# 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

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.**

or equivalently

## 4 An example

The following example is taken from [Milnor&Stasheff1974, §11]. If is of the form where , for example if , then

with

## 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).

## 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 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].

### 6.1 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]):

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

## 8 External links

- Wu class in nLab
- Wu class in the Wikipedia page on Stiefel-Whitney classes