Orientation character
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== Definition == | == Definition == | ||
<wikitex>; | <wikitex>; | ||
− | The following definition can be found in \cite{Davis&Kirk2001|Section 5}. | + | The following definition can be found in \cite{Davis&Kirk2001|Section 5}. Let $M$ be a topological manifold. The orientation character is a homomorphism |
− | + | ||
− | Let $M$ be a topological manifold. The orientation character is a homomorphism | + | |
$$ w \colon \pi_1(M) \to \Zz/2 $$ | $$ w \colon \pi_1(M) \to \Zz/2 $$ | ||
which may be defined as follows. Take the [[orientation covering]] $p:\hat M\to M$ and let $x\in M$ be the base point with lifts $\tilde x_1$ and $\tilde x_{-1}$ in $\hat M$. For a loop | which may be defined as follows. Take the [[orientation covering]] $p:\hat M\to M$ and let $x\in M$ be the base point with lifts $\tilde x_1$ and $\tilde x_{-1}$ in $\hat M$. For a loop | ||
$\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$ based at $x$, let $\tilde \gamma \colon [0, 1] \to M$ be the lift of $\gamma$ with $\tilde{\gamma}(0) = \tilde x_1$ and define $w$ on the homotopy class of $\gamma$ by | $\gamma \colon ([0, 1], \{0, 1\}) \to (M, x)$ based at $x$, let $\tilde \gamma \colon [0, 1] \to M$ be the lift of $\gamma$ with $\tilde{\gamma}(0) = \tilde x_1$ and define $w$ on the homotopy class of $\gamma$ by | ||
$$w([\gamma]) := \left\{ \begin{array}{rl} 1~~ & \text{if $\tilde{\gamma}(1) = \tilde x_1$,} \\ -1~~ & \text{if $\tilde{\gamma}(1) = \tilde x_{-1}$.} \end{array} \right. $$ | $$w([\gamma]) := \left\{ \begin{array}{rl} 1~~ & \text{if $\tilde{\gamma}(1) = \tilde x_1$,} \\ -1~~ & \text{if $\tilde{\gamma}(1) = \tilde x_{-1}$.} \end{array} \right. $$ | ||
− | The orientation character $w\colon\pi_1(M)\to \Zz/2$ induces a map $M\to B\Zz/2$ unique up to homotopy. By the first definition of the orientation character this map classifies the orientation covering. So $w$ is trivial if and only if $\hat M\cong M\times\Zz/2$, and therefore $M$ orientable. | + | The orientation character $w\colon\pi_1(M)\to \Zz/2$ induces a map $M \to B\Zz/2$ from $M$ to the [[Wikipedia:Classifying space|classifying space]] of $\Zz/2$, which is unique up to homotopy. By the first definition of the orientation character this map classifies the orientation covering. So $w$ is trivial if and only if $\hat M\cong M\times\Zz/2$, and therefore $M$ is orientable. |
</wikitex> | </wikitex> | ||
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Under the "evaluation" homomorphism of the Universal Coefficient Theorem $H^1(M;\mathbb{Z}/2) \to \hom(H_1(M;\mathbb{Z}), \mathbb{Z}/2)$, the first Stiefel-Whitney class $w_1(M)$ corresponds to a homomorphism | Under the "evaluation" homomorphism of the Universal Coefficient Theorem $H^1(M;\mathbb{Z}/2) \to \hom(H_1(M;\mathbb{Z}), \mathbb{Z}/2)$, the first Stiefel-Whitney class $w_1(M)$ corresponds to a homomorphism | ||
$w^{\prime\prime} \colon H_1(M;\mathbb{Z}) \to \mathbb{Z}/2$. | $w^{\prime\prime} \colon H_1(M;\mathbb{Z}) \to \mathbb{Z}/2$. | ||
− | {{beginthm|Proposition}}\label{sw} Let $\rho \colon \pi_1(M) \to H_1(M;\mathbb{Z})$ be the [[Hurewicz homomorphism]]. Then the orientation character of $M$ is given by the composotion $w = w^{\prime\prime} \circ \rho$.{{endthm}} | + | {{beginthm|Proposition}}\label{sw} Let $\rho \colon \pi_1(M) \to H_1(M;\mathbb{Z})$ be the [[Wikipedia:Hurewicz theorem|Hurewicz homomorphism]]. Then the orientation character of $M$ is given by the composotion $w = w^{\prime\prime} \circ \rho$.{{endthm}} |
