Wu class

(Difference between revisions)
Jump to: navigation, search
Line 1: Line 1:
{{Authors|Knapp}}{{MediaWiki:Being reviewed}}
+
{{Authors|Karlheinz Knapp}}{{MediaWiki:Being reviewed}}
==Introduction==
==Introduction==
<wikitex>;
<wikitex>;

Revision as of 14:33, 31 January 2014

An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 15:43, 18 February 2014 and the changes since publication.

The user responsible for this page is Karlheinz Knapp. No other user may edit this page at present.

This Definitions page is being reviewed under the supervision of the editorial board. Hence the page may not be edited at present. As always, the discussion page remains open for observations and comments.

Contents

1 Introduction

The Wu class of a manifold M 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}

with

\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 unoriented 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].<\ol>

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

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox