Orientation character

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1 Definition

Let M be a connected topological manifold throughout. The following definition can be found in [Davis&Kirk2001, Section 5]. The orientation character is a homomorphism

\displaystyle  w \colon \pi_1(M) \to \{ \pm 1 \} =: C_2

which may be defined as follows. Take the orientation covering p:\widehat M\to M and let x\in M be the base point with lifts \tilde x_1 and \tilde x_{-1} in \widehat 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

\displaystyle 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 C_2 induces a map M \to BC_2 from M to the classifying space of C_2, which is unique up to homotopy. By the definition of the orientation character, this map classifies the orientation covering. So w is trivial if and only if \widehat M \cong M\times C_2, and therefore M is orientable.

2 Alternative descriptions

Let \widetilde M be the universal covering of M. Then \pi_1(M) acts on \widetilde M by deck transformations.

Proposition 2.1. Let w\colon \pi_1(M)\to C_2 be the orientation character. Then w([\gamma])=1 if and only if the action of [\gamma] on \widetilde M is orientation preserving.

Proof. If M is orientable, then an orientation on M induces an orientation on \widetilde M and every deck transformation is orientation preserving. If M is non-orientable, then the standard orientation of the orientation covering \widehat M of M induces an orientation on \widetilde M. By construction the non-trivial deck transformation of \widehat M is orientation-reversing. Therefore, a deck transformation of \widetilde M\to M is orientation preserving if and only if it acts trivially on \widehat M.


If the manifold M^n carries a smooth structure, the orientation character can be additionally characterised in terms of tangential data. Define a map w^\prime \colon \pi_1(M) \to C_2 in the following way: Let x \in M be the base point. Pick a chart \varphi \colon U \to V \subset \mathbb{R}^n around x. Let \gamma be a based loop in M. The standard orientation [e_1, \dots, e_n] of \mathbb{R}^n corresponds to an orientation o of T_xM under the differential of \varphi^{-1} (cf. Orientation of manifolds). Fibre transport along \gamma yields another orientation o^\prime of T_xM. Now set

\displaystyle w^\prime([\gamma]) := \left\{ \begin{array}{rl} 1~~ & \text{if $o^\prime=o$,} \\ -1~~ & \text{if $o^\prime=-o$.} \end{array} \right.

Using the comparison between homological and tangential orientation (see Orientation of manifolds), one can show the following:

Proposition 2.2. The map w^\prime is a well-defined group homomorphism and coincides with the orientation character w.

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 H^1(M;\Zz/2) \to \hom(H_1(M;\Zz), \Zz/2), the first Stiefel-Whitney class w_1(M) corresponds to a homomorphism H_1(M;\Zz/2) \to \Zz/2 which we also call w_1(M).

Proposition 2.3. Let \rho \colon \pi_1(M) \to H_1(M;\mathbb{Z}) be the Hurewicz homomorphism. Then the orientation character of M coincides with the composition w_1(M) \circ \rho via the canonical isomorphism C_2 \cong \Zz/2.

Proof. 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)) \cong C_2 is generated by the classifying map of the Möbius bundle; denote this class by [\eta]. Hence, for a based loop \gamma in M with homotopy class [\gamma] \in \pi_1(M), we see the pullback \gamma^*TM is non-trivial if and only if \tau_*([\gamma]) = [\eta]. By Proposition 2.1, we see that w([\gamma]) = 1 if and only if \tau_*([\gamma]) = [\eta]. Applying the Universal Coefficient Theorem for BO(n), we see that the universal first Stiefel-Whitney class w_1, which generates H^1(BO(n); \Zz/2), has the property that w_1(\rho([\eta])) = 1. Hence, we see that w([\gamma]) = 1 if and only if \tau_*([\gamma]) = [\eta] if and only if w_1(\rho(\tau_*[\gamma])) = 1 if and only if w_1(M)(\rho([\gamma])) = 1.


3 Examples

  1. Since \mathbb RP^n is orientable if and only if n is odd, the orientation character w\colon\pi_1(\Rr P^n)\to C_2 is trivial if and only if n is odd. If n is even, w\colon\pi_1(\Rr P^n) \to C_2 is an isomorphism.
  2. The open Möbius strip has fundamental group \Zz and is non-orientable. Therefore, the orientation character is given by the surjection \Zz\twoheadrightarrow\Zz/2 \cong C_2.
  3. 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}C_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.
  4. Let n \geq 2. The preimage of an embedded disk D^n\subseteq M^n in the orientation covering \widehat 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
    \displaystyle \xymatrix{\pi_1(S^{n-1})\ar[r]\ar[d]&\pi_1(M\backslash \text{int}(D^n))\ar[d]\ar[ddr]&\\ \pi_1(N\backslash \text{int}(D^n))\ar[r]\ar[drr]&\pi_1(M\#N)\ar@{-->}[dr]&\\&&C_2}

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. Both the Hurewicz homomorphism and the first Stiefel-Whitney class are homotopy invariants (for the second point, see the Manifold Atlas page Wu class), so 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 (X, [X], w) is a connected Poincaré complex, then w is the only homomorphism y\colon\pi_1(X) \to C_2 such that X is a Poincaré complex with orientation character y.

Proof. Suppose (X,[X],y) is a connected Poincaré complex. We first introduce some notation:

  1. \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.
  2. \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.
  3. 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] \colon C^{n-*}(\widetilde{X}) \to C_*(\widetilde{X})^y, we obtain an isomorphism of \mathbb{Z}[\pi_1(X)]-modules

\displaystyle H_n(C^{n-*}(\widetilde{X})) \xrightarrow{\cong} H_0(C_*(\widetilde{X})^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.


Corollary 4.2. If (X, [X], w) is a connected Poincaré complex, then w depends only on the homotopy type of X.

Proof. Let (X,[X],w) and (X',[X'],w') be connected Poincaré complexes and f:X\rightarrow X' be a homotopy equivalence. Then (X,[X],w'\circ f_*\colon\pi_1(X)\to C_2) is a Poincaré complex and w and w'\circ f_* have to agree.


Remark 4.3. The above statements are also true for connected Poincaré pairs.

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 connected Poincaré complex coincides with the first Stiefel-Whitney class of its Spivak normal fibration.

5 References

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