Oriented cover
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== Definition == | == Definition == | ||
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− | An ''' | + | An '''orientable cover $(\widetilde{X},\pi,w)$''' of a (connected) space $X$ with an orientation character $w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$ is a regular covering of $X$ with group of covering translations $\pi$, together with an orientation character $w:\pi \to \Zz_2$ such that |
$$\xymatrix{ | $$\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 orientation. |
</wikitex> | </wikitex> | ||
== Examples == | == Examples == | ||
<wikitex>; | <wikitex>; | ||
− | The two most important examples of oriented covers are the universal cover $(\widetilde{X},\pi_1(X),w(X))$ and the [[Orientation covering|orientation double cover]] $(X^w,\Zz_2,\id_{\Zz_2})$. These correspond to the two extreme cases of factoring the orientation character via $\pi_1(X)$ and $\Zz_2$ respectively. Every oriented cover is a regular cover of $X^w$ and has $\widetilde{X}$ as a regular cover. | + | The two most important examples of oriented covers are the universal cover $(\widetilde{X},\pi_1(X),w(X))$ and the [[Orientation covering|orientation double cover]] $(X^w,\Zz_2,\id_{\Zz_2})$. These correspond to the two extreme cases of factoring the orientation character via $\pi_1(X)$ and $\Zz_2$ respectively. Every oriented cover is a regular cover of $X^w$ and has $\widetilde{X}$ as a regular cover - this is a consequence of the fact that if $H_1, H_2$ are normal subgroups of $G$ with $H_1$ a subgroup of $H_2$ then $H_1$ is a normal subgroup of $H_2$. |
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+ | == Convention == | ||
+ | <wikitex>; | ||
+ | In the case that $X$ is already orientatable, the orientation double cover $(X^w,\Zz_2,\id_{\Zz_2})$ of $X$ consists of two disjoint copies of $X$. Any orientation of $X^w$ that we choose should involve giving the two copies of $X$ opposite orientations, otherwise this will not correspond to a non-zero fundamental class with respect to twisted coefficients. | ||
</wikitex> | </wikitex> | ||
Revision as of 15:17, 21 May 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
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 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 .
4 Convention
In the case that is already orientatable, the orientation double cover of consists of two disjoint copies of . Any orientation of that we choose should involve giving the two copies of opposite orientations, otherwise this will not correspond to a non-zero fundamental class with respect to twisted coefficients.
5 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