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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.
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
as the formal sum
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 ,
4 An example
The following example is taken from [Milnor&Stasheff1974, §11]. If is of the form where , for example if , then
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
If , then is the total Stiefel-Whitney classes of ([Milnor&Stasheff1974, §8]) and with the tangent bundle of , we have , the total Wu class of . In general and define multiplicative characteristic classes, translating Whitney sum into cup product, i.e. they satisfy a Whitney product type formula
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].
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
Here is the Umkehr map of defined by via Poincaré duality. In the case , this reduces to generalizing (2).
- 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 un-oriented 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].
- For an appearance of the Wu class in surgery theory see [Madsen&Milgram1979, Ch. 4].
- 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]):
- [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
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- [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