Oriented cover
m (An oriented cover should contain a choice of orientation as part of the data!) |
m |
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$$\xymatrix{ | $$\xymatrix{ | ||
w(X): \pi_1(X) \ar[r] & \pi \ar[r]^-{w} & \Zz_2. | w(X): \pi_1(X) \ar[r] & \pi \ar[r]^-{w} & \Zz_2. | ||
− | }$$ An '''oriented cover''' is an orientable cover together with a choice of | + | }$$ An '''oriented cover''' is an orientable cover together with a choice of lift $\widetilde{b}$ of a basepoint $b\in X$ to $\widetilde{X}$. |
+ | </wikitex> | ||
+ | |||
+ | == Lifts correspond to orientations == | ||
+ | <wikitex>; | ||
+ | Let $(\widetilde{M},\pi,w)$ be an oriented cover of a connected manifold $M^m$. A choice of lift $\widetilde{b}\in\widetilde{M}$ corresponds to a choice of $w$-twisted fundamental class $[\widetilde{M}]\in H_m(M,M^w)$. Given a lift $\widetilde{b}$ a fundamental class $[\widetilde{M}]\in H_m(M,M^w)$ is uniquely determined by setting its restriction to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\})$ to be $1\in \pi$ and extending equivariantly. Conversely, given a fundamental class $[\widetilde{M}]\in H_m(M,M^w)$ define $\widetilde{b}$ to be the lift such that the restriction of $[\widetilde{M}]\in H_m(M,M^w)$ to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\})$ is $1\in \pi$. | ||
</wikitex> | </wikitex> | ||
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== Convention == | == Convention == | ||
<wikitex>; | <wikitex>; | ||
− | In the case that $ | + | In the case that the manifold $M$ is already orientatable, the orientation double cover $(M^w,\Zz_2,\id_{\Zz_2})$ consists of two disjoint copies of $M$. Any orientation of $M^w$ that we choose must involve giving the two copies of $M$ opposite orientations, otherwise it will not be in the image of the map $H_m(M,\Z^w) \to H_m(\widetilde{M})$. |
</wikitex> | </wikitex> | ||
Revision as of 17:36, 21 May 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Tex syntax error.
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
Tex syntax errorwith an orientation character is a regular covering of
Tex syntax errorwith 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
In the case that the manifold is already orientatable, the orientation double cover consists of two disjoint copies of . Any orientation of that we choose must involve giving the two copies of opposite 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
Tex syntax error.
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
Tex syntax errorwith an orientation character is a regular covering of
Tex syntax errorwith 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
In the case that the manifold is already orientatable, the orientation double cover consists of two disjoint copies of . Any orientation of that we choose must involve giving the two copies of opposite 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
Tex syntax error.
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
Tex syntax errorwith an orientation character is a regular covering of
Tex syntax errorwith 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
In the case that the manifold is already orientatable, the orientation double cover consists of two disjoint copies of . Any orientation of that we choose must involve giving the two copies of opposite 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