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Let be a connected topological manifold throughout. The following definition can be found in [Davis&Kirk2001, Section 5]. The orientation character is a homomorphism
which may be defined as follows. Take the orientation covering and let be the 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
The orientation character induces a map from to the classifying space of , which is unique up to homotopy. By the definition of the orientation character, this map classifies the orientation covering. So is trivial if and only if , and therefore is orientable.
2 Alternative descriptions
Let be the universal covering of . Then acts on by deck transformations.
Proposition 2.1. Let be the orientation character. Then if and only if the action of on is orientation preserving.
Proof. If is orientable, then an orientation on induces an orientation on and every deck transformation is orientation preserving. If is non-orientable, then the standard orientation of the orientation covering of induces an orientation on . By construction the non-trivial deck transformation of is orientation-reversing. Therefore, a deck transformation of is orientation preserving if and only if it acts trivially on .
If the manifold carries a smooth structure, the orientation character can be additionally characterised in terms of tangential data. Define a map in the following way: Let be the base point. Pick a chart around . Let be a based loop in . The standard orientation of corresponds to an orientation of under the differential of (cf. Orientation of manifolds). Fibre transport along yields another orientation of . Now set
Using the comparison between homological and tangential orientation (see Orientation of manifolds), one can show the following:
Proposition 2.2. The map is a well-defined group homomorphism and coincides with the orientation character .
In addition to this geometric characterisation, the orientation character also admits a description in terms of characteristic classes: Under the "evaluation" homomorphism of the Universal Coefficient Theorem , the first Stiefel-Whitney class corresponds to a homomorphism which we also call .
Proposition 2.3. Let be the Hurewicz homomorphism. Then the orientation character of coincides with the composition via the canonical isomorphism .
Proof. All 1-manifolds are orientable, so suppose that . Let classify the tangent bundle of and consider the induced map . Now is generated by the classifying map of the Möbius bundle; denote this class by . Hence, for a based loop in with homotopy class , we see the pullback is non-trivial if and only if . By Proposition 2.1, we see that if and only if . Applying the Universal Coefficient Theorem for , we see that the universal first Stiefel-Whitney class , which generates , has the property that . Hence, we see that if and only if if and only if if and only if .
- Since is orientable if and only if is odd, the orientation character is trivial if and only if is odd. If is even, is an isomorphism.
- The open Möbius strip has fundamental group and is non-orientable. Therefore, the orientation character is given by the surjection .
- If is an embedding of a manifold of the same dimension (possibly with boundary), then the orientation character of is given as the composition . This follows from the fact that the atlas of with local orientations induces by restriction an atlas of with local orientations. Therefore, the orientation covering of is the orientation covering of restricted to .
- Let . The preimage of an embedded disk in the orientation covering is a disjoint union of two disks. The orientation covering of a connected sum along is the "double connected sum" of the orientation coverings along the preimages of in the orientation coverings. How to pair the disks is determined by the local orientations. From this it follows that the orientation character of is given by the pushout
4 The orientation character via Poincaré duality
Even though the original definition is very geometric, the orientation character is already completely determined by the homotopy type of a given closed manifold. Both the Hurewicz homomorphism and the first Stiefel-Whitney class are homotopy invariants (for the second point, see the Manifold Atlas page Wu class), so we already know this for differentiable manifolds by Proposition 2.3. In general this is most easily seen in the more abstract setting of Poincaré complexes, see [Lück2001, Section 3.1].
Lemma 4.1. If is a connected Poincaré complex, then is the only homomorphism such that is a Poincaré complex with orientation character .
Proof. Suppose is a connected Poincaré complex. We first introduce some notation:
- denotes as a right -module, where the -action is given by .
- , where the tensor product is equipped with the diagonal action.
- For a left -chain complex , we define analogous to (2.).
Applying to the -chain equivalence , we obtain an isomorphism of -modules
Note that the left hand side is independent of the orientation character . Therefore, is isomorphic to as a -module and and have to agree.
Corollary 4.2. If is a connected Poincaré complex, then depends only on the homotopy type of .
Proof. Let and be connected Poincaré complexes and be a homotopy equivalence. Then is a Poincaré complex and and have to agree.
Remark 4.3. The above statements are also true for connected Poincaré pairs.
Proposition 2.3 generalises to the following:
Proposition 4.4 [Byun1999, Lemma 4.3]. Using the Universal Coefficients Theorem and the Hurewicz map, the orientation character of a connected Poincaré complex coincides with the first Stiefel-Whitney class of its Spivak normal fibration.
- [Byun1999] Y. Byun, Tangent fibration of a Poincaré complex, J. London Math. Soc. (2) 59 (1999), no.3, 1101–1116. MR1709099 (2000f:57023)
- [Davis&Kirk2001] J. F. Davis and P. Kirk, Lecture notes in algebraic topology, American Mathematical Society, 2001. MR1841974 (2002f:55001) Zbl 1018.55001
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- The Wikipedia page about the orientation character