# Wu class

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 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.
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## 1 Introduction

The Wu class of a manifold $M$${{Authors|Karlheinz Knapp}} ==Introduction== ; The Wu class of a manifold M is a characteristic class allowing a computation of the Stiefel-Whitney classes of M by knowing only H^{\ast}(M;\Zz/2) and the action of the Steenrod squares. ==Definition== ; Let M be a closed topological n-manifold, [M]\in H_{n}(M;\Zz/2) its fundamental class, Sq^{k} the k-th Steenrod square and \left\langle \cdot~,~\cdot \right\rangle:H^{i}(M;\Zz/2)\times H_{i}(M;\Zz /2)\longrightarrow \Zz/2 the usual Kronecker pairing. This pairing, together with the Poincaré duality isomorphism a\mapsto a\cap \lbrack M], induces isomorphisms \textup{Hom}(H^{n-k}(M;\Zz/2), \Zz/2) \cong H_{n-k}(M;\Zz /2)\cong H^{k}(M;\Zz/2), under which the homomorphism x\mapsto \left\langle Sq^{k}(x),[M]\right\rangle from H^{n-k}(M;\Zz/2) to \Zz/2 corresponds to a well defined cohomology class v_{k}\in H^{k}(M;\Zz/2). This cohomology class is called the k-th Wu class of M (\cite[§11]{Milnor&Stasheff1974}). We may rewrite its definition equivalently as an identity $$\label{W1} \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).$$ Define the total Wu class v\in H^{\ast }(M;\Zz/2)=H^{0}(M;\Zz/2)\oplus H^{1}(M;\Zz /2)\oplus ... \oplus H^{n}(M;\Zz/2), as the formal sum v:=1+v_{1}+v_{2}+...+v_{n}. Using the total Steenrod square, Sq:=Sq^{0}+Sq^{1}+Sq^{2}+ \dots \colon H^{\ast}(M;\Zz /2)\longrightarrow H^{\ast }(M;\Zz/2), equation (\ref{W1}) translates into the following formula $$\label{W2} \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. ==Relation to Stiefel-Whitney classes== ; From now on all manifolds are supposed to be smooth. The following theorem of Wu Wen-Tsun ({{cite|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: {{beginthm|Theorem|}} The total Stiefel-Whitney class of M, w=w(M):=1+w_{1}(M)+w_{2}(M)+...+w_{n}(M), is given by w(M)=Sq(v), or equivalently w_{k}(M)=\sum_{i=0}Sq^{i}(v_{k-i}). {{endthm}} For a proof see \cite[§11]{Milnor&Stasheff1974}. Since Sq is a ring automorphism of 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 v=Sq^{-1}(w(M)). The formula w(M)=Sq(v) may be used to extend the definition of the Stiefel-Whitney classes to Poincaré complexes. ==An example== ; The following example is taken from \cite[§11]{Milnor&Stasheff1974}. 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 v=(1+x+x^{2}+x^{4}+\dots)^{m+1}~~\text{and}~~w(M)=(1+x)^{m+1} with Sq(x)=x+x^{2} \quad \text{and} \quad Sq^{-1}(x)=x+x^{2}+x^{4}+ \dots~. ==A generalization== ; The following example is taken from {{cite|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 \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 (\cite[§8]{Milnor&Stasheff1974}) 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 \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 ^{i}\cdot T_{i} is a rational polynomial with denominators prime to , hence its reduction to mod cohomology is well defined. Then Atiyah and Hirzebruch proved: {{beginthm|Theorem|{{cite|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). {{endthm}} 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 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 (\ref{W2}). ==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 \cite[Théorém IV.10]{Thom1954}, a corollary of (1) is: Homotopy equivalent manifolds are un-oriented bordant.
3. Inserting the Stiefel-Whitney classes of M for x in \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 ({{cite|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 {{cite|Stong&Yoshida1987}}.
5. For an appearance of the Wu class in surgery theory see \cite{Madsen&Milgram1979|Ch. 4}.
=== Remarks === ; # Most of the above has analogues for odd primes, e.g. see {{cite|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 \xi (see \cite[§8]{Milnor&Stasheff1974}): 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!. == References == {{#RefList:}} == External links == * [http://ncatlab.org/nlab/show/Wu+class Wu class in nLab] * [[Wikipedia:Stiefel-Whitney_class#Wu_classes|Wu class]] in the Wikipedia page on Stiefel-Whitney classes [[Category:Definitions]]M$
is a characteristic class allowing a computation of the Stiefel-Whitney classes of $M$$M$ by knowing only $H^{\ast}(M;\Zz/2)$$H^{\ast}(M;\Zz/2)$ and the action of the Steenrod squares.

