1
vote
1answer
36 views

Construction of a covering space as a fibre bundle

In a direct proof of the equivalence of categories between the covering maps $p:(\hat X, \hat x) \rightarrow (X,x)$ of a topological space $(X,x)$ for sufficiently beautiful $X$ and the ...
0
votes
1answer
46 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
6
votes
1answer
105 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 ...
2
votes
0answers
75 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
1
vote
0answers
52 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
0
votes
1answer
46 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
7
votes
1answer
128 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 ...
0
votes
2answers
39 views

Liftings in Topology

I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$. Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they ...
2
votes
0answers
31 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
4
votes
1answer
54 views

True or False: Topological Group and $S^1 \vee S^1$

$i.$ $S^1 \vee S^1$ can be embedded in a topological group $ii.$ $S^1 \vee S^1$ can be covered by a topological group I think $i.$ is true since we can embed the wedge sum into $\mathbb{R}^2$, which ...
1
vote
0answers
46 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
0
votes
1answer
39 views

What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought ...
4
votes
1answer
66 views

Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
1
vote
0answers
51 views

Show That a Lift Always Exists

I've been considering this problem: Suppose that $X$ is a topological space and that $H_1(X)$ is a finite group of odd order. Show that if $p:\tilde{Y}\rightarrow Y$ is a covering space of index ...
0
votes
1answer
41 views

Are deck transformations homotopic to the identity?

Suppose that $p: X \to Y$ is the universal covering of some connected and locally path connected space $Y$, and that $\phi$ is a deck transformation. Is $\phi$ homotopic to the identity on $X$? If so, ...
0
votes
0answers
58 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
3
votes
1answer
84 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
3
votes
3answers
96 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
0
votes
1answer
37 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
2
votes
0answers
42 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
3
votes
1answer
27 views

Help proving a space is closed in order to show a space is properly discontinuous

This stems from exercise 6, section 81 in Munkres. Let $X$ be a locally compact Hausdorff space; let $G$ be a group of homeomorphisms of $X$ such that the action of $G$ is fixed-point free. Suppose ...
1
vote
0answers
25 views

Composition of covering maps is a covering map if the inverse image is finite. [duplicate]

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 $\forall z$, then $p$ is a covering map. $\textbf{My Attempt:}$ Let $U$ be an arbitrary open ...
0
votes
1answer
37 views

Why is this induced homomorphism not surjective?

Let $p: Y \to X$ be a covering with fix base point. I have already shown that the induced homomorphism $p_{*}=\pi_1(Y,y_0) \to \pi_1(X,x_0)$ is injective. However, since we are not calling it an ...
0
votes
0answers
39 views

lifting a closed curve

Is it always true (because of covering spaces has homotopy lifting property)? loop $f$ lifts to a closed curve if and only if any curve freely homotopic to $f$ lifts to a closed curve. or we have to ...
3
votes
1answer
68 views

Showing that every map $f : S^2 \rightarrow S^1$ is homotopic to the trivial map

Problem: Prove that every map $f : S^2 \rightarrow S^1$ is homotopic to the trivial map. Hint: Use the covering space $E: \mathbb{R} \rightarrow S^1$. If you can show that every map $f: ...
1
vote
1answer
57 views

Why is $p$ regular?

This is with regards to this lemma. Lemma: Let $p: \tilde{X} \rightarrow X$ be a covering map. Assume $\tilde{X}$ is connected and locally path-connected. Then $p$ is a regular covering $\iff$ for ...
0
votes
0answers
25 views

let $p: E\rightarrow B$ conitnuous and surjective. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices is unique.

let $p: E\rightarrow B$ coninuous and surjective and suppose that $U$ is an open set of $B$ that is evenly covered by p and U. Show that if U is connected,then the partition of $p^{-1}(U)$ into slices ...
3
votes
2answers
67 views

How can i find this universal cover? [closed]

I have $X = \{(x,y,z) | x^2 + y^2 = 1, z = 0\}$ and $Y = \{(x,y,z)| (x-1)^2 + y^2 + z^2 = 1\}$. What is the universal cover of $X \cup Y$?
2
votes
0answers
52 views

Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
1
vote
2answers
110 views

lifting a product of commutators of standard generators on 2-manifolds

I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf I don't understand this part: "(...) we can easily ...
3
votes
0answers
40 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
1
vote
1answer
40 views

Let $p$ be a covering space and $X, Y$ be path connected. Show there exists a map $q$ such that $q\circ p=1_{X}$ iff $p$ is a homeomorphism.

