# Tagged Questions

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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $W ... 1answer 39 views ### What are the deck transformations of this threefold cover of the figure 8? Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators$a$and$b$and the relations$a^2, b^2, aba^-1, bab^-1$. I thought ... 1answer 64 views ### Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and$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$... 0answers 49 views ### Show That a Lift Always Exists I've been considering this problem: Suppose that$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$is a covering space of index ... 1answer 41 views ### Are deck transformations homotopic to the identity? Suppose that$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$? If so, ... 0answers 57 views ### Finite covering space with compact spce. Prove that if$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$let us ... 1answer 83 views ### Monodromy representation of Airy equation Let$K=\Bbb{C}(z)$with the usual derivation and consider the Airy dierential equation$y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ... 3answers 91 views ### covering map$S^n \rightarrow P^n$is not null homotopic Here is the problem: Prove that the covering projection$S^n \rightarrow P^n$is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ... 1answer 35 views ### Lifting property of a covering space A common example of a covering space is$p:\mathbb R \rightarrow S^1 $,$p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ... 0answers 40 views ### lifting injective maps to injective maps in principal bundles Let$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 ... 1answer 26 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 ... 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 ... 1answer 36 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 ... 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 ... 1answer 66 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: ... 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 ... 0answers 23 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 ... 2answers 66 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? 0answers 46 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 ... 2answers 109 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 ... 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 ... 1answer 39 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 ... 0answers 57 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 ... 1answer 64 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 ... 2answers 135 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 ... 4answers 287 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 ... 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 ... 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 ... 0answers 71 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} ... 0answers 66 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 ... 1answer 51 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 ... 0answers 27 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 ... 0answers 24 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 ... 1answer 40 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]$$I couldn't understand the relation between them and the covering space,so ... 1answer 46 views ### Covering of a Topological Group(Use of fundamental theorem of covering spaces) Suppose we have two path-connected spaces G and H. Suppose also that G is a topological group with an identity element e and there is a covering$$ p: H \rightarrow G$$The problem asks that ... 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 ... 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 ... 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 ... 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 ... 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 ... 0answers 61 views ### Image of the map on homology induced by a covering Let$X$and$Y$are two compact connected oriented 2dim smooth manifolds, and$\pi\colon X\to Y$is an unramified covering of degree$d$. Consider the induced map$\pi_* \colon H_1 (X,\mathbb Z) \to ...
Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?