Oriented cover
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[edit] 1 Introduction
Lemma 1.1. The space is orientable if and only if the orientation character factors through .
Proof. Consider the diagram
If there exists a such that , then . Conversely if then the map
for any representative of is well defined and factors .
In light of this we make the following definition.
[edit] 2 Definition
An orientable cover of a (connected) space with an orientation character is a regular covering of with group of covering translations , together with an orientation character such that
[edit] 3 Lifts correspond to orientations
Let be an oriented cover of a connected manifold . A choice of lift corresponds to a choice of -twisted fundamental class . Given a lift a fundamental class is uniquely determined by setting its restriction to to be and extending equivariantly. Conversely, given a fundamental class define to be the lift such that the restriction of to is .
[edit] 4 Examples
The two most important examples of oriented covers are the universal cover and the orientation double cover . These correspond to the two extreme cases of factoring the orientation character via and respectively. Every oriented cover is a regular cover of and has as a regular cover - this is a consequence of the fact that if are normal subgroups of with a subgroup of then is a normal subgroup of . Thus for any oriented cover of the following diagram commutes:
[edit] 5 Orientations of the orientation double cover
Let be a connected manifold and let be the orientation double cover. Following [Ranicki2002, Proposition 4.48] there is a short exact sequence of -modules
Applying we obtain another short exact sequence which induces the following exact sequence in homology:
In other words, a -twisted fundamental class of a connected manifold can be thought of as a fundamental class of the orientation double cover that projects to zero in . In the case that the manifold is already orientatable, the orientation double cover consists of two disjoint copies of and a -twisted orientation corresponds to an orientation of where the two copies of are given opposite orientations. In the case that is non-orientable so a -twisted orientation is precisely an orientation of the cover .
[edit] 6 References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001