Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [covering-spaces]

For questions about or involving covering spaces in algebraic topology.

Let $\pi : E \to B$ be a continuous surjective map between topological spaces $E$ and $B$. We say that $\pi$ is a covering map if for every $x \in B$, there is an open neighbourhood $U$ of $x$ such that $\pi^{-1}(U)$ is a union of disjoint open sets in $E$, each of which is mapped homeomorphically onto $U$ by $\pi$.

We call $E$ a covering space of $B$ and often refer to $B$ as the base space.

The open neighbourhoods referred to in the definition are often called evenly covered neighbourhoods.

The fibres of $\pi$ are homeomorphic, so they all have the same cardinality; this cardinality is often called the number of sheets of the covering.

Reference: Covering space.

1696 questions
1
vote
1 answer

Basic covering space question

Given a path connected metric space $X$ and a cover $\tilde{X}$ which is also a path connected metric space with covering map $E$, then is $E$ a local isometry?
Sean
  • 743
0
votes
1 answer

X with contractible universal cover then any map $S^n \rightarrow X$ can be extended to $D^{n+1} \rightarrow X$

If a space X has with contractible universal cover then any map $S^n> \rightarrow X$ can be extended to $D^{n+1} \rightarrow X$ $D^{n+1}$ is the ball with dimension $n+1$ How should I approach this proof? Ihave drawn a diagram with the spaces, the…
southernKid33
  • 375
0
votes
1 answer

Covering of a submanifold

Let $f:\tilde{M}\to M$ be a $d$-fold covering map and $N\subset M$ be a codimension 2 submanifold. What is the number of path-connected components of $f^{-1}(N)$ (assuming $N$ is connected)?
usr1988
  • 35
  • 3
0
votes
0 answers

open covering of $S^1\times S^1$

Show that the map $\psi_{m,n}:S^1×S^1→S^1×S^1$given by $(z,w)→(z^m,w^n)$ where $mn≠0$ is a covering map. My attempt is $\psi_n:S^1\to S^1$ given by $z\to z^n$ is a covering map for $n\neq0$.Similarly $\psi_m:S^1\to S^1$ given by $z\to z^m$ is a…
Christiano
  • 49
0
votes
1 answer

Covering group of a connected group is simply connected

Let M is a Riemannian manifold and $\tilde{M}$ is its universal covering and G is connected subgroup of the isometries of M . I know that there is a covering group $\tilde{G}$ such that acts on $\tilde{M}$ . My question is " is $\tilde{G}$ simply…
Rojan Bakhtiari
  • 1
0
votes
1 answer

Constructing a universal covering space

Let $X$ be the topological space consisting of the standard 2-sphere together with a line segment from the north pole to the south pole. Compute $\pi_{1}(X)$ and construct the universal covering space of $X$. By van Kampen theorem, this figure can…
Jack
  • 2,017
0
votes
2 answers

I need to find and open cover of A is there a finite subcover for B?

Let $A= (0, 1]$ and let $B={(\frac{n+2}{2^n}, 2^{1/n}): n \in \Bbb N)}$ Please help I am having trouble on where to start because I'm still having a hard time on understanding open covers and subcovers.
M.Matthews
  • 79
0
votes
0 answers

Which is correct between cover or covering?

Let $I$ be closed n-dimensional intervals, $$I=\{\mathbf{x}: a_j\le x_j \le b_j, \quad j=1, \cdots, n\}$$ and $S$ be a countable collection of intervals $I_k$, $$S=\{I_j,\quad j=1, 2, \cdots\}$$ In this case, if $\displaystyle E\subset\cup_{I_j\in…
Danny_Kim
  • 3,423
-1
votes
1 answer

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?
Carlos
  • 1
1
2