Wu class
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Contents |
1 Introduction
The Wu class of a manifold is a characteristic class allowing a computation of the Stiefel-Whitney classes of
by knowing only
and the action of the Steenrod squares.
2 Definition
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![[M]\in H_{n}(M;\Zz/2)](/images/math/9/3/4/9345d0a191a038e84bcec9b1764bfe6e.png)
![Sq^{k}](/images/math/f/8/8/f883cf734cdc739c7ef394a5c5c336e2.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![\displaystyle \left\langle \cdot~,~\cdot \right\rangle:H^{i}(M;\Zz/2)\times H_{i}(M;\Zz /2)\longrightarrow \Zz/2](/images/math/a/a/5/aa5fb2097da9123155af5dddf44becef.png)
![a\mapsto a\cap \lbrack M]](/images/math/7/c/d/7cdea1ce2260dd4ec586438c1f8e5cd4.png)
![\textup{Hom}(H^{n-k}(M;\Zz/2), \Zz/2) \cong H_{n-k}(M;\Zz /2)\cong H^{k}(M;\Zz/2),](/images/math/5/c/d/5cd3dda6cc91e307b2a76108cd79e54c.png)
under which the homomorphism from
to
corresponds to a well defined cohomology class
. This cohomology class is called the
-th Wu class of
([Milnor&Stasheff1974, §11]). We may rewrite its definition equivalently as an identity
![\left\langle v_{k}\cup x,[M]\right\rangle =\left\langle Sq^{k}(x),[M]\right\rangle \quad \quad \text{ for all }x\in H^{n-k}(M;\Zz/2).](/images/math/8/6/3/863cd20a89c4ac2cd739a96dac9edac2.png)
![\displaystyle v\in H^{\ast }(M;\Zz/2)=H^{0}(M;\Zz/2)\oplus H^{1}(M;\Zz /2)\oplus ....\oplus H^{n}(M;\Zz/2),](/images/math/4/a/9/4a915b0ee05d2380a21be3480d1725d8.png)
as the formal sum
![v:=1+v_{1}+v_{2}+...+v_{n}.](/images/math/c/b/0/cb0dc3e85f9888ab68c2f45f2aadc0a2.png)
![\displaystyle Sq:=Sq^{0}+Sq^{1}+Sq^{2}+ \dots \colon H^{\ast}(M;\Zz /2)\longrightarrow H^{\ast }(M;\Zz/2),](/images/math/3/0/3/303f97d2162f46cdc6249c38f113484f.png)
![\left\langle v\cup x,[M]\right\rangle =\left\langle Sq(x),[M]\right\rangle \quad \quad \text{for all} x\in H^{\ast }(M;\Zz/2),](/images/math/c/0/e/c0e8ed4066d130b358a5483c4a7e8a98.png)
which may also be used as a definition of the total Wu class of . From the definition it is clear that the Wu class is defined even for a Poincaré complex
3 Relation to Stiefel-Whitney classes
From now on all manifolds are supposed to be smooth. The following theorem of Wu Wen-Tsun ([Wu1950]) allows a computation of the Stiefel-Whitney classes of
using only
and the action of the Steenrod squares:
Theorem 3.1.
The total Stiefel-Whitney class of![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle w=w(M):=1+w_{1}(M)+w_{2}(M)+...+w_{n}(M),](/images/math/e/4/4/e44be5229d794ff6a9be8f5448c8d83a.png)
![w(M)=Sq(v),](/images/math/f/7/c/f7c9646ea7b4329f4c93efd9672b3f86.png)
![w_{k}(M)=\sum_{i=0}Sq^{i}(v_{k-i}).](/images/math/a/e/7/ae795f958be1100062db87f4dd3c4fb2.png)
![Sq](/images/math/f/e/d/fed8fadd31fda7012c7602767d70d887.png)
![\displaystyle H^{\ast \ast }(X;\Zz/2):=\prod_{i\geq 0}H^{i}(X;\Zz/2),](/images/math/4/8/a/48a2f8a6ba7601bde5fe9b694e25d07e.png)
![Sq^{-1}](/images/math/e/a/8/ea8e228a5e8b945c809fd282a9f45d9e.png)
![H^{\ast \ast }(M;\Zz/2)=H^{\ast }(M;\Zz/2)](/images/math/9/b/9/9b95901c7b758b385ec024b693c803c0.png)
![\displaystyle v=Sq^{-1}(w(M)).](/images/math/7/8/b/78bd87f88eb9846364e7ebc6af852976.png)
![w(M)=Sq(v)](/images/math/8/d/1/8d188f68b7f048dc6607792f4ad44185.png)
4 An example
The following example is taken from [Milnor&Stasheff1974, §11]. If is of the form
where
, for example if
, then
![v=(1+x+x^{2}+x^{4}+\dots)^{m+1}~~\text{and}~~w(M)=(1+x)^{m+1}](/images/math/3/8/5/3851094e30986e8d3583f5822065f4b1.png)
![Sq(x)=x+x^{2} \quad \text{and} \quad Sq^{-1}(x)=x+x^{2}+x^{4}+ \dots .](/images/math/0/5/0/0505d8a9c3ed29623dffa0e588ddc7eb.png)
5 A generalization
The following example is taken from [Atiyah&Hirzebruch1961]. Let be a natural ring automorphism of
and
the Thom isomorphism of a real vector bundle
on
. Define
![\underline{\lambda }=\underline{\lambda }(\xi ):=\Phi _{\xi }^{-1}\circ \lambda \circ \Phi _{\xi }(1)~~\text{and}~~\textup{Wu}(\lambda ,\xi ):=\lambda ^{-1}\circ \underline{\lambda }=\lambda ^{-1}\circ \Phi _{\xi }^{-1}\circ \lambda \circ \Phi _{\xi }(1). <label>W5</label>](/images/math/3/8/c/38cff604c4790ebbb3d9a29d34e0ef7a.png)
![\lambda =Sq](/images/math/9/4/b/94b6135c3439d375689a7e825e4bb62c.png)