## 2 Definition

Let $M$$M$ be a closed topological $n$$n$-manifold, $[M]\in H_{n}(M;\Zz/2)$$[M]\in H_{n}(M;\Zz/2)$ its fundamental class, $Sq^{k}$$Sq^{k}$ the $k$$k$-th Steenrod square and
$\displaystyle \left\langle \cdot~,~\cdot \right\rangle:H^{i}(M;\Zz/2)\times H_{i}(M;\Zz /2)\longrightarrow \Zz/2$
the usual Kronecker pairing. This pairing, together with the Poincaré duality isomorphism $a\mapsto a\cap \lbrack M]$$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$$x\mapsto \left\langle Sq^{k}(x),[M]\right\rangle$ from $H^{n-k}(M;\Zz/2)$$H^{n-k}(M;\Zz/2)$ to $\Zz/2$$\Zz/2$ corresponds to a well defined cohomology class $v_{k}\in H^{k}(M;\Zz/2)$$v_{k}\in H^{k}(M;\Zz/2)$. This cohomology class is called the $k$$k$-th Wu class of $M$$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).$$\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),$$\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$$M$. From the definition it is clear that the Wu class is defined even for a Poincaré complex $M.$$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)$$w_{i}(M)$ of $M$$M$ using only $H^{\ast }(M;\Zz/2)$$H^{\ast }(M;\Zz/2)$ and the action of the Steenrod squares:

Theorem 3.1.

The total Stiefel-Whitney class of $M$$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$$Sq$ is a ring automorphism of
$\displaystyle H^{\ast \ast }(X;\Zz/2):=\prod_{i\geq 0}H^{i}(X;\Zz/2),$
$Sq^{-1}$$Sq^{-1}$ is defined on $H^{\ast \ast }(M;\Zz/2)=H^{\ast }(M;\Zz/2)$$H^{\ast \ast }(M;\Zz/2)=H^{\ast }(M;\Zz/2)$ and we may write
$\displaystyle v=Sq^{-1}(w(M)).$
The formula $w(M)=Sq(v)$$w(M)=Sq(v)$ may be used to extend the definition of the Stiefel-Whitney classes to Poincaré complexes.

## 4 An example

The following example is taken from [Milnor&Stasheff1974, §11]. If $H^{\ast }(M;\Zz/2)$$H^{\ast }(M;\Zz/2)$ is of the form $\Zz/2[x]/(x^{dm+1}),$$\Zz/2[x]/(x^{dm+1}),$ where $x\in H^{d}(X;\Zz/2),$$x\in H^{d}(X;\Zz/2),$ $d\geq 1,$$d\geq 1,$ $n=d\cdot m$$n=d\cdot m$, for example if $M = \mathbb{C}P^m$$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$$\lambda$ be a natural ring automorphism of $H^{\ast \ast }(X;\Zz/2)$$H^{\ast \ast }(X;\Zz/2)$ and $\Phi _{\xi }$$\Phi _{\xi }$ the Thom isomorphism of a real vector bundle $\xi$$\xi$ on $X$$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$$\lambda =Sq$, then $\underline{\lambda }=w$$\underline{\lambda }=w$ is the total Stiefel-Whitney classes $w(\xi )$$w(\xi )$ of $\xi$$\xi$ ([Milnor&Stasheff1974, §8]) and with $\xi =\tau M,$$\xi =\tau M,$ the tangent bundle of $X=M$$X=M$, we have $\textup{Wu}(Sq,\tau M)=v$$\textup{Wu}(Sq,\tau M)=v$, the total Wu class of $M$$M$. In general $\xi \mapsto \underline{\lambda }(\xi )$$\xi \mapsto \underline{\lambda }(\xi )$ and $\xi \mapsto \textup{Wu}(\lambda ,\xi )$$\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]]$$f(x) \in \Zz/2[[x]]$, which is given by its value on the universal line bundle. The generalized Wu class $\textup{Wu}(\lambda ,\xi )$$\textup{Wu}(\lambda ,\xi )$ is defined as a commutator class, thus measuring how $\lambda$$\lambda$ and $\Phi _{\xi }$$\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$$K$ -Theory and rational cohomology is formulated. This relation is more than only formal: Let $T_{i}$$T_{i}$ be the $i$$i$-th Todd polynomial, then $2^{i}\cdot T_{i}$$2^{i}\cdot T_{i}$ is a rational polynomial with denominators prime to $2,$$2,$ hence its reduction to mod $2$$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}}.$$x/Sq^{-1}(x)=1+\sum_{j\geq 0}x^{2^{j}}.$ For a continuous map $f:M\rightarrow N$$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_{!}$$f_{!}$ is the Umkehr map of $f$$f$ defined by $f_{\ast }$$f_{\ast }$ via Poincaré duality. In the case $f:M\rightarrow \ast$$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 ,$$\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$$v$ and $w=Sq(v)$$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$$n$-manifold determine its un-oriented bordism class [Thom1954, Théorém IV.10], a corollary of (1) is: Homotopy equivalent manifolds are un-oriented bordant.
3. Inserting the Stiefel-Whitney classes of $M$$M$ for $x$$x$ in
$\displaystyle \left\langle v\cup x,[M]\right\rangle =\left\langle Sq(x),[M]\right\rangle,$

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

4. Conditions on the Wu classes $v_{s}$$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$$\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!.$$\binom{x}{i}=x(x-1)\dots(x-i+1)/i!.$