Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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Every continuous $f:\mathbb{R}P^5\rightarrow (S^1\vee S^1)\times T^3$ is homotopic to a constant map.

A practice exam question: Show that every continuous map $f:\mathbb{R}P^5\rightarrow (S^1\vee S^1)\times T^3$ is homotopy equivalent to a constant map. I'm not even sure where to start with this one....
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Prove that the homeomorphisms generate action such the projection to quotient is a covering space.

I need to prove that the action in the homeomorphism $\varphi:S^3\to S^3$ defined by $$ \varphi(z_1,z_2) = \Bigl( e^{\frac{2\pi i}{n}}z_1, e^{\frac{2\pi mi}{n}}z_2 \Bigr) $$ is a covering space on the ...
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Covering transformation and orientation ( in Synge Theorem )

Case 1: $M_1$ is a compact orientable manifold. And $$ \pi_1:\tilde M_1\rightarrow M_1 $$ is the universal covering of $M_1$. Introduce on $\tilde M_1$ the orientation and metric such that $\pi_1$ ...
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Automorphisms of a covering of $p:E \rightarrow B$ where $E$ is connected [duplicate]

The book I am reading states the following A covering map $p:E \rightarrow B$ has an automorphism group $AUT(p)$. AN automorphism is a homeomorphism $\alpha:E \rightarrow E$ such that $p \circ \alpha ...
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Describe the covering space $S^1 \vee S^1$ corresponding to the commutator subgroup $[F_2,F_2] \leq F_2 \cong \pi_1(S^1 \vee S^1)$\\

What familiar group is the group of desk transformation isomorphic to? Describe isomorphism explicitly. Use the covering space to deduce that $[F_2,F_2] \cong F_{\infty}$ Let that the covering space $...
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Monodromy and an equivalence of categories for covering spaces

I just came across the following remark on Terry Tao's blog which I found cryptic: If $M$ is a connected topological manifold, and $p$ is a point in $M$, the (topological) fundamental group $\pi_1(M,...
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Continuous map $S^1\to S^1$ with $|$fiber$|$ constant is a covering map

Recall that for any positive integer $n$, the map $S^1\to S^1$, $z\mapsto z^n$ is a degree $n$ covering map. I am considering the following possible generalization: suppose that a continuous map $f:S^...
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Riemannian covering space

Suppose $M^n \subset \mathbb{R}^{n+1}$ is a connected, oriented and complete smooth Riemannian manifold with mean curvature $H\geq 0$. (We may even suppose $M$ is the graph of a smooth function). Do ...
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Coverage of a cirlce

Suppose I have a coordinate system with one main circle and additional circles. I want to determine the area, the main circle is covering, which is not covered by the other circles. Further, it might ...
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Can a space with $\pi_1(X)=F_2$ have a nontrivial covering space which is homeomorphic to $X$?

Any finite sheeted connected cover of the circle is again homeomorphic to a circle. On the group level, this is consistent with the fact that subgroups of $\mathbb Z$ are isomorphic to $\mathbb Z$. ...
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Prove a map is a covering map!

Let $a\in\mathbb{C}$ and $f:\mathbb{C} \to \mathbb{C}$ with $f(z)=z^3+a \bar{z}$. If $\Sigma$ denotes the critical set of $f$ then I want to know whether the restriction of $f$ to $\mathbb{C}\setminus\...
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Let $ X = \mathbb{R}P^2 \vee S^1$. Describe all covering spaces for X that have degree 2 or 3.

We know that the fundamental group of $\mathbb{R}P^2 \vee S^1$ is $\langle a,b\mid b^2\rangle$, we will find covering space of X corresponds to (a) subgroup $\langle a\rangle$ of $\langle a,b \mid b^2\...
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Covering space that admit no section

I'm stuck on the following problem from a notes. Let $B=\{(a_0,...,a_{n-1})\in\mathbb{C}^n:P_a(x)=x^n+a_{n-1}x^{n-1}+...+a_0\ \text{ is square free}\}\ $ and $E=\{(x,a)\in\mathbb{C}\times B:P_a(x)=0\}$...
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Prove that $X$ is not homotopic to any graph if its covering space is homotopic to a product space

Let $p: \widetilde{X} \to X$ be a covering map. Assume that $\widetilde{X}$ is path connected, and that $X$ is path connected and locally path connected. Assume that $\widetilde{X}$ is homotopy ...
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Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$

$\newcommand{\Z}{\mathbb Z}$ Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
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1 vote
1 answer
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Prove a compact subset of a covering space contains a finite number of preimages

Let $p:\widetilde{X} \to X$ be a covering map, $K$ a compact subset of $\widetilde{X}$, and $x$ an element of $X$. Prove that $K\cap p^{-1}(\{x\})$ is finite. Let $x\in X$ be fixed. Since $p$ is a ...
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What is the relationship between universal covering spaces and fundamental groups?

