For questions about or involving covering spaces in algebraic topology.

learn more… | top users | synonyms (1)

0
votes
1answer
22 views

Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
2
votes
1answer
71 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
0
votes
3answers
29 views

Is a covering map on compact metric space, $k$ - to- $1$ at all points?

Let $X$,$Y$ be topological space, surjective map $\varphi:X\rightarrow Y$ is called a covering map if there is an open cover $\{U_{\alpha}\}$ of $Y$ such that for every $\alpha$, ...
10
votes
1answer
410 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
2
votes
1answer
35 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let ...
15
votes
4answers
856 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
2
votes
1answer
34 views

What is the intuition behind covering spaces?

I've come to study this definition and become interested on the intuition behind it mainly because of the study of spinors, motivated by Quantum Mechanics. The definition of covering space is as ...
2
votes
1answer
23 views

Given that $H^1(X)=0$ on a connected space, show that all maps to $X\to S^1$ are null homotopic

Let $X$ be a path-connected, locally path-connected topological space, with $H^1(X)=0$. I would like to show that any map $f:X\to S^1$ is null homotopic, but I haven't really made any progress. ...
0
votes
1answer
17 views

Question about covering map

Let $(X, d_1),(Y, d_2)$ be metric space, $f:X\rightarrow Y$ is called covering map, if for evry $y\in Y$, there is open set $U$ of $y$ such that $f^{-1}(U)$ is a union of disjoint open sets in $X$, ...
1
vote
0answers
23 views

Computing the group of deck transformations w.r.t. a polynomial

Let $p: \mathbb{C}\backslash Y' \to \mathbb{C}\backslash X'$ be a polynomial where $Y'$ is the set of branch points and $X'$ is the image of $Y'$ under $p$. If $\deg p = n$ then $p$ is an unbranched ...
1
vote
2answers
84 views

A detail about reconstructing covering space from the action $\pi_1(X,x_0)\to S_{p^{-1}(x_0)}$ in Hatcher's book

I'm struggling understanding a small sentece from Hatcher's Algebraic topology book (available online for free). In page 70 Hatcher wants to reconstruct the covering $p:\tilde X\to X$ from the ...
0
votes
1answer
64 views

Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
0
votes
0answers
25 views

Free action of a discrete group gives a covering space

I'd like to find a short proof of the following seemingly basic fact. Suppose a discrete group $G$ acts freely on a manifold $X$ with the quotient $X/G$ being compact. Then $X$ is a covering space of ...
3
votes
2answers
310 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
0answers
62 views

limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
0
votes
0answers
16 views

Deck group of a connected n-fold cover must have at most n elements

Let $p:Y\to X$ be an $n$-fold covering map, with $Y$ connected. Show that $Deck(p)$ has at most $n$ elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct ...
1
vote
0answers
31 views

Fundamental group of a covering space

I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$. However, I do not understand what is implied about the fundamental groups of ...
6
votes
2answers
49 views

Is the Fundamental Group of space with contractible universal cover torsion free?

Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle $K$ torsion free? We have the following presentation: $$\pi_1(K) = \langle a,b: aba = b ...
2
votes
1answer
343 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 ...
0
votes
2answers
49 views

Size of the deck transformation group

If $p\colon Y\to X$ is a $k$-fold covering map, and $Y$ is path-connected, what is the size of Deck($p$), the deck transformation group? I was attempting to prove that the answer is $\leq k$, but ...
1
vote
1answer
38 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
1
vote
2answers
64 views

Two lifts of a local homeomorphism

Just learning about sheaves. Suppose I have a sheaf $\mathscr{F}$ on a topological space. (The sheaf can take values in sets, let's say.) Let $E \overset{\pi}{\to}X$ be the etale space of this ...
0
votes
1answer
23 views

Möbius strip covering space

how can we describe the universal covering space of the Möbius strip? the Möbius strip is a square $[0,1]\times [0,1]$ with identifications $(0,y)\sim (1,1-y)$. So my guess is that the universal ...
1
vote
0answers
15 views

Is this covering group $G'$ of $G$ unique?

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). ...
0
votes
0answers
25 views

what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover ...
0
votes
0answers
35 views

Fundamental group and universal cover for this quotient space

For a non-zero integer $p$ define the topological space $L_p$ by: \begin{equation} L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p} \end{equation} Check that $L_1\cong\mathbb{D}^2$, and more ...
0
votes
2answers
34 views

Covering spaces of wedge sum of circles

Let's suppose I'd like to characterise all connected covering spaces of wedge sum of circles.When it comes to a torus, a sphere or $ S^{1} \vee S^{2} $ it is easy to point out exactly how do all ...
0
votes
0answers
46 views

degree of $f\circ g$

Let $f,g : \mathbb{S}^1 \to \mathbb{S}^1$ be two continuous maps where $\mathbb{S}^1$ is the unit circle. Prove that $\deg (f \circ g) = \deg f \deg g$. I don't know at all how to do it : I first set ...
2
votes
1answer
21 views

How to lift maps going into the base of a fiber bundle?

