Questions tagged [covering-spaces]

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16 votes
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What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...
Thom's user avatar
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3 votes
0 answers
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Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

$\DeclareMathOperator\Aut{Aut}\newcommand{\alg}{\mathrm{alg}}\newcommand{\an}{\mathrm{an}}$Edited after Noam Elkies' comment: From what I understand (very little actually), there exist elliptic curves ...
Sylvain JULIEN's user avatar
2 votes
0 answers
161 views

Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?

Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
wonderich's user avatar
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26 votes
2 answers
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Teaching the fundamental group via everyday examples

This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys. What ...
23 votes
5 answers
2k views

Does anyone know a basepoint-free construction of universal covers?

Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...
Kim's user avatar
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10 votes
1 answer
2k views

Is a space with no covering spaces simply connected?

Suppose $X$ is a path connected space such that every connected covering space of $X$ is trivial (1-fold.) Must $X$ be simply connected? Intuitively, the answer seems to be no (imagine taking a disk,...
David Cohen's user avatar
6 votes
1 answer
1k views

A generalization of covering spaces to fiber bundles with totally path-disconnected fibers

There is a classical theorem about covering spaces and the actions of the fundamental group. Theorem 1: Let $B$ be a non-empty locally path-connected and path-connected space. The category of ...
Andrej Bauer's user avatar
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6 votes
1 answer
411 views

Why, if the geometric realisation of a simplicial map $p$ is a (topological) covering map, must $p$ be a (simplicial) covering map?

I essentially am asking for an explanation of the comment under this post by Tom Goodwillie. In the "Kerodon", Lurie defines a simplicial covering map as follows: A map $p:E\to X$ of ...
FShrike's user avatar
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5 votes
0 answers
238 views

characteristic classes of a covering space with symmetric group action

Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space $$ S_n\to M\to M/S_n $$ where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...
QSR's user avatar
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3 votes
1 answer
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cohomology of the orbit space of a group action

Let $M$ be a manifold. Let a finite group $G$ act on $M$ discretely. Let $F$ be a field. Suppose the induced action of $G$ on the cohomology algebra $H^*(M,F)$ is known. We want to obtain $H^*(M/G;F)$...
Shiquan Ren's user avatar
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3 votes
2 answers
1k views

Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
user avatar
3 votes
0 answers
152 views

Making extensions $L/K$ aware of the Galois group coming from $K/k$

Although inspired by my question on math.SE https://math.stackexchange.com/q/1902190/214353 this is not a crosspost. What happened is that after an answer and some comments I realized more clearly ...
მამუკა ჯიბლაძე's user avatar
2 votes
0 answers
130 views

Symmetric group-cocycle descends to symmetric product

Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
KuSi's user avatar
  • 53
2 votes
2 answers
483 views

covering theory with compact open topology

In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected. Under ...
Paul Pfeiffer's user avatar
2 votes
2 answers
523 views

Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $...
H1ghfiv3's user avatar
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0 votes
0 answers
88 views

Automorphism group of a Lie group $G$ vs that of a double covering group $\tilde{G}$: same or not? [duplicate]

Previously there are some counterexamples given in Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not? such that the automorphism group of a Lie group 𝐺 is not ...
wonderich's user avatar
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