For questions about or involving covering spaces in algebraic topology.

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14
votes
1answer
618 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 ...
6
votes
1answer
828 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: ...
5
votes
2answers
572 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 ...
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 ...
8
votes
2answers
1k views

Covering space Hausdorff implies base space Hausdorff

There is an exercise problem in Hatcher's Algebraic Topology book asking to show that if $p:\tilde{X}\rightarrow X$ is a covering space with $p^{-1}(x)$ finite and nonempty for all $x\in X$, then ...
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
607 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 ...
3
votes
1answer
947 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 ...
3
votes
3answers
615 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 ...
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 ...
2
votes
1answer
443 views

$G$ topological group, $H$ discrete normal subgroup, $p$ projection, form Covering Space.

Let $G$ be a topological group. Let $H$ be a discrete normal subgroup of $G$. Let $p : G \to G/H$ be the projection map. Show that $(G, p, G/H)$ form a covering space. Here is what I have so far: ...
2
votes
0answers
138 views

Hyperbolic Universal Covering Space

I have been working with Ricci flow in the euclidean and hyperbolic space but have been having considerable trouble determining how to generate a universal covering space for complex hyperbolic ...
1
vote
0answers
79 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 ...
5
votes
3answers
725 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 ...
7
votes
1answer
533 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
660 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 ...
9
votes
0answers
147 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
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
151 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
334 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
3answers
182 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 ...
4
votes
2answers
321 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 ...
3
votes
1answer
483 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 ...
3
votes
2answers
233 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 ...
2
votes
1answer
520 views

this map $p(z)=z^n$ is a covering map?

For any positive integer $n$ the map $p:S^1\to S^1$, defined by $p(z)=z^n$ is a covering map. I think it's easy, but as I'm a really beginner I'm struggling to prove it. By the way, what kind of ...
9
votes
2answers
773 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
302 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
870 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 ...
5
votes
0answers
193 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
1answer
372 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 ...
4
votes
1answer
124 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
2answers
627 views

Universal cover of a figure eight?

An example in my lecture notes says, 'draw a simply connected covering space over the figure eight'. Howerver, after googling, wikipedia tells me that ''The universal cover of the figure eight can ...
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, ...
4
votes
1answer
659 views

Covering space homeomorphism

In the course of an exercise from Hatcher's topology text, I came to the following point. Given $p: \tilde{X} \to X$ the universal cover for $X$, and a continuous map $h: \tilde{X} \to \tilde{X}$ ...
4
votes
2answers
447 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
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 ...
3
votes
2answers
309 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 ...
3
votes
1answer
474 views

Domain is locally path-connected if image of a covering map is locally path-connected.

Show that if $p:X\rightarrow Y$ is a covering map and $Y$ is locally path-connected, then so is $X$. How do you go about proving this? I can think of two ways of doing this, either by definition of ...
2
votes
1answer
130 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
2answers
143 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
3answers
390 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
2answers
643 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
332 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 ...
2
votes
1answer
344 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
1answer
673 views

Composition of coverings of path connected spaces

Do there exist covering maps $p:X\rightarrow Y$ and $q:Y\rightarrow Z$ such that $X$ is path connected and the composition $q\circ p$ fails to be a covering map?
2
votes
1answer
458 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 ...