# 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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### 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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1 vote
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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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1 vote
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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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### 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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### 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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### 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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### 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 ...
1 vote
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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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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{...