Intereting Posts

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induction proof of fibonacci number
Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
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Independence and Conditional Independence between random variables
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Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$
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Another proof of uniqueness of identity element of addition of vector space
simply connected covering of a path connected space
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Eigenvectors of harmonic series matrix

I’d like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah’s paper “Elliptic operators, discrete groups, and von Neumann algebras.” http://www.maths.ed.ac.uk/~aar/papers/atiyah_elld.pdf

Suppose a discrete group $G$ acts freely on a manifold $X$ with the quotient $X/G$ being compact. Then $X$ is a covering space of $X/G$ with covering map given by the quotient map $p: X\rightarrow X/G$.

Thanks.

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- Topology knowledge for CW complexes $\oplus$ Reference request
- Why can't differentiability be generalized as nicely as continuity?
- Is the product of covering maps a covering map?
- Top homology of a manifold with boundary
- Klein bottle covered by the torus

The discreteness assumption is meaningless since given a continuous group action $G\times X\to X$, it remains continuous if we equip $G$ with discrete topology. Now, a counter-example to the claim is the action of the group of the additive group of real numbers on itself

$$

{\mathbb R}_{d} \times {\mathbb R} \to {\mathbb R}

$$

via addition. Here the subscript $d$ means the discrete topology, otherwise ${\mathbb R}$ is equipped with the standard topology. The quotient space is a single point, hence, compact. The action itself is clearly not a covering action, but is free. I think, Atiyah forgot to add **proper**, in the sense of Palais:

A continuous action $G\times X\to X$ is **proper**, in the sense of Palais, if for any two points $x, y\in X$ there exists a pair of neighborhoods $U_x, U_y$ of these points such that

$$

\{g\in G: g U_x\cap U_y\ne \emptyset\}

$$

is a relatively compact subset of $G$. When $G$ is discrete, this just means that this subset is finite.

The correct statement is then:

If $G\times X\to X$ is a Palais-proper, free, action of a discrete group on a Hausdorff space, then the projection X\to X/G$ is a covering map.

Related: Proper and free action of a discrete group.

Dave Glickenstein’s notes give a short proof, see page 9.

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