# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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### Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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### Problem 1.3.12 from Hatcher

Let a and b be the generators of $\pi_1$ $(S^1 \vee S^1)$ corresponding to the two $S^1$ summands. Draw a picture of the covering space of $S^1 \vee S^1$ corresponding to the normal subgroup generated ...
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### Colorings of covers of graphs

Let $G$ be a finite graph. Let $Cov_k(G)$ be the set of all $k$-sheeted covers of $G$. Note that if $G$ is $\chi$-colorable, then so is every graph in $Cov_k(G)$. I am wondering how the average ...
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### Does a deck transformation have a homotopy that lifts to it?

I have a closed connected manifold $X$, consider the universal cover $p: \tilde X \rightarrow X$. If I recall correctly, any homotopy $F: X \times I \rightarrow X$ with $F(\cdot, 0) = id_X$ lifts to a ...
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### Does maps between fundamental groups induces a continuous map between spaces?

Main question is this: Suppos $M$ is a manifold and $G$ is a finite group. If there is a group homomorphism $\phi:\pi_1 M\to G$, is there a continuous map $f:M\to BG$, where $BG$ is a classifying ...
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Let $(X,A)$ and $(Y,B)$ pairs satisfying the homotopy extension property (or a CW-pairs if necessary) with $A$ and $B$ contractible. Let $f:(X,A)\to (Y,B)$ a map of pairs and let $p_X:\widetilde{X}\to ... 1answer 51 views ### The fundamental group of the Lattice - (R x Z) U (Z x R) I am trying to show that the identity map$ id:S_h \vee S_v \rightarrow S_h \vee S_v$does not lift to L =$(\mathbb{R} \otimes \mathbb{Z}) \cup (\mathbb{Z} \otimes \mathbb{R}) $via the covering ... 2answers 103 views ### This covering map is homeomorphism Suppose that$f:\mathbf RP^2\rightarrow X$is a covering map and$X$is a CW-complex. Show that$f$is homoemorphism. We know covering map is continuous and onto,so we should show that$f$is one-... 0answers 29 views ### Are there more intuitive and/or illustrative reason for why covering maps do not compose besides the classical counterexample? In an introductory topology class, one learns about covering maps$p : E \to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no.... 1answer 18 views ###$\mathrm{LMod}(X)$has finite index in$\mathrm{Mod}(X)$I am reading about the Birman-Hilden conjecture, specifically the thesis of Rebbeca Winarski and there is some basic point I am not getting. We consider a covering space$p: Y \rightarrow X$between ... 1answer 42 views ### Show that a finite generated subgroup$H$of$G$with index 3 exists which is not normal in$G$. > Suppose that$a$&$b$are the generators of a free group$G$.Show that a finite generated subgroup$H$of$G$with index 3 exists which is not normal in$G$. The way i try to solve the ... 0answers 39 views ### Covering spaces for$S^1 \vee S^1\vee S^1$When I read the HATCHER book(algebraic topolog) a question i found on a pdf and the truth is i cant find a way to solve it and realize it! the question which i saw after reading page 57/58/59( ... 0answers 29 views ### Degree of universal cover of simple Lie group I have seen a statement that if$\mathfrak{g}$is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra$\mathfrak{g}$. Equivalently, the simply connected group with Lie ... 0answers 11 views ### Lie group map whose differential is an isomorphism is a covering map While trying to read the proof in Fulton and Harris of their “Second Principle,” I ran across something that I do not understand. They seem to claim that if$f: G\rightarrow H$is a map of Lie groups ... 0answers 21 views ### Irregular covering of 2-holed torus$S_2$I need to find an irregular covering of the 2-holed torus. If a covering is regular, then$p_*(\pi_1(E,e_0))$is a normal subgroup of$\pi_1(S_2,b_0)$, where$E$denotes the covering space. So an ... 2answers 45 views ### Let$q\colon E\to X$be a covering map. If$E$is compact then$X$is compact and$q$is a finite-sheeted covering. Here's what I have so far: Since$q$is continuous and surjective then$X$is compact. For all$x\in X$there exists an evenly covered neighborhood$U_x$, so$\{U_x \colon x\in X\}$is an open cover ... 