Orientation covering
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Contents |
1 Construction
The orientation covering of a topological manifold is a canonical two-fold covering is called the For more information and a discussion placing the orientation cover in a broader setting, see [Dold1995, VIII § 2].
Theorem 1.1 c.f. [Dold1995, VIII 2.11]. Let be a -dimensional topological manifold. There is an oriented manifold and a -fold covering called the orientation covering. If is a smooth, resp. piecewise linear, manifold then and the covering map are smooth, resp. piecewise linear.
Proof. As a set is the set of pairs , where is a local orientation of at , either given by a generator of or by an orientation of in the smooth case (for the equivalence of these data see the atlas page on orientation of manifold). The map assigns to . Since there are precisely two local orientations, the fibers of this map have cardinality .
Next we define a topology on this set. Let be a chart of (smooth, if is smooth). We orient by the standard orientation given by the standard basis , , ..., , from which we define 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 atlas page on orientation of manifolds). We call the standard local orientation at by . Using the chart we transport this standard orientation to by the induced map on homology or the differential in the case of tangential orientations. The local orientations given by this orientation of is 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 over all charts of (smooth charts, if is smooth). By construction these open subsets are homeomorphic to an open subset of , and so we obtain an atlas of . In the smooth case this is a smooth atlas making a smooth manifold. The map is by construction a -fold covering, smooth, if is smooth. By construction is oriented in a tautological way. Thus we have constructed a -fold covering of by an oriented manifold , which is smooth, if is smooth.
2 Properties
In this section we record the key properties of the orientation cover are given in Proposition \ref{prop:properties} below. The orientation covering of a manifold is very closely related to the orientation character of . This is a homomorphism
which may be defined as follows. Fix a base-point with lifts and in . For a loop based at , let be the lift of with and define on the homotopy class of by
Proposition 2.1. Let be the orientation covering of a topological manifold .
- is orientable if and only if and is the projection to .
- Converely, if is connected then is non-orientable if and only if is connected.
- is orientable if and only if is the zero homomorphism.
- By construction, the deck transformation of orientation covering is orientation reversing.
- If is an oriented manifold and is a -fold covering with orientation reversing deck transformation, then is isomorphic to the orientation covering.
Proof. (1.) If is orientable, we pick an orientation and see that is the disjoint union of the space
and its complement, so it is isomorphic to the trivial covering . In turn if the orientation covering is trivial it decomposes into to open (and thus oriented) subsets homeomorphic to and so is orientable.
(2.) Follows immediately from (1.) since a two-fold cover of a connected space is non-trivial if and only if the total space of the covering is disconnected.
(3.) Follows from (1.) since classifies the orientation cover: see the page orientation character.
(4.) Is true by construciton as stated.
(5.) We have a map by mapping toTex syntax errorp
Tex syntax error. It is easily checked that his is an isomorphism of these two coverings.
3 Examples
We give a list of basic in interesting orientation double coverings.
- If is orientable then is isomorphic to the projection .
- If is even, is non-orienable and with orientation cover . The deck transformation of the orientation covering is the antipodal map on .
- The orientation cover of the Klein bottle is the projection from the 2-torus; .
- The orientation of the open Möbius strip is the cylinder; .
4 References
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
5 External links
- The Encylopedia of Mathematics article on orientation.
- The Wikipedia page on the orientability.