Orientation character
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Contents |
1 Definition
Let 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](/images/math/d/1/3/d131cd758a2178a0e507b97bde1c6591.png)
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
![\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.](/images/math/f/8/7/f876d7ea45e9d6ecec88e6185fb5d558.png)
The orientation character induces a map
from
to the classifying space of
,
which is unique up to homotopy. By the 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 the action of
on
is orientation preserving.
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 trivially on
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
![\displaystyle w^\prime([\gamma]) := \left\{ \begin{array}{rl} 1~~ & \text{if $o^\prime=o$,} \\ -1~~ & \text{if $o^\prime=-o$.} \end{array} \right.](/images/math/2/9/a/29a864a178dc830d62741382219a360c.png)
Using the comparison between homological and tangential orientation (see Orientation of manifolds), one can show the following:
Proposition 2.2.
The map is a well-defined group homomorphism and coincides with the orientation character
.
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
which we also call
.
Proposition 2.3.
Let be the Hurewicz homomorphism.
Then the orientation character of
coincides with the composition
via the canonical isomorphism
.
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 Möbius bundle;
denote this class by
. Hence, for a based loop
in
with homotopy class
,
we see the pullback
is non-trivial if and only if
.
By Proposition 2.1, we see that
if and only if
.
Applying the Universal Coefficient Theorem for
, we see that the 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
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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 an isomorphism.
- The open Möbius strip has fundamental group
and is non-orientable. Therefore, the orientation character is given by the surjection
.
- 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
.
- Let
. The preimage of an embedded disk
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. Both the Hurewicz homomorphism and the first Stiefel-Whitney class are homotopy invariants (see 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 is a connected Poincaré complex, then
is the only homomorphism
such that
is a Poincaré complex with orientation character
.
Proof.
Suppose is a connected 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
![\displaystyle H_n(C^{n-*}(\tilde{X})) \xrightarrow{\cong} H_0(C_*(\tilde{X})^y) \cong \mathbb{Z}^y.](/images/math/1/9/1/191890f17210d15c3269300f8e3764d3.png)
Note that the left hand side is independent of the orientation character . Therefore,
is isomorphic to
as a
-module and
and
have to agree.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
Corollary 4.2.
If is a connected Poincaré complex, then
depends only on the homotopy type of
.
Proof.
Let and
be connected Poincaré complexes and
be a homotopy equivalence. Then
is a Poincaré complex and
and
have to agree.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
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
- [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
6 External links
- The Wikipedia page about the orientation character