<!--{{beginthm|Proposition}}\label{sw} Upon precomposition with the Hurewicz homomorphism $\pi_1(M) \to H_1(M;\mathbb{Z})$, the homomorphism $w^{\prime\prime}$ coincides with the orientation character $w$.{{endthm}}--> | <!--{{beginthm|Proposition}}\label{sw} Upon precomposition with the Hurewicz homomorphism $\pi_1(M) \to H_1(M;\mathbb{Z})$, the homomorphism $w^{\prime\prime}$ coincides with the orientation character $w$.{{endthm}}--> | ||
{{beginproof}} | {{beginproof}} | ||
− | All 1-manifolds are orientable, so suppose that $n = dim(M) > 1$. Let $\tau\colon M \to BO(n)$ classify the tangent bundle of $M$ and consider the induced map $\tau_* \colon \pi_1(M) \to \pi_1(BO(n))$. Now $\pi_1(BO(n)) \cong \pi_0(O(n)) = \Z/2$ is generated by the classifying map of the Mobius bundle, denote this class by $[\eta]$. Hence for a based loop $\gamma \in M$ with homotopy class $[\gamma] \in \pi_1(M)$, we see that the tangent bundle of $M$ restricted to $\gamma$ is non-trivial if and only if $\tau_*([\gamma]) = [\eta]$. | + | All [[1-manifolds]] are orientable, so suppose that $n = \textup{dim}(M) > 1$. Let $\tau\colon M \to BO(n)$ classify the tangent bundle of $M$ and consider the induced map $\tau_* \colon \pi_1(M) \to \pi_1(BO(n))$. Now $\pi_1(BO(n)) \cong \pi_0(O(n)) = \Z/2$ is generated by the classifying map of the [[Wikipedia:Mobius strip|Mobius bundle]], denote this class by $[\eta]$. Hence for a based loop $\gamma \in M$ with homotopy class $[\gamma] \in \pi_1(M)$, we see that the tangent bundle of $M$ restricted to $\gamma$ is non-trivial if and only if $\tau_*([\gamma]) = [\eta]$. |
By Proposition \ref{x} we see that $w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$. Applying the universal coefficient theorem in $BO(n)$, we see that universal first Stiefel-Whitney class $w_1$, which generates $H^1(BO(n); \Z/2)$, has the property that $w_1(\eta) = 1$. | By Proposition \ref{x} we see that $w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$. Applying the universal coefficient theorem in $BO(n)$, we see that universal first Stiefel-Whitney class $w_1$, which generates $H^1(BO(n); \Z/2)$, has the property that $w_1(\eta) = 1$. | ||
− | Hence we see that $w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$ if and only if $w_1(\tau_*[\gamma]) = 1$ if and only if $w_1(M)([\gamma]) = 1$. Since $w \colon \pi_1(M) \to | + | Hence we see that $w([\gamma]) = 1$ if and only if $\tau_*([\gamma]) = [\eta]$ if and only if $w_1(\tau_*[\gamma]) = 1$ if and only if $w_1(M)([\gamma]) = 1$. Since $w \colon \pi_1(M) \to \Zz/2$ and $w_1(M) \circ \rho \colon \pi_1(M) \to \Zz/2$ are both homomorphisms we see that they agree. |
{{endproof}} | {{endproof}} | ||
</wikitex> | </wikitex> | ||
− | |||
== Examples == | == Examples == | ||
− | <wikitex | + | <wikitex>; |
− | # Since $\mathbb RP^n$ is orientable if and only if $n$ is odd, the orientation character $w\colon\pi_1\Rr P^n\to \Zz/2$ is trivial if and only if $n$ is odd. If $n$ is even $w\colon\pi_1\Rr P^n\cong\Zz/2\to \Zz/2$ is the identity. | + | # Since $\mathbb RP^n$ is orientable if and only if $n$ is odd, the orientation character $w\colon\pi_1(\Rr P^n)\to \Zz/2$ is trivial if and only if $n$ is odd. If $n$ is even $w\colon\pi_1(\Rr P^n) \cong \Zz/2\to \Zz/2$ is the identity. |
