Orientation covering

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An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 18:40, 7 March 2014 and the covering&diff=cur&oldid=11510 changes since publication.

The user responsible for this page is Matthias Kreck. No other user may edit this page at present.

Let M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_oy3tXx be a n-dimensional topological manifold. We construct an oriented manifold \hat M and a 2-fold covering p : \hat M \to M called the orientation covering. The non-trivial deck transformation of this covering is orientation-reversing. As a set \hat M is the set of pairs (x, o_x), where o_x is a local orientation of M at x given by a generator of the infinite cyclic group H_n(M, M-x;\mathbb Z). The map p assignes x to (x,o_x). Since there are precisely two local orientations, the fibres of this map have cardinality 2.

Next we define a topology on this set. Let \varphi : U \to V\subset \mathbb R^n be a chart of M. We orient \mathbb R^n by the standard orientation given by the standard basis e_1, e_2, ..., e_n, from which we define a a continuous local orientation by identifying the tangent space with \mathbb R^n. 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 x \in \mathbb R^n by sto_x. Using the chart we transport this standard orientation to U by the induced map on homology. The local orientations given by this orientation of U determine a subset of \hat M, which we require to be open. Doing the same starting with the non-standard orientation of \mathbb R^n we obtain another subset, which we also call open. We give \hat M 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 \mathbb R^n, and so we obtain an atlas of \hat M. The map p is by construction a 2-fold covering. By construction \hat M is oriented in a tautological way and the non-trivial deck transformation of the covering is orientation reversing.

Thus we have constructed a 2-fold covering of M by an oriented manifold \hat M, which is smooth, if M is smooth. This covering is called the orientation covering.

If M is smooth one can use the local tangential orientation of T_xM 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 \hat M is a smooth manifold and p a local diffeomorphism.

For more information and a discussion placing the orientation covering in a wider setting, see [Dold1995, VIII § 2].

Contents

1 Characterization of the orientation covering

One can easily characterize the orientation covering:

Proposition 1.1. If N is an oriented manifold and p: N \to M is a 2-fold covering with orientation reversing non-trivial deck transformation, then it is isomorphic to the orientation covering.

Proof. We have a map N \to \hat M by mapping y \in N to (p(y), orientation \,\, induced \,\, by \,\, p). This is an isomorphism of these two coverings.

\square

If M is orientable, we pick an orientation and see that \hat M is the disjoint union of \{(x,o_x)| \,\, o_x \,\, is \,\, the \,\, local \,\, orientation \,\, given \,\, by \,\, the \,\, orientation \,\, of \,\, M\} and its complement, so it is isomorphic to the trivial covering M \times \mathbb Z/2. In turn if the orientation covering is trivial it decomposes \hat M into two open (and thus oriented) subsets homeomorphic to M and so M is orientable. Thus we have shown:

Proposition 1.2. M is orientable if and only if the orientation covering is trivial. If M is connected, M is non-orientable if and only if \hat M is connected. In particular, any simply-connected manifold is orientable.

2 Relation to the orientation character

We assume now that M is connected. The orientation character is a homomorphism w: \pi_1(M) \to \{ \pm 1\}, which attaches +1 to a loop S^1 \to M if and only if the pull back of the orientation covering is trivial. By the classification of coverings this implies that w is trivial if and only if M is orientable.

3 Examples

Here are some examples of orientation coverings.

  1. If M is orientable then p \colon \hat M \to M is isomorphic to the projection M \times \mathbb Z/2 \to M.
  2. If n is even, \mathbb R P^n is non-orientable and the orientation cover is the canonical projection S^n \to \mathbb R P^n. The deck transformation of the orientation covering is the antipodal map on S^n.
  3. The orientation cover of the Klein bottle K^2 is the canonical projection from the 2-torus; p \colon T^2 \to K^2.
  4. The orientation cover of the open Möbius strip Mö is the canonical projection from the cylinder; p \colon S^1 \times \Rr \to Mö.

4 References

5 External links

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