Oriented cover
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | 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 $X$. |
{{beginthm|Lemma}} | {{beginthm|Lemma}} | ||
− | The | + | The space $\widetilde{X}$ is orientable if and only if the orientation character $w(X)$ factors through $\pi$. |
{{endthm}} | {{endthm}} | ||
{{beginproof}} Consider the diagram | {{beginproof}} Consider the diagram | ||
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In light of this we make the following definition. | In light of this we make the following definition. | ||
</wikitex> | </wikitex> | ||
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== Definition == | == Definition == | ||
<wikitex>; | <wikitex>; | ||
− | 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 lift $\widetilde{b}$ of a basepoint $b\in X$ to $\widetilde{X}$. |
</wikitex> | </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 [[Poincaré duality|$w$-twisted fundamental class]] $[\widetilde{M}]\in H_m(M;\Z^w)$. Given a lift $\widetilde{b}$ a fundamental class $[\widetilde{M}]\in H_m(M;\Z^w)$ is uniquely determined by setting its restriction to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\}; \Z)$ to be $1\in \pi$ and extending equivariantly. Conversely, given a fundamental class $[\widetilde{M}]\in H_m(M;\Z^w)$ define $\widetilde{b}$ to be the lift such that the restriction of $[\widetilde{M}]\in H_m(M;\Z^w)$ to $H_n(\widetilde{M},\widetilde{M}-\{\widetilde{b}\};\Z)$ is $1\in \pi$. | ||
+ | </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 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 $H_2$. Thus for $(\overline{X},\pi,w)$ any oriented cover of $X$ the following diagram commutes: |
+ | $$\xymatrix{ && \pi_1(\widetilde{X})=\{1\} \ar[dl] \ar[dd] && \\ & \pi_1(\overline{X})\ar[dr]^-{p_*} \ar[dl]&&&\\ \pi_1(X^w)\ar[rr] && \pi_1(X) \ar[dd]^{\id} \ar[dr]^-{q} \ar[rr]^{w(X)} && \Z_2\\ &&&\pi \ar[ur]^-{w} & \\ && \pi_1(X)\ar[ur]&& | ||
+ | }$$ | ||
+ | </wikitex> | ||
+ | |||
+ | == Orientations of the orientation double cover == | ||
+ | <wikitex>; | ||
+ | Let $M^m$ be a connected manifold and let $(M^w,\Zz_2,\id_{\Zz_2})$ be the orientation double cover. Following \cite{Ranicki2002|Proposition 4.48} there is a short exact sequence of $\Z[\Z_2]$-modules | ||
+ | $$\xymatrix@R=1mm{0 \ar[r] &\Z^{-} \ar[r] & \Z[\Z_2] \ar[r] & \Z \ar[r] &0 \\ & 1 \ar@{|->}[r] & 1-T, && \\ &&a+bT \ar@{|->}[r] & a+b. &} $$ | ||
+ | Applying $-\otimes_{\Z[\Z_2]} S(M^w)$ we obtain another short exact sequence which induces the following exact sequence in homology: | ||
+ | $$\xymatrix{\ldots \ar[r] & H_{n+1}(M) \ar[r] & H_n(M;\Z^w) \ar[r] & H_n(M^w) \ar[r]^-{p_*} & H_n(M) \ar[r] & \ldots}$$ | ||
+ | As $M$ is $m$-dimensional we have that $H_{m+1}(M)=0$ and so by the long exact sequence $$ H_m(M;\Z^w)\cong \ker(p_*:H_m(M^w)\to H_m(M)).$$ | ||
+ | In other words, a $w$-twisted fundamental class of a connected manifold $M$ can be thought of as a fundamental class of the orientation double cover that projects to zero in $H_m(M)$. 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 $M$ and a $w$-twisted orientation corresponds to an orientation of $M^w$ where the two copies of $M$ are given opposite orientations. In the case that $M$ is non-orientable $H_m(M;\Z)=0$ so a $w$-twisted orientation is precisely an orientation of the cover $M^w$. | ||
</wikitex> | </wikitex> | ||
− | == References == | + | == References== |
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Latest revision as of 17:10, 16 June 2014
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 1 Introduction
Lemma 1.1. The space 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.
[edit] 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
[edit] 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 .
[edit] 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 . Thus for any oriented cover of the following diagram commutes:
[edit] 5 Orientations of the orientation double cover
Let be a connected manifold and let be the orientation double cover. Following [Ranicki2002, Proposition 4.48] there is a short exact sequence of -modules
Applying we obtain another short exact sequence which induces the following exact sequence in homology:
Tex syntax erroris -dimensional we have that and so by the long exact sequence
Tex syntax errorcan be thought of as a fundamental class of the orientation double cover that projects to zero in . In the case that the manifold
Tex syntax erroris already orientatable, the orientation double cover consists of two disjoint copies of
Tex syntax errorand a -twisted orientation corresponds to an orientation of where the two copies of
Tex syntax errorare given opposite orientations. In the case that
Tex syntax erroris non-orientable so a -twisted orientation is precisely an orientation of the cover .
[edit] 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