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 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
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 .
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 .
5 Convention
Tex syntax erroris already orientatable, the orientation double cover consists of two disjoint copies of
Tex syntax error. Any orientation of that we choose must involve giving the two copies of
Tex syntax erroropposite orientations, otherwise it will not be in the image of the map .
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
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 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
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 .
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 .
5 Convention
Tex syntax erroris already orientatable, the orientation double cover consists of two disjoint copies of
Tex syntax error. Any orientation of that we choose must involve giving the two copies of
Tex syntax erroropposite orientations, otherwise it will not be in the image of the map .
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
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 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
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 .
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 .
5 Convention
Tex syntax erroris already orientatable, the orientation double cover consists of two disjoint copies of
Tex syntax error. Any orientation of that we choose must involve giving the two copies of
Tex syntax erroropposite orientations, otherwise it will not be in the image of the map .
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