# The open [[Wikipedia:Mobius_strip|Möbius strip]] has fundamental group $\Zz$ and is non-orientable. Therefore, the orientation character is given by the projection $\Zz\twoheadrightarrow\Zz/2$. | # The open [[Wikipedia:Mobius_strip|Möbius strip]] has fundamental group $\Zz$ and is non-orientable. Therefore, the orientation character is given by the projection $\Zz\twoheadrightarrow\Zz/2$. | ||
# If $i\colon N\to M$ is an embedding of a manifold of the same dimension (possibly with boundary), then the orientation character of $N$ is given as the composition $\pi_1(N)\xrightarrow{i_*} \pi_1(M)\xrightarrow{w}\Zz/2$. This follows from the fact that the atlas of $M$ with local orientations induces by restriction an atlas of $N$ with local orientations. Therefore, the orientation covering of $N$ is the orientation covering of $M$ restricted to $N$. | # If $i\colon N\to M$ is an embedding of a manifold of the same dimension (possibly with boundary), then the orientation character of $N$ is given as the composition $\pi_1(N)\xrightarrow{i_*} \pi_1(M)\xrightarrow{w}\Zz/2$. This follows from the fact that the atlas of $M$ with local orientations induces by restriction an atlas of $N$ with local orientations. Therefore, the orientation covering of $N$ is the orientation covering of $M$ restricted to $N$. | ||
− | # The preimage of $D^n\subseteq M^n$ in the orientation covering $\hat M$ is a disjoint union of two disks. The orientation covering of a connected sum $M\#N$ along $D^n$ is the "double connected sum" of the orientation coverings along the preimages of $D^n$ in the orientation coverings. How to pair the disks is determined by the local orientations. From this it follows that the orientation character of $M\#N$ is given by the pushout | + | # The preimage of $D^n\subseteq M^n$ in the orientation covering $\hat M$ is a disjoint union of two disks. The orientation covering of a [[Connected sum|connected sum]] $M\#N$ along $D^n$ is the "double connected sum" of the orientation coverings along the preimages of $D^n$ in the orientation coverings. How to pair the disks is determined by the local orientations. From this it follows that the orientation character of $M\#N$ is given by the pushout |
$$\xymatrix{\pi_1(S^{n-1})\ar[r]\ar[d]&\pi_1(M\backslash D^n)\ar[d]\ar[ddr]&\\ | $$\xymatrix{\pi_1(S^{n-1})\ar[r]\ar[d]&\pi_1(M\backslash D^n)\ar[d]\ar[ddr]&\\ | ||
\pi_1(N\backslash D^n)\ar[r]\ar[drr]&\pi_1(M\#N)\ar@{-->}[dr]&\\&&\Zz/2}$$ | \pi_1(N\backslash D^n)\ar[r]\ar[drr]&\pi_1(M\#N)\ar@{-->}[dr]&\\&&\Zz/2}$$ | ||
</wikitex> | </wikitex> | ||
− | |||
== The orientation character via Poincaré duality== | == The orientation character via Poincaré duality== | ||
− | <wikitex | + | <wikitex>; |
Even though the original definition is very geometric, the orientation | Even though the original definition is very geometric, the orientation | ||
character is already completely determined by the homotopy type of a | character is already completely determined by the homotopy type of a | ||
− | given closed manifold. We already know this for differentiable manifolds by Proposition \ref{sw}. In general this is most easily seen in the more abstract setting of Poincaré complexes, see \cite{Lück2001|Section 3.1}. | + | given closed manifold. We already know this for differentiable manifolds by Proposition \ref{sw}. In general this is most easily seen in the more abstract setting of [[Poincaré complexes]], see \cite{Lück2001|Section 3.1}. |
{{beginthm|Lemma}} | {{beginthm|Lemma}} | ||
