Oriented cover
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== Introduction == | == Introduction == | ||
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− | This page is based on \cite{Ranicki2002|Definition 4.56}. Let $p:\widetilde{X} \to X$ be a regular covering of a connected space with orientation character $w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$. Let $\pi$ denote the group of covering translations. Since $\widetilde{X}$ is a regular cover $\pi_1(\widetilde{X})$ is a normal subgroup of $\pi_1(X)$ and $\pi \cong \pi_1(X)/\pi_1(\widetilde{X})$ (See \cite{Hatcher2002|Proposition 1.39}). Let $q: \pi_1(X) \to \pi$ denote the quotient map. The orientation character of the cover factors as $$w(\widetilde{X}) = w(X)\circ p_*,$$ which corresponds to the intuition that the cover $\widetilde{X}$ is orientable if all loops in $\widetilde{X}$ project to orientable loops in $X$. | + | This page is based on \cite{Ranicki2002|Definition 4.56}. Let $p:\widetilde{X} \to X$ be a regular covering of a connected space with [[Orientation character|orientation character]] $w(X)\in H^1(X;\Zz_2) = \Hom(\pi_1(X),\Zz_2)$. Let $\pi$ denote the group of covering translations. Since $\widetilde{X}$ is a regular cover $\pi_1(\widetilde{X})$ is a normal subgroup of $\pi_1(X)$ and $\pi \cong \pi_1(X)/\pi_1(\widetilde{X})$ (See \cite{Hatcher2002|Proposition 1.39}). Let $q: \pi_1(X) \to \pi$ denote the quotient map. The orientation character of the cover factors as $$w(\widetilde{X}) = w(X)\circ p_*,$$ which corresponds to the intuition that the cover $\widetilde{X}$ is orientable if all loops in $\widetilde{X}$ project to orientable loops in $X$. |
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== Examples == | == Examples == | ||
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− | The two most important examples of oriented covers are the universal cover $(\widetilde{X},\pi_1(X),w(X))$ and the 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. |
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Revision as of 17:03, 19 April 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 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