Let $p\colon X\rightarrow Y$ be a covering map where $X$ and $Y$ are path connected. Show that there exists a map $q\colon Y\rightarrow X$, such that $q\circ p=1_{X}$ if and only if $p$ is a ...
3
votes
0answers
58 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
4
votes
1answer
77 views

Show that if B is simply-connected, then p is a homeomorphism.

Let $p: E \rightarrow B$ be a covering map with $E$ path-connected. Show that if $B$ is simply-connected, then $p$ is a homeomorphism. I'm checking to see if my solution is flawed. Since $p$ is a ...
1
vote
2answers
136 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
12
votes
4answers
300 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 ...
0
votes
1answer
28 views

Verify $p _0 : [0,1] \mapsto S^1 , p_0(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map.

I want to verify that the restriction to the interval $[0,1]$ of the map $p : \mathbb{R} \mapsto S^1 $ given by $ p(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map. I tried as follows. Take $s ...
3
votes
1answer
62 views

Is Orbit Criterion an abstract nonsense? Different induced fundamental groups.

(Orbit Criterion) Let $p:\tilde X \to X$ be a covering map. If $\tilde q, \tilde q' \in \tilde X$ are two points in the same fiber $p^{-1}(q)$, there exists a covering transformation taking $\tilde ...
1
vote
0answers
77 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
0
votes
0answers
68 views

exercise 17 of hatcher page 80 chapter 1.3

Given a group $G$ and a normal subgroup $ N$, show that there exists a normal covering space $\widetilde{X} \rightarrow X $ with $\pi_{1}(X)\approx G ,\pi_{1}(\widetilde{X})\approx N $, and deck ...
1
vote
1answer
52 views

Why does the intersection change to a union in $r^{-1}(\bigcap r(V_i\cap W_i))=\bigcup V_i\cap W_i$?

Let $q: X\to Y$ and $r:Y\to Z$ be covering maps, $p=r\circ q$. If $r^{-1}(z)$ is finite for each $z$ in $Z$, $p$ is a covering map. There is a proof on ask a topologist, but I can't follow why ...
0
votes
0answers
29 views

is there a specific way to find deck transformation and its related group?

is there a specific way to find deck transformation and its related group? this question came to my mind at the first time I studied deck transformation and related topics of covering space. it ...
0
votes
0answers
25 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
0
votes
1answer
41 views

using covering space technique,prove that $[G:H \cap K] \leq [G:H][G:K]$.

using covering space technique,prove that if $G$ is a group with subgroups $H$ and $K$ then $$[G:H \cap K] \leq [G:H][G:K]$$ I couldn't understand the relation between them and the covering space,so ...
1
vote
1answer
48 views

Covering of a Topological Group(Use of fundamental theorem of covering spaces)

Suppose we have two path-connected spaces $G$ and $H$. Suppose also that $G$ is a topological group with an identity element $e$ and there is a covering $$ p: H \rightarrow G $$ The problem asks that ...
1
vote
1answer
45 views

what is the Cayley complex of dihedral group $D_{4}$?

what is the Cayley complex of dihedral group $D_{4}$? I am aware of Cayley graph of $D_{4}$,can you explain to me how I should I attach 2-cell complexes to the loops to make it covering space? I ...
0
votes
3answers
50 views

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,show that $p^{-1}(A)$ is path connected.

if $p:\widetilde{X}\rightarrow X$ is a covering space and $\widetilde{X}$ is path connected ,also $A\subset X$ is a path connected subset,show that $p^{-1}(A)$ is path connected. I suppose that ...
0
votes
0answers
80 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 ...
0
votes
1answer
41 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 ...
1
vote
1answer
56 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 ...