Let $p:E\to B$ be a fiber bundle and $f:B'\to B$ a map. Under what conditions does a lift $f':B'\to E$ exist? In the context of covering spaces, I remember a necessary and sufficient condition is that ...
1
vote
3answers
53 views

Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
0
votes
0answers
51 views

Universal cover of a CW complex corresponding to an identification space

I am looking at a past exam paper for my introductory algebraic topology course, and am asked, for each of the following identification spaces, to find a CW complex homeomorphic to the space, draw the ...
0
votes
1answer
17 views

Covering maps as bundles

One geometric way to see a continuous map (or any set function really) is as a "fiber bundle" with the usual picture of a comb - the base space indexes the fibers of the map and there's a nice picture ...
0
votes
0answers
26 views

Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”

(Closely related) This question centers around section 6.5 of Borceux and Janelidze's Galois Theories. Definition 1. Let $\mathcal M$ be a class of arrows in a category (in our case $\mathsf{Top}$). ...
3
votes
1answer
73 views

An example of lifting a group action to the universal cover.

Through a previous question, I understood how we can lift the action of a group $G$ on a topological space $X$ to an action of a covering group $G'$ of $G$ on the universal cover $\tilde{X}$ in such a ...
0
votes
0answers
17 views

Which is correct between cover or covering?

Let $I$ be closed n-dimensional intervals, $$I=\{\mathbf{x}: a_j\le x_j \le b_j, \quad j=1, \cdots, n\}$$ and $S$ be a countable collection of intervals $I_k$, $$S=\{I_j,\quad j=1, 2, \cdots\}$$ In ...
7
votes
1answer
87 views

Field $K$ with $\operatorname{Gal}(\overline{K}/K)\simeq\widehat{F_2}$

Is there a field K such that $\operatorname{Gal}(\overline{K}/K)$ is the profinite free group with two generators? For one generator I know that for all the $\mathbb{F}_p$ we have ...
3
votes
1answer
30 views

Proof of the quotient map $\pi : X\to X/G$ is a covering map only if the action of $G$ is properly discontinuous.

The following is a theorem from Munkres' Topology and there's a part in the proof of the theorem that I don't understand. I've written the part in bold. $X/G$ is the orbit space obtained from $X$ by ...
1
vote
0answers
15 views

Maximal analytic continuation gives rise to a covering

Suppose that $a$ is a point on a connected Riemann surface $X$ and $\varphi \in \mathcal{O}_a$ admits an analytic continuation along every curve in $X$ starting at $a$. Let $(Y, p, f, b)$ be the ...
2
votes
1answer
25 views

Branch points of $f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}(z + \frac{1}{z})$

Problem: find the branch points of the function $$ f : \mathbb{C} \to \mathbb{P}^1 : z \mapsto \frac{1}{2}\bigg(z + \frac{1}{z}\bigg). $$ My try: The zeros are $i$ and $-i$, but I don't see why $f|V$ ...
0
votes
1answer
24 views

Materials on some construction involving classification of covering spaces

Let $X_u \rightarrow X $ be a universal covering. Let $S $ be any set with a group $\pi(X,x_0) $ acting on it from the right side. Then we get space $S \times X_u $ with a group action $ S \times X_u ...
5
votes
1answer
121 views

Universal covering and double cover functors

Cross-posted on MO Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know ...
4
votes
1answer
645 views

covering spaces and the fundamental groupoid

Briefly, my question is whether there is a basepoint-free statement of the basic theorem on covering spaces. For a nice space $X$, I would hope that there is an equivalence of categories between ...
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 ...
0
votes
1answer
55 views

The fundamental group of preimage of covering map

Define i: B $\to$ Y is an inclusion, p: X $\to$ Y is a covering map. Define $D=p^{-1}(B)$. We assume here B and Y are locally path-connected and semi-locally simply connected. Then if B,Y, X are ...
1
vote
0answers
24 views

Construction of K(G,1) for arbitrary group G

In Example 1B.7 of Hatcher's 'Algebraic Topology', he attempts to construct a $K(G,1)$ space for any group G. His construction goes as follows: Consider the $\Delta$-complex (denote it as $EG$) ...
2
votes
1answer
62 views

Is it possible to have a connected manifold that is a double cover of a 2-sphere?

I have come up with a branched covering, but it necessarily has two branch points. From that I'm assuming that it can't be done, possibly related to the hairy ball theorem, but I don't know how to ...
4
votes
0answers
33 views

How to find fundamental groups and covering spaces of $\mathbb{RP}^2\vee \mathbb{RP}^2$?

The following is an exercise I was assigned in homotopy theory. Defined $X = \mathbb{RP}^2\vee \mathbb{RP}^2$. a) Find $\pi_1(X)$. b) Find the universal cover of $X$. c) Find all of its connected ...
0
votes
0answers
31 views

Non-Example of covering space

Let $X:= \mathbb{R_1} \sqcup \mathbb{R_2}$ where $\sqcup$ means the disjoint union. Now $X$ is a topological space with the "Disjoint topology", in which the opens are disjoint union of opens of ...
1
vote
1answer
44 views

Universal covering map and induced maps on homology and homotopy

I had an additional question regarding universal covering maps. If $p:U\rightarrow X$ is a universal covering map for space $X$ does it induce isomorphisms on homology $H_i$ for $i>1$. Or if ...
1
vote
1answer
43 views

Showing there is no covering map $\mathbb{R}P^2\to T^2$

$\newcommand{\R}{\mathbb{R}}$ I'm being asked to show there is no covering map $\R P^2\to T^2$ (the torus) by showing that every map $\R P^2\to T^2$ is null-homotopic, by lifting to the universal ...