# Tagged Questions

For questions about or involving covering spaces in algebraic topology.

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I proved one of Hatcher's propositions on my own and my proof is quite a bit different than his. The Unique Lifting Property says: Given a covering space $p:\tilde{X} \rightarrow X$ and a map $f: ... 1answer 25 views ### Homotopy between two paths implies triviality of the loop they form If$X$admits a universal covering space and$\alpha$and$\gamma$are to homotopic paths between$x$and$p(y)$, then$\alpha*\gamma^{-1}$is nullhomotopic? 1answer 31 views ### Interpretation of points in covering spaces as homotopy classes of paths [on hold] If$p:\widetilde{X} \to X$is a covering map,$y \in \widetilde{X}$determines a homotopy class of paths in$X$joining the base point$x_0$to the point$p(y)$. But a homotopy class of paths in$X$... 1answer 22 views ### Question About Covering Space Classification Theorem I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup$p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$we ... 1answer 28 views ### Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover) Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let$X$be a path-connected space with$\pi := \pi_1(X,*)$abelian and ... 0answers 26 views ### If$p: E \rightarrow X$is a covering map with$E$connected and$|p^{-1}(x_{0})|=k$for some$x_{o}$then$|p^{-1}(x)|=k$for all$x \in E$. Prove that if$p:E \rightarrow X$is a covering map with$E$connected and$p^{-1}(x_{0})$has$k$elements for some$x_{0} \in X$, then$p^{-1}(x)$has$k$elements for every$x \in X$. Is my proof ... 1answer 40 views ### Group of deck transformations acts properly discontinuously Let$M$be a connected (smooth Riemannian) manifold which admits a universal cover$\tilde{M}$. Let$\Gamma$be the group of deck transformations on$\tilde{M}$. I want to show that$\Gamma$acts ... 1answer 23 views ### Let$G$a simple connected topological group and$H$a normal discrete subgroup, then$\pi_1(G/H,e) = H.$I know that$G$is a covering space for$G/H$and there is a injection between the fundamental group of$G$and$G/H.$How to proceed to show that$\pi_1(G/H,e) = H?$. 2answers 39 views ### When is the universal cover of a Riemannian manifold complete? Let$(M,g)$be a connected Riemannian manifold which admits a universal cover$(\tilde{M}, \tilde{g})$, where$\tilde{g}$is the Riemannian metric such that the covering is a Riemannian covering. I ... 0answers 20 views ### Descending group actions to coverings Let$X$be a path-connected space with universal cover$\widetilde{X}$, let$Y$be another covering of$X$$$\widetilde{X} \hspace{1cm} \\ \searrow \\ \downarrow\hspace{.5cm} Y\\ \hspace{.25cm}\... 1answer 50 views ### Does a group action lifted to the universal cover commute with the fundamental group action? Question: Let \varphi \colon G \to \text{Homeo}(X) be a group action on a topological space X with basepoint x_0 and universal covering \pi \colon \widetilde{X} \to X. Then the subgroup of ... 1answer 71 views ### Prove that two n-sheeted covering space of S^{1} are isomorphic. We have a Blaschke product B(z) \colon S^1 \to S^1 of order n and the map f \colon S^1 \to S^1, f(z)=z^n. Both maps are regular on S^1. We have already proved that both are n-sheeted ... 0answers 40 views ### Existence of the lift of a curve Let (M, g) be a complete Riemannian manifold and let (\tilde{M}, \tilde{g}) its universal cover. Let \pi : \tilde{M} \to M be the covering map. Let \gamma : I \to (M, g) be a smooth curve ... 3answers 105 views ### Why bother showing S^{1} covers itself? I've just been introduced to covering spaces, and one of the examples I've been shown is that p: S^{1} \to S^{1}, p(z)=z^{n} is a covering map for every n. My question is: why would you care? ... 1answer 41 views ### Existence of a universal cover of a manifold. Suppose M is a manifold, topological or smooth etc. As a topological space M is required to be primarily locally homeomorphic to \Bbb R^n, with some added things that don't come along with this, ... 0answers 37 views ### Hatcher Covering Spaces Ex. 11 & 31 and Surjectivity of the Covering Map I am confused by the statements of a couple of the exercises in Section 1.3 of Hatcher. I think they need additional hypotheses that are not reflected in Hatcher's errata. Exercise 11: Construct ... 0answers 54 views ### Is this what universal covering spaces are used for? From the perspective of real analysis, we have:$$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$Something ... 1answer 48 views ### Is a Blaschke product/rational function a covering map for a n-sheeted covering of S^{1}? We have a Blaschke product B(z) of order n (you can think of it as a rational function with n zeros and n poles), the zeros are obviously inside \mathbb{D}. Why is B(z) \colon S^{1} \to S^{... 0answers 15 views ### Covering spaces of surface sphere glued to the mobius strip at one point on its boundary. I have determined the universal covering space, but I am having trouble finding two-sheeted and three sheeted covering spaces. Any help would be greatly appreciated on how to approach this! 1answer 74 views ### The Euler Characteristic of \mathbf RP^2 is a Fraction. Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For X a finite CW complex and p:\tilde X\to X an n-sheeted covering space, show that \chi(\tilde X)=n\chi(X). Here \chi ... 