− | If $(X, [X], w)$ is a | + | If $(X, [X], w)$ is a Poincaré complex, then $w$ is the only homomorphism $y\colon\pi_1(X) \to \Zz/2$ such that $X$ is a Poincaré complex with orientation character $y$. |
{{endthm}} | {{endthm}} | ||
{{beginproof}} | {{beginproof}} | ||
Suppose $(X,[X],y)$ is a Poincaré complex. We first introduce some notation: | Suppose $(X,[X],y)$ is a Poincaré complex. We first introduce some notation: | ||
− | # $\mathbb{Z}^y$ denotes $\mathbb{Z}$ as a right $\mathbb{Z}\pi_1(X)$-module, where the $\pi_1(X)$-action is given by $z \cdot g = y(g) \cdot z$. | + | # $\mathbb{Z}^y$ denotes $\mathbb{Z}$ as a right $\mathbb{Z}[\pi_1](X)$-module, where the $\pi_1(X)$-action is given by $z \cdot g = y(g) \cdot z$. |
− | # $\mathbb{Z}\pi_1(X)^y := \mathbb{Z}^y \otimes_{\mathbb{Z}\pi_1(X)}\mathbb{Z}\pi_1(X)$, where the tensor product is equipped with the diagonal action. | + | # $\mathbb{Z}[\pi_1](X)^y := \mathbb{Z}^y \otimes_{\mathbb{Z}[\pi_1](X)}\mathbb{Z}[\pi_1](X)$, where the tensor product is equipped with the diagonal action. |
− | # For a left $\mathbb{Z}\pi_1(X)$-chain complex $C_*$, we define $C_*^y$ analogous to (2.). | + | # For a left $\mathbb{Z}[\pi_1(X)]$-chain complex $C_*$, we define $C_*^y$ analogous to (2.). |
− | Applying $H_*$ to the $\mathbb{Z}\pi_1(X)$-chain equivalence $? \cap [X] | + | Applying $H_*$ to the $\mathbb{Z}[\pi_1](X)$-chain equivalence $? \cap [X] |
\colon C^{n-*}(\tilde{X}) \to C_*(\tilde{X})^y$, we obtain an | \colon C^{n-*}(\tilde{X}) \to C_*(\tilde{X})^y$, we obtain an | ||
− | isomorphism of $\mathbb{Z}\pi_1(X)$-modules | + | isomorphism of $\mathbb{Z}[\pi_1](X)$-modules |
$$H_n(C^{n-*}(\tilde{X})) \xrightarrow{\cong} H_0(C_*(\tilde{X})^y) | $$H_n(C^{n-*}(\tilde{X})) \xrightarrow{\cong} H_0(C_*(\tilde{X})^y) | ||
\cong \mathbb{Z}^y.$$ | \cong \mathbb{Z}^y.$$ | ||
− | Note that the left hand side is independent of the orientation character $y$. Therefore, $\Zz^y$ is isomorphic to $\Zz^w$ as a $\Zz\pi_1(X)$-module and $y$ and $w$ have to agree. | + | Note that the left hand side is independent of the orientation character $y$. Therefore, $\Zz^y$ is isomorphic to $\Zz^w$ as a $\Zz[\pi_1](X)$-module and $y$ and $w$ have to agree. |
{{endproof}} | {{endproof}} | ||
{{beginthm|Corollary}} | {{beginthm|Corollary}} | ||
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{{endthm}} | {{endthm}} | ||
{{beginproof}} | {{beginproof}} | ||
− | Let $(X,[X],w)$ and $(X',[X'],w')$ be Poincaré complexes and $f:X\rightarrow X'$ be a homotopy equivalence. Then $(X,[X],w'\circ f_*\colon\ | + | Let $(X,[X],w)$ and $(X',[X'],w')$ be Poincaré complexes and $f:X\rightarrow X'$ be a homotopy equivalence. Then $(X,[X],w'\circ f_*\colon\pi_1(X)\to\Zz/2)$ is a Poincaré complex and $w$ and $w'\circ f_*$ have to agree. |
{{endproof}} | {{endproof}} | ||
{{beginthm|Remark}} | {{beginthm|Remark}} |
Revision as of 11:46, 3 April 2013
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Contents |
1 Definition
The following definition can be found in [Davis&Kirk2001, Section 5]. Let be a topological manifold. The orientation character is a homomorphism
which may be defined as follows. Take the orientation covering and let be the base point with lifts and in . For a loop based at , let be the lift of with and define on the homotopy class of by
The orientation character induces a map from to the classifying space of , which is unique up to homotopy. By the first definition of the orientation character this map classifies the orientation covering. So is trivial if and only if , and therefore is orientable.