What is the relationship between universal covering spaces and fundamental groups? I know, for example, if $p: \widetilde{X}\to X$ is a covering map then the induced homomorphism $p_\ast : \pi_1(\...
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1 vote
1 answer
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Discontinuous lifts of continuous maps

While recapitulating the theory of covering maps, I thought that it would a most inconvenient thing if the lift of a continuous map defined by continuation along paths to a covering space should be ...
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6 votes
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Example of two non isomorphic universal covers.

I know that if a connected and locally path-connected topological space $X$ admits an universal cover, than it's unique up to isomorphism ("isomorphism" is meant in the category of (...
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2 votes
1 answer
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Continuity of Lifts and Covering Spaces

Suppose that $A$ and $B$ are covering spaces of a $X$. Suppose that $f: A \to X$ and $g: B \to X$ are the associated covering maps. If I wanted to construct a lift $h: A \to B$ such that $g \circ h = ...
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3 votes
1 answer
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Checking Existence of Coverings

Let T denote the torus. I am currently trying to determine the existence of a few covering maps: $\mathbb{R}^2 \to S^2$, $\mathbb{R}^2 \to T$, $S^2 \to \mathbb{R}^2$, $S^2 \to T$, $T \to S^2$, $T \to \...
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4 votes
2 answers
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Let $p :X \to Y$ be a covering map and $Y$ path connected. Show that the cardinality of $p^{-1}(\{y\})$ is the same for every $y \in Y$.

Let $p :X \to Y$ be a covering map and $Y$ path connected. Show that the cardinality of $p^{-1}(\{y\})$ is the same for every $y \in Y$. Let $y_0, y_1 \in Y$, then $\exists \alpha :I \to Y$ such that ...
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2 votes
1 answer
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Alternative way to prove that the powers $\mathbb{C}^* \to \mathbb{C}^*, z \mapsto z^k$ and $\exp: \mathbb{C} \to \mathbb{C}^*$ are covering maps

Prove that the power map, $\mathbb{C}^* \to \mathbb{C}^*, \: z \mapsto z^k$ and the exponential map $\mathbb{C} \to \mathbb{C}^*, \: z \mapsto \exp(z)$ are covering maps using the fact that they are ...
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2 votes
1 answer
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About the fact that $f(z) = e^z$ is a covering map

I am reading a proof about the complex exponential map $f:\Bbb C \to \Bbb C \setminus \{0\}, f(z)=e^z$ being a covering map and I can see that it's a surjective map, but the proof then goes on and ...
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Sheets form a neighborhood basis of the covering space.

I'm trying to understand the proof of Proposition 1.33 (Hatcher, Algebraic topology, online). The author proves that $\tilde{f}$ is continous at $y$, by showing that the preimage of a generic sheet ...
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1 answer
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What is the universal cover of a torus minus one point?

What is the universal cover of a torus minus one point? Same as this question but I did not find the answer to be understandable. I know a universal covering must be simply connected, i.e. have ...
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2 votes
1 answer
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Non simply-connected covering space of two other non simply-connected covering spaces

Let $X$ be a space with fundamental group $\mathbb{Z}$ which is path-connected, locally path-connected, and semilocally simply-connected, and let $x\in X$. We have two covering spaces $p_1: (\tilde{X}...
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1 answer
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How many connected covering spaces of $\mathbb{R}P^2 \times \mathbb{R}P^2$ are there? How many morphisms are there between them?

This is one of exercises I have after a chapter about the fundamental theorem of covering spaces (see its statement). Category $\mathrm{Set}^{\Pi_1(X)}$ there denotes the category of functors from the ...
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Why this map is not a cover map?

My teacher says that a trivial example of a non cover map is $f:[-1,1]\rightarrow[0,1]$ defined as $$ f(x)=|x|$$ $f$ is surjective and continuous. Also, for every $x\in[0,1]$ we can find a ...
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1 vote
1 answer
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The conjugation action $\mathbb{H}^*\times \mathbb{H} \rightarrow \mathbb{H}$ restricted to unit-quaternions yields an orthogonal representation

Consider the action $\mathbb{H}^*\times\mathbb{H} \rightarrow \mathbb{H}, (h,h')\mapsto hh'h^{-1}$. Show that it preserves the orthogonal-decomposition $\mathbb{R}\bigoplus $Im$\mathbb{H}$, and ...
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Is the canonical projection $p: X \times X \rightarrow X$ a covering map of the topological space $X$?

Is the canonical projection $p: X \times X \rightarrow X$ a covering map of the topological space $X$? I would say it satisfies the definition. For instance: $p: \mathbb{R}^2 \rightarrow \mathbb{R}$.
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Find isometry on the cylindrical surface

Assuming that $\sigma$ is an equidistant transformation on the cylindrical surface and $\pi$ is the covering mapping from Euclid plane to the cylindrical surface which transform $\left(x, y\right)$ to ...
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Every closed subspace of a paracompact space $X$ is paracompact.

