

Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
2
votes
1answer
118 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 ...
5
votes
3answers
744 views

proving that a covering map with certain domain and range is homeomorphism

Let $p:E\to B$ be a covring map, with $E$ path connected. Show that if B is simply connected, then $p$ is a homeomorphism. Well I don't know exactly what can I do here, maybe I have to start with ...
9
votes
0answers
153 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
7
votes
1answer
543 views

Induced map on homology from a covering space isomorphism

Suppose $S^1 \times \mathbb{R}P^2$ covers some space. Why is it that any covering space isomorphism $h$ induces the identity map on $H_1$? I don't see how to prove this except maybe from looking at ...
11
votes
1answer
676 views

Prove that a covering map is a homeomorphism

I got stuck in the following exercise: Let $p:\widetilde{X}\rightarrow X$ be a covering map with $\widetilde{X}$ connected and $p^{-1}(x)$ finite, for every $x\in X$. Show that if there exists a ...
7
votes
1answer
231 views

Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
6
votes
1answer
152 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
5
votes
2answers
335 views

When a covering map is finite and connected, there exists a loop none of whose lifts is a loop.

I've read the following exercise. Let $p:\tilde X\to X$ be finite connected covering map. Show that there exists a loop in $X$ none of whose lifts is a loop. I can't understand why it's supposed ...
4
votes
2answers
325 views

Is a polynomial also a covering map?

Question: Let $p(z)$ be a polynomial over $\mathbb{C}$. Is it true that $p:\mathbb{C} \to \mathbb{C}$ is a covering map ? Partial answer: Let us look first at the points where $p'(z)\ne 0$. There the ...
4
votes
3answers
190 views

Why spherical coordinates is not a covering?

Maybe this is an idiot question and I'm committing a trivial mistake. Let $\phi (\theta, \varphi) = (\cos \theta \sin \varphi, \sin \theta\sin \varphi, \cos \varphi)$ be the usual covering of the ...
3
votes
2answers
241 views

If a connected open set is evenly covered, then its preimage is uniquely partitioned into slices

This is from Topology by Munkres: Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$...
3
votes
1answer
498 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 ...
9
votes
2answers
788 views

Is a covering space of a manifold always a manifold

Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both ...
8
votes
2answers
306 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
6
votes
2answers
887 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 $y$-...
5
votes
0answers
197 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 (...
4
votes
2answers
457 views

Does a morphism between covering spaces define a covering?

My question involves topological spaces $X$, $Y$ and $Z$, two coverings $p : Y \rightarrow X$ and $q: Z \rightarrow X$ of $X$ and a morphism $f: Y \rightarrow Z$ of coverings, i.e. a map which ...
4
votes
1answer
126 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 ...
4
votes
3answers
1k views

Another Question in Hatcher

First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again. The problem is 1.3.15. It reads ...
4
votes
1answer
392 views

Covering of a CW-complex is a CW-complex

Let $X$ be a CW- complex, with filtration $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X$. Let $p\colon E \to X$ be a covering space. Prove that $E$ is a CW complex with filtration given ...
4
votes
1answer
185 views

Domain is Hausdorff if image of covering map is Hausdorff

Suppose that $p:X\rightarrow Y$ is a covering map. Show that if $Y$ is Hausdorff, then so is $X$. I have an answer but I'm not sure if it's right? By definition of Hausdorff, $\forall x,y, \in Y, x\...
3
votes
1answer
175 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
3
votes
2answers
319 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
2
votes
1answer
348 views

Question on covering spaces

Let $q:X\to Y$ and $r:Y \to Z$ be covering maps;let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering space. I'm confused that $p=r\circ q$ is obvious ...
2
votes
2answers
651 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 ...
2
votes
1answer
469 views

Monodromy correspondence

Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real ...
2
votes
2answers
149 views

Universal Cover of a Surface with Boundary. What does Cantor set on Boundary Correspond to?

I am trying to understand in more detail the answer to: Universal Cover of a Surface (with Boundary) It is mentioned that the universal cover of a hyperbolic surface $S$ with geodesic boundary is a ...
2
votes
1answer
132 views

about path connected covering spaces.

Let $p:E\rightarrow X$ be a covering space. It is well known that if $X$ is connected, then all the fibers have the same cardinality. This can be seen as a simple consequence of the fact that the ...
2
votes
1answer
339 views

Alternative definition of covering spaces.

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and $B$...
1
vote
1answer
1k 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 ...
1
vote
3answers
153 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 (\...
1
vote
2answers
411 views

Covering space and Fundamental group

Let $p:E\to X$ be a covering space and $\pi_1(E)$ be a fundamental group of $E$. Can you give me a recept for calculating a fundamental group $\pi_1(X)$ (may be for some special cases)? Thanks a lot!
5
votes
1answer
55 views

Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
3
votes
1answer
93 views

simply connected covering of a path connected space

Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set. Show that the inclusion induced homomorphism $i_{\sharp} : \...
3
votes
2answers
89 views

Classifying Covering Spaces using First Cohomology

I am familiar with the classification of covering spaces of a space $X$ in terms of subgroups of $\pi_1(X)$ (up to conjugation). However, if $X$ is a manifold, I know that $H^1(X; G)$ classifies G-...
2
votes
1answer
360 views

Is the product of covering maps a covering map?

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times ...