2 Alternative descriptions
Let be the universal covering of . Then acts on by deck transformations.
Proposition 2.1. Let be the orientation character, then if and only if acts orientation preserving on .
Proof. If is orientable, then an orientation on induces an orientation on and every deck transformation is orientation preserving. If is non-orientable, then the standard orientation of the orientation covering of induces an orientation on . By construction the non-trivial deck transformation of is orientation-reversing. Therefore, a deck transformation of is orientation preserving if and only if it acts trivial on .
2.1 The orientation character for smooth manifolds
If the manifold carries a smooth structure, the orientation character can be additionally characterised in terms of tangential data. Define a map in the following way: Let be the base point. Pick a chart around . Let be a based loop in . The standard orientation of corresponds to an orientation of under the differential of (cf. Orientation of manifolds). Fibre transport along yields another orientation of . Now set
Proof. This follows from the comparison between homological and tangential orientation. See Orientation of manifolds.
In addition to this geometric characterisation, the orientation character also admits a description in terms of characteristic classes: Under the "evaluation" homomorphism of the Universal Coefficient Theorem , the first Stiefel-Whitney class corresponds to a homomorphism .
Proof. All 1-manifolds are orientable, so suppose that . Let classify the tangent bundle of and consider the induced map . Now is generated by the classifying map of the Mobius bundle, denote this class by . Hence for a based loop with homotopy class , we see that the tangent bundle of restricted to is non-trivial if and only if . By Proposition 2.2 we see that if and only if . Applying the universal coefficient theorem in , we see that universal first Stiefel-Whitney class , which generates , has the property that . Hence we see that if and only if if and only if if and only if . Since and are both homomorphisms we see that they agree.
3 Examples
- Since is orientable if and only if is odd, the orientation character is trivial if and only if is odd. If is even is the identity.
- The open Möbius strip has fundamental group and is non-orientable. Therefore, the orientation character is given by the projection .
- If is an embedding of a manifold of the same dimension (possibly with boundary), then the orientation character of is given as the composition . This follows from the fact that the atlas of with local orientations induces by restriction an atlas of with local orientations. Therefore, the orientation covering of is the orientation covering of restricted to .
- The preimage of in the orientation covering is a disjoint union of two disks. The orientation covering of a connected sum along is the "double connected sum" of the orientation coverings along the preimages of in the orientation coverings. How to pair the disks is determined by the local orientations. From this it follows that the orientation character of is given by the pushout
4 The orientation character via Poincaré duality
Even though the original definition is very geometric, the orientation character is already completely determined by the homotopy type of a given closed manifold. We already know this for differentiable manifolds by Proposition 2.3. In general this is most easily seen in the more abstract setting of Poincaré complexes, see [Lück2001, Section 3.1].
Lemma 4.1. If is a Poincaré complex, then is the only homomorphism such that is a Poincaré complex with orientation character .
Proof. Suppose is a Poincaré complex. We first introduce some notation:
- denotes as a right -module, where the -action is given by .
- , where the tensor product is equipped with the diagonal action.
- For a left -chain complex , we define analogous to (2.).
Applying to the -chain equivalence , we obtain an isomorphism of -modules
Note that the left hand side is independent of the orientation character . Therefore, is isomorphic to as a -module and and have to agree.
Corollary 4.2. If is a Poincaré complex, then depends only on the homotopy type of .
Proof. Let and be Poincaré complexes and be a homotopy equivalence. Then is a Poincaré complex and and have to agree.
Remark 4.3. The above statements are also true for Poincaré pairs in general.
Proposition 2.3 generalises to the following:
Proposition 4.4 [Byun1999, Lemma 4.3]. Using the Universal Coefficients Theorem and the Hurewicz map, the orientation character of a Poincaré complex coincides with the first Stiefel-Whitney class of its Spivak normal fibration.
5 References
- [Byun1999] Y. Byun, Tangent fibration of a Poincaré complex, J. London Math. Soc. (2) 59 (1999), no.3, 1101–1116. MR1709099 (2000f:57023) Zbl 0935.57032
- [Davis&Kirk2001] J. F. Davis and P. Kirk, Lecture notes in algebraic topology, American Mathematical Society, 2001. MR1841974 (2002f:55001) Zbl 1018.55001
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020