# Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

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### Proof of relation between Betti numbers of base manifold and its covering space

For a covering $p\colon\widetilde{M}\to M$ with compact $\widetilde{M}$, how to show that $b_i(M)\leq b_i(\widetilde{M})$ for $0<i<n=\dim(M)$. If I understand this inequality correctly it says ...
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### $SO(3)$ double covers $L(4,1)$

Let $P^2$ be the real projective plane. I am trying to show that its unit tangent bundle (for a fixed arbitrary metric on $P^2$) is a lens space $L(4,1)$. It seems that this paper (https://www.maths....
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### Can the Klein bottle cover the torus? Show the Klein bottle can't be covered by a space $X$ with $\pi_1(X)=\mathbb{Z}/3\mathbb{Z}$.

I'm brushing up on covering spaces for an upcoming exam, and I came across the following problems related to the Klein bottle: Can there exist a cover of the torus by the Klein bottle? and Let $X$ ...
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### Calculating $\pi_2(X\cup_\alpha e_\alpha)$ using Hurewics theorem and covering spaces

Consider the CW-complex $X$ obtained by wedging two circles. Denote by $a$ and $b$ the generators of $\pi_1(X)$. On $X$, attach two discs with attaching maps \begin{align*}S^1\stackrel{a^5(ab)^{-2}}{\...
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### What is a g-frame?

I am reading A quick trip through knot theory (link to pdf!) by R.H. Fox, in particular the section of branched covering (section 8 pag. 26 into the document). When he describes his algorithm to find ...
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### Construct a normal covering space.

Let $X$ be a connected and locally connected space. Let $(C,q)$ be a connected covering space over $X$. Then, can we construct a normal covering space $(E,p)$ over $X$ such that there exists a ...
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### Covering space of an interval is trivializable.

We say a covering space $(E,p)\to X$ is trivializable, if there is a space $F$ (with the discrete topology), and a homeomorphism between $\varphi:E\to X\times F$ such that $p=pr_1\circ \varphi$, where ...
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### What are all the topological spaces that the $S^1$ can cover up to homeomorphism?

I'm currently studying $G$-coverings for the first time and I came across an interesting question. First of all, I know that all connected coverings of $S^1:=\{z\in\mathbb{C}\mid |z|=1\}$ are of the ...
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### Question regarding isomorphism between quotient of fundamental groups.

I'm trying to solve the following problem that can be found in Kosniowski's a first course in algebraic topology. This problem is under the chapter on Borsuk Ulam theorem problem $20.7$ d. Question ...
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### Universal covering

Let $M=S^1 \times \mathbb{R}^n$ with the metric $g=-d\psi^2+q$, where $d \psi^2$ is the standard metric on $S^1$ and $q$ is the euclidean metric on $\mathbb{R}^n$. I know that the universal covering ...
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### Can isomorphic groups act on a topological space in different ways?

The question arises from the fact that each topological manifold $X$ is homeomorphic to its universal cover $X_0$ quotiented by the action of the fundamental group $\pi_1(X)$. It is natural to ask ...
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