Wu class

Latest revision as of 04:11, 3 February 2021

1 Introduction

The Wu class of a manifold ${{Authors|Knapp}}{{MediaWiki:Being reviewed}} ==Introduction== <wikitex>; 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. </wikitex> ==Definition== <wikitex>; 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 \begin{equation} \textup{Hom}(H^{n-k}(M;\Zz/2), \Zz/2) \cong H_{n-k}(M;\Zz /2)\cong H^{k}(M;\Zz/2), \end{equation} 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 \begin{equation} \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). \end{equation} 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 \begin{equation} v:=1+v_{1}+v_{2}+...+v_{n}. \end{equation} 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 \begin{equation} \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), \end{equation} 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 $M.$ </wikitex> ==Relation to Stiefel-Whitney classes== <wikitex>; 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 \begin{equation} w(M)=Sq(v), \end{equation} or equivalently \begin{equation} w_{k}(M)=\sum_{i=0}Sq^{i}(v_{k-i}). \end{equation} {{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 the class of topological manifolds. </wikitex> ==An example== <wikitex>; 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 \begin{equation} v=(1+x+x^{2}+x^{4}+\dots)^{m+1}~~\text{and}~~w(M)=(1+x)^{m+1} \end{equation} with \begin{equation} Sq(x)=x+x^{2} \quad \text{and} \quad Sq^{-1}(x)=x+x^{2}+x^{4}+ \dots . \end{equation} </wikitex> ==A generalization== <wikitex>; 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 \begin{equation} \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). <label>W5</label> \end{equation} 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 \begin{equation} \underline{\lambda }(\xi \oplus \eta )=\underline{\lambda }(\xi )\cup \underline{\lambda }(\eta )\text{ \ and \ }\textup{Wu}(\lambda ,\xi \oplus \eta )=\textup{Wu}(\lambda ,\xi )\cup \textup{Wu}(\lambda ,\eta ). \end{equation} 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}}]}} \begin{equation} \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). \end{equation} {{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 \begin{equation} f_{!}(\lambda (x)\cup \textup{Wu}(\lambda ^{-1},\tau M))=\lambda (f_{!}(x))\cup \textup{Wu}(\lambda ^{-1},\tau N). \end{equation} 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}). </wikitex> ==Applications== <wikitex>; <ol> <li> 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. <li> 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 unoriented bordant. <li> 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. <li> 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}.<\ol> === 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}): \begin{equation} 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} \end{equation} where $\binom{x}{i}=x(x-1)\dots(x-i+1)/i!.$ </wikitex> == References == {{#RefList:}} [[Category:Definitions]]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

$$\displaystyle \left\langle \cdot~,~\cdot \right\rangle:H^{i}(M;\Zz/2)\times H_{i}(M;\Zz /2)\longrightarrow \Zz/2$$

$$\displaystyle \textup{Hom}(H^{n-k}(M;\Zz/2), \Zz/2) \cong H_{n-k}(M;\Zz /2)\cong H^{k}(M;\Zz/2),$$

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)$\left\langle v_{k}\cup x,[M]\right\rangle =\left\langle Sq^{k}(x),[M]\right\rangle \quad \quad \text{ for all }x\in H^{n-k}(M;\Zz/2).$

$$\displaystyle v\in H^{\ast }(M;\Zz/2)=H^{0}(M;\Zz/2)\oplus H^{1}(M;\Zz /2)\oplus ... \oplus H^{n}(M;\Zz/2),$$

as the formal sum

$$\displaystyle v:=1+v_{1}+v_{2}+...+v_{n}.$$

$$\displaystyle Sq:=Sq^{0}+Sq^{1}+Sq^{2}+ \dots \colon H^{\ast}(M;\Zz /2)\longrightarrow H^{\ast }(M;\Zz/2),$$

(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

$$\displaystyle w_{k}(M)=\sum_{i=0}Sq^{i}(v_{k-i}).$$

$$\displaystyle H^{\ast \ast }(X;\Zz/2):=\prod_{i\geq 0}H^{i}(X;\Zz/2),$$

$$\displaystyle v=Sq^{-1}(w(M)).$$

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 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

• [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 $\mathcal{N}$, 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 $i$-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