Wu class

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1 Introduction

The Wu class of a manifold M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5b5d05 is a characteristic class allowing a computation of the Stiefel-Whitney classes of M by knowing only H^{\ast}(M;\Zz/2) and the action of the Steenrod squares.

2 Definition

Let M be a closed topological n-manifold, [M]\in H_{n}(M;\Zz/2) its fundamental class, Sq^{k} the k-th Steenrod square and
\displaystyle \left\langle \cdot~,~\cdot \right\rangle:H^{i}(M;\Zz/2)\times H_{i}(M;\Zz /2)\longrightarrow \Zz/2
the usual Kronecker pairing. This pairing, together with the Poincaré duality isomorphism a\mapsto a\cap \lbrack M], induces isomorphisms
\displaystyle  \textup{Hom}(H^{n-k}(M;\Zz/2), \Zz/2) \cong H_{n-k}(M;\Zz /2)\cong H^{k}(M;\Zz/2),

under which the homomorphism x\mapsto \left\langle Sq^{k}(x),[M]\right\rangle from H^{n-k}(M;\Zz/2) to \Zz/2 corresponds to a well defined cohomology class v_{k}\in H^{k}(M;\Zz/2). This cohomology class is called the k-th Wu class of M ([Milnor&Stasheff1974, §11]). We may rewrite its definition equivalently as an identity

(1)\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).
Define the total Wu class
\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),

as the formal sum

\displaystyle  v:=1+v_{1}+v_{2}+...+v_{n}.
Using the total Steenrod square,
\displaystyle Sq:=Sq^{0}+Sq^{1}+Sq^{2}+ \dots \colon H^{\ast}(M;\Zz /2)\longrightarrow H^{\ast }(M;\Zz/2),
equation (1) translates into the following formula
(2)\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),

which may also be used as a definition of the total Wu class of M. From the definition it is clear that the Wu class is defined even for a Poincaré complex M.

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 w_{i}(M) of M using only H^{\ast }(M;\Zz/2) and the action of the Steenrod squares:

Theorem 3.1.

The total Stiefel-Whitney class of M,
\displaystyle w=w(M):=1+w_{1}(M)+w_{2}(M)+...+w_{n}(M),
is given by
\displaystyle  w(M)=Sq(v),

or equivalently

\displaystyle  w_{k}(M)=\sum_{i=0}Sq^{i}(v_{k-i}).
For a proof see [Milnor&Stasheff1974, §11]. Since Sq is a ring automorphism of
\displaystyle H^{\ast \ast }(X;\Zz/2):=\prod_{i\geq 0}H^{i}(X;\Zz/2),
Sq^{-1} is defined on H^{\ast \ast }(M;\Zz/2)=H^{\ast }(M;\Zz/2) and we may write
\displaystyle  v=Sq^{-1}(w(M)).
The formula w(M)=Sq(v) may be used to extend the definition of the Stiefel-Whitney classes to the class of topological manifolds.

4 An example

The following example is taken from [Milnor&Stasheff1974, §11]. If H^{\ast }(M;\Zz/2) is of the form \Zz/2[x]/(x^{dm+1}), where x\in H^{d}(X;\Zz/2), d\geq 1, n=d\cdot m, for example if M = \mathbb{C}P^m, then

\displaystyle  v=(1+x+x^{2}+x^{4}+\dots)^{m+1}~~\text{and}~~w(M)=(1+x)^{m+1}


\displaystyle Sq(x)=x+x^{2} \quad \text{and} \quad Sq^{-1}(x)=x+x^{2}+x^{4}+ \dots~.

5 A generalization

The following example is taken from [Atiyah&Hirzebruch1961]. Let \lambda be a natural ring automorphism of H^{\ast \ast }(X;\Zz/2) and \Phi _{\xi } the Thom isomorphism of a real vector bundle \xi on X. Define

\displaystyle  \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).

