Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 15 / 4

For every oriented open set $U$ and nowhere-vanishing orientation form $\omega$, let ${\hat{U}}_{\omega }$ contain the points $(p,o_p)$ for which $o_p$ is the orientation determined by ${\omega }_p$. These sets form a basis of a topology for the desired topology on the orientation double cover.

They cover the set of all pairs: every point has an oriented coordinate chart, and changing the sign of its orientation form gives the other orientation. To check the basis intersection axiom, consider ${\hat{U}}_{\omega }$ and ${\hat{V}}_{\eta }$. On $U\cap V$ there is a unique smooth nonzero function $h$ with $\eta =h\omega$. Their intersection is the sheet overSince $W$ is open, this intersection is ${\hat{W}}_{\omega {|}_W}$, another basis set. Thus intersections are open without requiring the two local orientations to agree on every component of the overlap.

The restriction $\pi :{\hat{U}}_{\omega }\to U$ is bijective. Its inverse sends a relatively open set $V\subset U$ to the basis set ${\hat{V}}_{\omega {|}_V}$, so the inverse is continuous. Conversely the projection of the intersection of ${\hat{U}}_{\omega }$ with any other basis set is an open agreement set like $W$, so the projection is continuous as well. Hence each sheet projects homeomorphically onto its oriented open set. In particular,with both sheets open. This establishes the two-sheeted covering space structure.

For a coordinate chart $x:U\to x(U)$ and either orientation sheet over $U$, take the chart $x\circ \pi$ on that sheet. On a nonempty overlap, its transition to $y\circ \pi$ is exactly $y\circ x^{-1}$, restricted to the open agreement set from part (a). It is smooth with smooth inverse. These charts therefore define a smooth structure for which $\pi$ is locally a diffeomorphism.

The topology is Hausdorff: points with distinct base points are separated by inverse images of disjoint base neighborhoods; two points over the same base point are separated by the two disjoint sheets over one coordinate neighborhood. A countable coordinate cover of $M$ gives a countable collection of two-sheeted charts, whose Euclidean chart bases show second countability. Thus the cover is a genuine smooth manifold, not just a collection of compatible charts. The construction using all base charts makes the smooth structure independent of the chosen atlas. It also makes $\pi :{\hat{U}}_{\omega }\to U$ a diffeomorphism for every oriented open $U$, since it is a bijective local diffeomorphism.

At $\hat{p}=(p,o_p)$, the derivative $d{\pi }_{\hat{p}}:T_{\hat{p}}\hat{M}\to T_{p}M$ is a linear isomorphism. Define the orientation of $T_{\hat{p}}\hat{M}$ to be the one mapped to $o_p$. On ${\hat{U}}_{\omega }$ it is represented by ${\pi }^{\ast }\omega$. Where two sheets overlap, their base orientation forms differ by a positive function, so their pullbacks determine the same orientation. This proves the canonical orientation of the cover:The deck involution $(p,o_p)\mapsto (p,-o_p)$ reverses this canonical orientation of the orientation double cover. No global orientation of the base was used.

A connected smooth manifold is path connected because it is locally path connected. Each connected component $C$ of the orientation double cover is itself path connected. Choose a point of $C$ over $p$. For any $q\in M$, lift a path from $p$ to $q$ starting at that chosen point, using the path lifting theorem. The lifted path lies in $C$ and ends above $q$. Therefore every component maps onto $M$.

Fix one fibre, containing exactly two points. Every component meets that fibre, and distinct components meet disjoint subsets of it, so there are at most two connected components.

If $M$ is orientable, a global continuous choice of tangent-space orientation $o$ gives two disjoint sheetsThey are open: on a sufficiently small oriented coordinate neighborhood the global choice agrees consistently with one of the two local sheets. Each projects homeomorphically to the connected base, so these are precisely two connected components.

Conversely suppose the cover is disconnected. It has exactly two components $C_{+},C_{-}$. Both project onto the base and each fibre contains only two points, so each component meets every fibre exactly once. The restriction $\pi :C_{+}\to M$ is consequently a bijective local diffeomorphism, hence a global diffeomorphism. Transporting the canonical orientation of $C_{+}$ through this map or, equivalently, using its smooth inverse section, gives a global orientation of $M$. Thus the connectedness criterion for the orientation double cover is