Let $M$ be a connected Hausdorff second countable topological space. I will call $M$ self-covering if it is its own $n$-fold cover for some $n>1$. For instance, the circle is its own double cover by $e^{2\imath\varphi}\mapsto e^{\imath\varphi}$.

**Questions:**

What would be a reasonable assumption on $M$ to ensure it is self-covering? Is every connected compact/closed topological manifold self-covering?

Assume that $M$ is a differentiable manifold, and let $\phi:M\to M$ be the local diffeomorphism providing the $n$-fold cover for $n>1$. Is it true that the modulus of the Jacobian $|\det J(d\phi(x))|<1$ everywhere aside from countably many isolated points $x$ where it can be 1? In other words, is a covering map a volume contraction?

Please, note that I am not a topologist; more accessible language and references would be appreciated.

infinite. $\endgroup$ – Francesco Polizzi Mar 24 '17 at 15:10isthe boundary of another manifold. $\endgroup$ – Oscar Randal-Williams Mar 24 '17 at 15:23