0answers 41 views ### Pushforward of canonical bundle restricted to divisor isomorphic to restriction of pushfoward of canonical bundle Consider the branched covering f \colon X \to \mathcal{Q}_7 of the 7-dimensional smooth projective quadric by a smooth connected projective variety X. Since we have the 6-dimensional quadric \... 1answer 67 views ### Correspondence \{principal G-bundles on M\}\leftrightarrow\{conjugacy classes of homomorphisms \pi_1(M)\to G\} Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ... 1answer 53 views ### Characterizing spaces with no nontrivial covers I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property? 1answer 37 views ### Preimage of a simply closed curve under the two-dimensional antipodal map Suppose p:S^2\to P^2 is the quotient antipodal map, and J is a simply closed curve in P^2, then p^{-1}(J) is either a simply closed curve in S^2, or two disjoint simply closed curves in S^2... 2answers 37 views ### A covering space of a Hausdorff space is Hausdorff Let p:Y\to X be a covering space. If X is Hausdorff, so is Y. Hello, I have a question to this task. I want to show that Y is a Hausdorff space. Hence for y_1, y_2\in Y with y_1\neq y_2 ... 0answers 51 views ### Covering maps of schemes. A curve X is modular if there is a finite covering X_0(N)\rightarrow X. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces? 0answers 37 views ### Making a principal bundle into a covering space Suppose \pi : P\rightarrow M is a principal G-bundle. I want to make this into a covering map by changing the topology of P. By local triviality we can find for each x\in M an open U\subset ... 0answers 29 views ### Surjective group homomorphism between quotient groups Assumptions: Assume that G is a topological group and Z_1,Z_2 are discrete, normal subgroups of G (hence central) and G / Z_1 and G / Z_2 denote the quotient groups. Assume moreover that ... 1answer 50 views ### Subgroup of Finite Index Containing a Given Finitely Generated Subgroup of a Free Group: Problem 12 in 1A in Hatcher. Problem 1A.12 (Hatcher) Let F be a finitely generated free group and H be a finitely generated subgroup of F. Let x\in F-H. Show that there is a finite index subgroup K of F such that H\... 1answer 29 views ### Covering map associated with open cover Let \left\{U_i \right\} be an open cover of X. On some online sources and some MSE questions, the map \coprod _iU_i\rightarrow X is given as an example for a local homeomorphism which is not a ... 0answers 15 views ### Representing Covering Spaces by Permutations: Proof Verification. \newcommand{\FG}{\pi_1} Given a covering projection p:\tilde X\to X, and x_0\in X, we can naturally define a \emph{right} action on F=p^{-1}(x_0). For each point \tilde x\in F, and each [\... 1answer 36 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 ... 1answer 43 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 ... 1answer 26 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. ... 3answers 32 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, \varphi^{-1}(U_{\... 1answer 20 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, ... 1answer 36 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 \phi:\... 1answer 71 views ### Quotient topology on unit sphere Let \sim be the equivalence relation$$a\sim b\iff a=b\text{ or }a=-b,$$for a,b on the unit sphere S^2. Let Q be the quotient space. How do I show that the quotient map is a covering ... 0answers 27 views ### Free action of a discrete group gives a covering space I'd like to find a short proof of the following seemingly basic fact. Suppose a discrete group G acts freely on a manifold X with the quotient X/G being compact. Then X is a covering space of ... 0answers 65 views ### limit of covering spaces Say we have X a manifold with a compact exhaustion of embedded submanifolds X=\cup K_n with K_n\subset K_{n+1}. Let H\subset \pi_1(X) a infinite index subgroup that is finitely generated, ... 0answers 19 views ### Deck group of a connected n-fold cover must have at most n elements Let p:Y\to X be an n-fold covering map, with Y connected. Show that Deck(p) has at most n elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct \tau_1,...,... 0answers 35 views ### Fundamental group of a covering space I understand the correspondence between the subgroups of a fundamental group \pi_1(X) and the covering spaces of X. However, I do not understand what is implied about the fundamental groups of ... 2answers 58 views ### Is the Fundamental Group of space with contractible universal cover torsion free? Some classmates and I were working on the following question - is the fundamental group of the Klein Bottle K torsion free? We have the following presentation:$$\pi_1(K) = \langle a,b: aba = b \... 1answer 39 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 ... 1answer 38 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 ... 2answers 68 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 sheaf.... 0answers 18 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). ... 0answers 34 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$GL(n,\...
For a non-zero integer $p$ define the topological space $L_p$ by: $$L_p=\mathbb{D}^2\sqcup\mathbb{S}^1\big{/}z\sim{z^p}$$ Check that $L_1\cong\mathbb{D}^2$, and more ...
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 ...