![\underline{\lambda }=w](/images/math/0/4/3/043589189128144af65dcf0ce2d2321d.png)
![w(\xi )](/images/math/6/0/5/605c5f5c24732562504574c70401c769.png)
![\xi](/images/math/f/d/e/fde23047ffd6f8f6f7cd3c61a4cc8878.png)
![\xi =\tau M,](/images/math/1/6/2/162fa6e72656cc8042e208ca1e07558f.png)
![X=M](/images/math/a/e/b/aeb70e9c4e2f9f7a5f083012c40d4e3a.png)
![\textup{Wu}(Sq,\tau M)=v](/images/math/2/1/2/2125efc45692a4e83f5df9072862ea81.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\xi \mapsto \underline{\lambda }(\xi )](/images/math/0/8/e/08eca29381b0f79f4b19e5ef8aa637be.png)
![\xi \mapsto \textup{Wu}(\lambda ,\xi )](/images/math/a/1/b/a1b67735a09fb53fd64368ae1900c8df.png)
![\underline{\lambda }(\xi \oplus \eta )=\underline{\lambda }(\xi )\cup \underline{\lambda }(\eta )\text{ \ and \ }\textup{Wu}(\lambda ,\xi \oplus \eta )=\textup{Wu}(\lambda ,\xi )\cup \textup{Wu}(\lambda ,\eta ).](/images/math/3/9/8/398dc9dd6bc02879b07adf7acdb20fb8.png)
Such a characteristic class is determined by a power series , which is given by its value on the universal line bundle. The generalized Wu class
is defined as a commutator class, thus measuring how
and
commute. This is similar to the situation considered in the (differential) Riemann-Roch formulas, in which the interaction between the Chern character and the Thom isomorphism in
-Theory and rational cohomology is formulated. This relation is more than only formal: Let
be the
-th Todd polynomial, then
is a rational polynomial with denominators prime to
hence its reduction to mod
cohomology is well defined. Then Atiyah and Hirzebruch proved:
Theorem 5.1 [[Atiyah&Hirzebruch1961]].
![\textup{Wu}(Sq,\xi )=\sum_{i\geq 0}2^{i}\cdot T_{i}(w_{1}(\xi ),w_{2}(\xi ), \dots ,w_{i}(\xi )) \quad \text{in}~~H^{\ast \ast }(X;\Zz/2).](/images/math/3/4/8/3487d25dfdcf28f103279b056b1b1910.png)
The proof is by comparing the power series belonging to the multiplicative characteristic classes on both sides of the equation, which turn out to be For a continuous map
between closed differentiable manifolds the analogue of the Riemann-Roch formula is
![f_{!}(\lambda (x)\cup \textup{Wu}(\lambda ^{-1},\tau M))=\lambda (f_{!}(x))\cup \textup{Wu}(\lambda ^{-1},\tau N).](/images/math/d/c/9/dc90dedc2722cc9047320cc8e964ae72.png)
Here is the Umkehr map of
defined by
via Poincaré duality. In the case
, this reduces to
generalizing (4).
6 Applications
- The definition of the total Wu class
and
show, that the Stiefel-Whitney classes of a smooth manifold are invariants of its homotopy type.
- Since the Stiefel-Whitney classes of a closed
-manifold determine its un-oriented bordism class [Thom1954, Théorém IV.10], a corollary of (1) is: Homotopy equivalent manifolds are unoriented bordant.
- Inserting the Stiefel-Whitney classes of
for
in
and using
one gets relations between Stiefel-Whitney numbers of
-manifolds. It is a result of Dold ([Dold1956]) that all relations between Stiefel-Whitney numbers of
-manifolds are obtained in this way.
- Conditions on the Wu classes
for nonbounding manifolds are given in [Stong&Yoshida1987]. 5. For an appearance of the Wu class in surgery theory see [Madsen&Milgram1979, Ch. 4].<\ol>
Remarks
- Most of the above has analogues for odd primes, e.g. see [Atiyah&Hirzebruch1961].
- Not directly related to the Wu class is Wu's explicit formula for the action of Steenrod squares on the Stiefel-Whitney classes of a vector bundle
(see [Milnor&Stasheff1974, §8]):
(13)where
7 References
- [Atiyah&Hirzebruch1961] M. F. Atiyah and F. Hirzebruch, Cohomologie-Operationen und charakteristische Klassen, Math. Z. 77 (1961), 149–187. MR0156361 (27 #6285) Zbl 0109.16002
- [Dold1956] A. Dold, Erzeugende der Thomschen Algebra
, Math. Z. 65 (1956), 25–35. MR0079269 (18,60c) Zbl 0071.17601
- [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Stong&Yoshida1987] R. Stong and T. Yoshida, Wu classes, Proc. Amer. Math. Soc. 100 (1987), no.2, 352–354. MR884478 (88e:57025) Zbl 0644.57011
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
- [Wu1950] W. Wu, Classes caractéristiques et
-carrés d'une variété, C. R. Acad. Sci. Paris, 230 (1950), 508–511. MR0035992 (12,42f) Zbl 0035.11002