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Let be a -dimensional topological manifold. We construct an oriented manifold and a -fold covering called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set is the set of pairs , where is a local orientation of at given by a generator of the infinite cyclic group . The map assignes to . Since there are precisely two local orientations, the fibres of this map have cardinality .
Next we define a topology on this set. Let be a chart of . We orient by the standard orientation given by the standard basis , , ..., , from which we define a a continuous local orientation by identifying the tangent space with . Since for a smooth manifold a tangential orientation defines a homological orientation, this also gives a homological orientation: see [Kreck2013, §3]. We call the standard local orientation at by . Using the chart we transport this standard orientation to by the induced map on homology. The local orientations given by this orientation of determine a subset of , which we require to be open. Doing the same starting with the non-standard orientation of we obtain another subset, which we also call open. We give the topology generated by these open subsets, where we vary about all charts. By construction each of these open subsets is are homeomorphic to an open subset of , and so we obtain an atlas of . The map is by construction a -fold covering. By construction is oriented in a tautological way and the non-trivial deck transformation of the covering is orientation reversing.
Thus we have constructed a -fold covering of by an oriented manifold , which is smooth, if is smooth. This covering is called the orientation covering.
If is smooth one can use the local tangential orientation of instead of the homological orientation to construct the orientation covering (for the equivalence of these data see the Manifold Atlas page Orientation of manifolds; [Kreck2013, §6]). Since a countable covering of a smooth manifold has a unique smooth structure such that the projection map is a local diffeomorphism, in the smooth case is a smooth manifold and a local diffeomorphism.
For more information and a discussion placing the orientation covering in a wider setting, see [Dold1995, VIII § 2].
1 Characterization of the orientation covering
One can easily characterize the orientation covering:
Proposition 1.1. If is an oriented manifold and is a -fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering.
Proof. We have a map by mapping to . This is an isomorphism of these two coverings.
If is orientable, we pick an orientation and see that is the disjoint union of and its complement, so it is isomorphic to the trivial covering . In turn if the orientation covering is trivial it decomposes into two open (and thus oriented) subsets homeomorphic to and so is orientable. Thus we have shown:
Proposition 1.2. is orientable if and only if the orientation covering is trivial. If is connected, is non-orientable if and only if is connected. In particular, any simply-connected manifold is orientable.
2 Relation to the orientation character
We assume now that is connected. The orientation character is a homomorphism , which attaches to a loop if and only if the pull back of the orientation covering is trivial. By the classification of coverings this implies that is trivial if and only if is orientable.
Here are some examples of orientation coverings.
- If is orientable then is isomorphic to the projection .
- If is even, is non-orientable and the orientation cover is the canonical projection . The deck transformation of the orientation covering is the antipodal map on .
- The orientation cover of the Klein bottle is the canonical projection from the 2-torus; .
- The orientation cover of the open Möbius strip is the canonical projection from the cylinder; .
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Kreck2013] M. Kreck, Orientation of manifolds, Bull. Man. Atl. (2013).