Oriented cover
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Contents |
1 Introduction
Lemma 1.1. The cover 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.
2 Definition
An oriented 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
3 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.
4 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