Given a covering space action of a group \(G\) on an orientable manifold \(M\) by orientation-preserving homeomorphisms, show that \(M/G\) is also orientable.
Solution
Let \(q:M\to M/G\) be the quotient map. Since the action is a covering-space action, \(q\) is a covering map and \(M/G\) is a manifold.
Fix \(y\in M/G\) and choose an evenly covered neighborhood \(U\) of \(y\). Let \(\widetilde U\) be one sheet over \(U\). Because \(M\) is oriented, \(\widetilde U\) inherits a local orientation. Transport this orientation to \(U\) using the homeomorphism \[ q|_{\widetilde U}:\widetilde U\xrightarrow{\cong}U. \] We must show this does not depend on the chosen sheet. Any two sheets over \(U\) differ by a unique deck transformation coming from an element \(g\in G\). By hypothesis every such \(g\) preserves the orientation of \(M\), so the two transported orientations on \(U\) agree.
Hence every sufficiently small \(U\subset M/G\) has a well-defined orientation, and these local orientations agree on overlaps. Thus \(M/G\) is orientable.