Every closed subspace of a paracompact space $X$ is paracompact. My attempt: Let $A\subset X$ be closed and $\{U_{\alpha}\}_{\alpha \in I}$ an open cover of $A$. This means that $U_\alpha = A \cap U_{\...
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Definition of fiberwise action

I'm studying Margaret Symington's paper on torus fibrations, and I came across this action: We have a regular Lagrangian Fibration $\pi : (M^{2n}, w^2) \longrightarrow B$ ($(M^{2n}, w^2)$ is a ...
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3 votes
1 answer
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Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$

Draw a cover of $S^1 \vee S^1$ whose $\pi_1$ is isomorphic to $\langle a^2,b^3,aba^{-1}b^{-1} \rangle \leq F_2$. This is a follow-up to my post from yesterday regarding the kernel K of a map $\Phi: ...
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1 vote
1 answer
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A claim from Hatcher's Algebraic Topology P. 94

In Hatcher's Algebraic Topology P. 94, there is a claim: For a map $f:A\rightarrow B$ between connected CW complexes, let $p:\tilde{M}_f \rightarrow M_f$ be universal cover of the mapping cylinder $...
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Is the Galois group of a Galois cover an actual Galois group of a field extension?

My question will probably sound extremely naive to people who knows something of this theory, but I'm starting to learn small bits of this extremely general machinery and I'm lost in finding suitable ...
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3 votes
2 answers
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Grothendieck's Galois theory implies Galois theory?

Szamuely's book Galois groups and fundamental groups formulates several variants of the main theorem of Galois theory. This is the usual formulation (dual isomorphism of posets between intermediate ...
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4 votes
1 answer
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A question from Hatcher's Algebraic Topology

I'm working on the following exercise from Hatcher's algebraic topology: Let $\phi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation $\phi(x,y) = (2x,y/2)$. This generates an action ...
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5 votes
2 answers
168 views

Is there a categorical characterization of covering maps?

Given a topological space $X$, some spaces $S$ will be covering spaces for $X$ and other spaces won't. Furthermore, some continuous maps $\pi: S \rightarrow X$ will be covering maps, and other maps ...
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How to understand the covering space of Mapping cylinder?

Let $\partial:E\rightarrow Y$ be a map between to topological spaces. Define $X$ to be the mapping cylinder $X=E\times [0,1] \bigsqcup Y / \sim : (x,1) \sim \partial(x), x\in E$. Then $X$ comes with a ...
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1 vote
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Fundamental group of a covering space of $S^1 \vee S^1$

I am currently working on a problem about the fundamental group of a covering space of $S^1 \vee S^1$, and I just want to see if I understand the mechanics of the question. Let the free group $F_2= \...
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5 votes
1 answer
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Fiber-preserving homotopy equivalences form a group up to homotopy.

Let $p\colon \tilde X \to X$ be a covering space. A homotopy equivalence $\tilde f\colon \tilde X \to \tilde X$ is fiber-preserving if $p(\tilde x) = p(\tilde y)$ implies $p\tilde f(\tilde x) = p\...
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0 votes
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Connected Components of Covering

Suppose $p:Y\to X$ is a covering map and that $X$ is locally connected. Let $Y'$ be a connected component of $Y$. I need to show that $p(Y')$ is a connected component of $X$. In general, I do not know ...
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2 votes
1 answer
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Does simply connected implies locally pathwise-connected?

I am currently studying a little bit of covering spaces theory and I have a question. Does simply connected implies locally pathwise-connected? I ask this because the lifting criterion states that a ...
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2 votes
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Monodromy representation of branched covers

From Hilden-Montesinos theorem, we know that every $3$-Manifold can be constructed as an irregular dihedral 3-fold cover of $S^3$ i.e. there is a represented Knot $K$ such that $ \omega: \pi_1(S^3 - K)...
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1 vote
1 answer
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Smooth covers and local sections

Let $\pi : \bar{M} \rightarrow N$ a smooth cover. Show that for each $q\in\bar{M}$ exists a local smooth section $\sigma : U \rightarrow \bar{M}$ of $\pi$ such that $q\in \sigma(U)$. Here there is my ...
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1 vote
1 answer
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Cohomology of covering spaces

Let $\pi: X\to Y$ be a finite covering map between compact differentiable manifolds such that $Y=X/G$. It is well-known that $$ H^q(Y,\mathbb R)= H^q(X,\mathbb R)^G. $$ Can we expect the converse ? ...
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Is the pushforward a vector bundle for a covering map?

Let $p : S \to M$ be a n-sheeted covering space where S and M are Riemann surfaces. Let $L$ be a line bundle over $S$. Then the pushforward sheaf $p_*(L)$ is a n rank vector bundle on $M$. If $p$ is ...
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prove that the inverse of an isomorphisim between two covering maps is an isomorphisim without assuming connectness

I am reading hatcher's algebraic topology. And the following is the definition of covering map in his book: (The definition of evenly covered) Giving a space $\widetilde{X}$ and a map $p: \widetilde{...
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