If \lambda =Sq, then \underline{\lambda }=w is the total Stiefel-Whitney classes w(\xi ) of \xi ([Milnor&Stasheff1974, §8]) and with \xi =\tau M, the tangent bundle of X=M, we have \textup{Wu}(Sq,\tau M)=v, the total Wu class of M. In general \xi \mapsto \underline{\lambda }(\xi ) and \xi \mapsto \textup{Wu}(\lambda ,\xi ) define multiplicative characteristic classes, translating Whitney sum into cup product, i.e. they satisfy a Whitney product type formula

\displaystyle  \underline{\lambda }(\xi \oplus \eta )=\underline{\lambda }(\xi )\cup \underline{\lambda }(\eta )\quad \text{ and } \quad \textup{Wu}(\lambda ,\xi \oplus \eta )=\textup{Wu}(\lambda ,\xi )\cup \textup{Wu}(\lambda ,\eta ).

Such a characteristic class is determined by a power series f(x) \in \Zz/2[[x]], which is given by its value on the universal line bundle. The generalized Wu class \textup{Wu}(\lambda ,\xi ) is defined as a commutator class, thus measuring how \lambda and \Phi _{\xi } 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 K -Theory and rational cohomology is formulated. This relation is more than only formal: Let T_{i} be the i-th Todd polynomial, then 2^{i}\cdot T_{i} is a rational polynomial with denominators prime to 2, hence its reduction to mod 2 cohomology is well defined. Then Atiyah and Hirzebruch proved:

Theorem 5.1 [Atiyah&Hirzebruch1961].

\displaystyle \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).

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 x/Sq^{-1}(x)=1+\sum_{j\geq 0}x^{2^{j}}. For a continuous map f:M\rightarrow N between closed differentiable manifolds the analogue of the Riemann-Roch formula is

\displaystyle  f_{!}(\lambda (x)\cup \textup{Wu}(\lambda ^{-1},\tau M))=\lambda (f_{!}(x))\cup \textup{Wu}(\lambda ^{-1},\tau N).

Here f_{!} is the Umkehr map of f defined by f_{\ast } via Poincaré duality. In the case f:M\rightarrow \ast, this reduces to \left\langle \textup{Wu}(\lambda ,\tau M)\cup x,[M]\right\rangle =\left\langle \lambda (x),[M]\right\rangle , generalizing (2).

6 Applications

  1. The definition of the total Wu class v and w=Sq(v) show, that the Stiefel-Whitney classes of a smooth manifold are invariants of its homotopy type.
  2. Since the Stiefel-Whitney classes of a closed n-manifold determine its un-oriented bordism class [Thom1954, Théorém IV.10], a corollary of (1) is: Homotopy equivalent manifolds are un-oriented bordant.
  3. Inserting the Stiefel-Whitney classes of M for x in
    \displaystyle  \left\langle v\cup x,[M]\right\rangle =\left\langle Sq(x),[M]\right\rangle,

    and using v=Sq^{-1}(w) one gets relations between Stiefel-Whitney numbers of n-manifolds. It is a result of Dold ([Dold1956]) that all relations between Stiefel-Whitney numbers of n-manifolds are obtained in this way.

  4. Conditions on the Wu classes v_{s} for nonbounding manifolds are given in [Stong&Yoshida1987].
  5. For an appearance of the Wu class in surgery theory see [Madsen&Milgram1979, Ch. 4].

6.1 Remarks

  1. Most of the above has analogues for odd primes, e.g. see [Atiyah&Hirzebruch1961].
  2. 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 \xi (see [Milnor&Stasheff1974, §8]):
\displaystyle  Sq^{k}(w_{m}(\xi ))=w_{k}\cup w_{m}+\binom{k-m}{1}w_{k-1}\cup w_{m+1}+ \dots + \binom{k-m}{k}w_{0}\cup w_{m+k}

where \binom{x}{i}=x(x-1)\dots(x-i+1)/i!.

7 References

8 External links

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