2answers 30 views ### pull back of smooth covering space is injective I need to prove this but I don't really know where to start: Let$p:M\to N$be a smooth covering space between smooth manifolds. Show that$p^*:\Omega(N)\to\Omega(M)$is injective. Where$\Omega(M)$... 1answer 30 views ### Cover of a simply connected and lpc space is trivial I have I question about a reduction step in the proof of the statement that a cover of simply connected and locally path-connected space is trivial (source: "Fundamental Groups and Galois Groups" by ... 0answers 16 views ### Covering with total space a CW complex Let$G$be a discrete group acting properly discontinuous on a CW complex$X$. Does$X/G$also have a CW structure? Further questions: If not, maybe with stronger conditions ($G$finite, each ... 1answer 19 views ### regular covering transformation can you help me with proving this statement? > Suppose that$F:S^1\longrightarrow S^1 $such that$F(z)=z^n$, is a covering space.prove that$F$is regular covering transformation Honestly..i ... 1answer 53 views ### covering spaces and equivalency of these three propositions [closed] This question is really important for me since the answer will give me the a way for solving similar proofs at algebraic topology lessons,so i need your help.. I need to prove these following ... 2answers 61 views ### Classification of covering spaces for spaces that are not locally path connected: counterexamples? The standard theory treats the case where the base space$B$is path connected, locally path connected, and semi-locally simply connected. While being path connected and semi-locally simply connected ... 1answer 37 views ### Fundamental group via deck transformations considering rotation on sphere Let$Z_m$act on$S^1$by multiplication with$e^{2\pi ki/m}$for$k \in Z_m$. Let$X = S^1 / Z_m$be the orbit space of this action. Then we have a universal cover$q:S^1 \rightarrow X$given by the ... 1answer 42 views ### Uniqueness of lift in covering space theory Let Y be a topological space and$\pi: X\rightarrow Y$a covering map. Take two lifts of the covering map$\tilde {\pi_1}$,$\tilde {\pi_2} : X\rightarrow X$such that they agree on$x_0 \in X$. ... 0answers 61 views ### Infinite cyclic cover corresponding to non-zero cohomology class$\alpha \in H^1(x,\mathbb Z)$I want to understand the following sentence: Let X a compact (complex) manifold which has a non-zero cohomology class$\alpha \in H^1(X,\mathbb Z)$. Let$\pi: \bar X\to X$be the corresponding ... 1answer 151 views ### Deck transformation group in algebraic geometry Let$f:X\to Y$be a finite morphism between (irreducible) varieties. We can define$\operatorname{Aut}(X/Y)$to be the automorphism of$X$commutes with$f$. For the case over$\mathbb C$, we can also ... 1answer 124 views ### Covering map corresponding to surjection Let$X$be a compact manifold with$\pi_1(W)=S_3$,$T$be an$n$-dimensional complex torus. Denote the universal cover of$W$by$\tilde{W}$. Let$S_3$act on$(T\times T\times T) \times \tilde{W}$... 1answer 59 views ### Index of covering in covering group In wikipedia I read: In mathematics, a covering group of a topological group$H$is a covering space$G$of$H$such that$G$is a topological group and the covering map$p : G \rightarrow H$is a ... 1answer 68 views ###$p:X\to Y$covering space if$q\circ p:X\to Z$and$q:Y\to Z$covering spaces and$Z$locally path-connected. I have done the problem, but I'm confused about why the local path-connectedness of$Z$is necessary. My solution: Let$p:X\to Y$and$q:Y\to Z$be such that$q$and$q\circ p$are covering spaces. ... 1answer 41 views ### What maps descend to homeomorphisms I was reading "A primer on Mapping Class Groups" by B. Farb and D. Margalit and I'm stuck at a point in the proof of Mod$(A)\cong \mathbb{Z}$where$A$is the annulus, where the restriction of$M$on$...
Let $S$ be a orientable closed surface with genus $g \geq 1$ and let $\gamma \subset S$ be an immersed curve. Does there exist a finite cover of $S$ where $\gamma$ lifts to a curve that is homotopic ...
I got a little bit confused with Hatcher's explanation of fundamental group action on the chain complex of the universal cover (Algebraic Topology, A. Hatcher, p. 328). Assume $X$ is a path-connected ...