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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Wu class} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_stiefelwhitney_classes}{Relation to Stiefel-Whitney classes}\dotfill \pageref*{relation_to_stiefelwhitney_classes} \linebreak \noindent\hyperlink{RelationToPontryaginClasses}{Relation to Pontryagin classes}\dotfill \pageref*{RelationToPontryaginClasses} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \noindent\hyperlink{to_higher_dimensional_chernsimons_theory}{To higher dimensional Chern-Simons theory}\dotfill \pageref*{to_higher_dimensional_chernsimons_theory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \emph{Wu classes} are a type of [[universal characteristic class]] in $\mathbb{Z}_2$-[[cohomology]] that refine the [[Stiefel-Whitney classes]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} For $X$ a [[topological space]] equipped with a class $E : X \to B SO(n)$ (a real [[vector bundle]] of some [[rank]] $n$), write \begin{displaymath} w_k \in H^k(X, \mathbb{Z}_2) \end{displaymath} for the [[Stiefel-Whitney classes]] of $X$. Moreover, write \begin{displaymath} \cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2) \end{displaymath} for the [[cup product]] on $\mathbb{Z}_2$-[[cohomology groups]] and write \begin{displaymath} Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2) \end{displaymath} for the [[Steenrod square]] operations. \begin{defn} \label{WuClassesBySteenrodSquares}\hypertarget{WuClassesBySteenrodSquares}{} The \textbf{Wu class} \begin{displaymath} \nu_k \in H^k(X,\mathbb{Z}_2) \end{displaymath} is defined to be the class that ``represents'' $Sq^k(-)$ under the cup product, in the sense that for all $x \in H^{n-k}(X, \mathbb{Z}_2)$ where $n$ is the dimension of $X$, we have \begin{displaymath} Sq^k(x) = \nu_k \cup x \,. \end{displaymath} \end{defn} (e.g. \hyperlink{MilnorStasheff74}{Milnor-Stasheff 74, p. 131-133}) \begin{remark} \label{}\hypertarget{}{} In other words this says that the lifts of Wu classes to [[integral cohomology]] ([[integral Wu structures]]) are \emph{[[characteristic element of a bilinear form|characteristic elements]]} of the [[intersection product]] on integral cohomology, inducing [[quadratic refinements]]. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_stiefelwhitney_classes}{}\subsubsection*{{Relation to Stiefel-Whitney classes}}\label{relation_to_stiefelwhitney_classes} The total [[Stiefel-Whitney class]] $w$ is the total [[Steenrod square]] of the total Wu class $\nu$. \begin{displaymath} w = Sq(\nu) \,. \end{displaymath} Solving this for the components of $\nu$ in terms of the components of $w$, one finds the first few Wu classes as [[polynomials]] in the Stiefel-Whitney classes as follows \begin{itemize}% \item $\nu_1 = w_1$; \item $\nu_2 = w_2 + w_1^2$ \item $\nu_3 = w_1 w_2$ \item $\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4$ \item $\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3$ \end{itemize} \ldots{} \hypertarget{RelationToPontryaginClasses}{}\subsubsection*{{Relation to Pontryagin classes}}\label{RelationToPontryaginClasses} \begin{prop} \label{InTermsOfPontryagin}\hypertarget{InTermsOfPontryagin}{} Let $X$ be an [[orientation|oriented]] [[manifold]] $T X : X \to B SO(n)$ with [[spin structure]] $\hat T X : X \to B Spin(n)$. Then the following classes in [[integral cohomology]] of $X$, built from [[Pontryagin classes]], coincide with Wu-classes under mod-2-reduction $\mathbb{Z} \to \mathbb{Z}_2$: \begin{itemize}% \item $\nu_4 = \frac{1}{2} p_1$ \item $\nu_8 = \frac{1}{8}(11 p_1^2 - 20 p_2)$ \item $\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3)$. \end{itemize} (all products are [[cup product|cup products]]). \end{prop} This is discussed in (\hyperlink{HopkinsSinger}{Hopkins-Singer, page 101}). \begin{cor} \label{DivisibilityOfCupSquare}\hypertarget{DivisibilityOfCupSquare}{} Suppose $X$ is 8 dimensional. Then, for $G \in H^4(X, \mathbb{Z})$ any integral 4-class, the expression \begin{displaymath} G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z}) \end{displaymath} is always even (divisible by 2). \end{cor} \begin{proof} By the basic properties of Steenrod squares, we have for the 4-class $G$ that \begin{displaymath} G \cup G = Sq^4(G) \,. \end{displaymath} By the definition \ref{WuClassesBySteenrodSquares} of Wu classes, the image of this integral class in $\mathbb{Z}_2$-coefficients equals the cup product with the Wu class \begin{displaymath} G \cup G - G \cup \frac{1}{2}p_1 = Sq^4(G) - G \cup \nu_4 = 0 \; mod \; 2. \,, \end{displaymath} where the first step is by prop. \ref{InTermsOfPontryagin}. \end{proof} \hypertarget{applications}{}\subsection*{{Applications}}\label{applications} \hypertarget{to_higher_dimensional_chernsimons_theory}{}\subsubsection*{{To higher dimensional Chern-Simons theory}}\label{to_higher_dimensional_chernsimons_theory} \begin{remark} \label{}\hypertarget{}{} The relation \ref{DivisibilityOfCupSquare} plays a central role in the definition of the [[higher dimensional Chern-Simons theory|7-dimensional Chern-Simons theory]] which is [[holographic principle|dual]] to the [[self-dual higher gauge theory]] on the [[M5-brane]]. In this context it was first pointed out in (\hyperlink{Witten}{Witten 1996}) and later elaborated on in (\hyperlink{HopkinsSinger}{Hopkins-Singer}). Specifically, in this context $G$ is the 4-class of the [[circle n-bundle with connection|circle 3-bundle]] underlying the [[supergravity C-field]], subject to the quantization condition \begin{displaymath} G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a \,, \end{displaymath} for some $a \in H^4(X, \mathbb{Z})$, which makes direct sense as an equation in $H^4(X, \mathbb{Z})$ if the [[spin structure]] on $X$ happens to be such $\frac{1}{2}p_1$ is further divisible by 2, and can be made sense of more generally in terms of [[twisted cohomology]] (which was suggested in (\hyperlink{Witten}{Witten 1996}) and made precise sense of in (\hyperlink{HopkinsSinger}{Hopkins-Singer}) ). For simplicity, assume that $\frac{1}{2}p_1$ of $X$ is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to [[ordinary differential cohomology]] and so we consider [[circle n-bundle with connection|differential cocycles]] $\hat a, \hat G \in \hat H^4(X)$ with \begin{equation} \hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,, \label{DifferentialQuantizationCondition}\end{equation} where the differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ is discussed in detail at \emph{[[differential string structure]]}. Now, after [[Kaluza-Klein mechanism|dimensional reduction]] on a 4-[[sphere]], the [[action functional]] of [[11-dimensional supergravity]] on the remaining 7-dimensional $X$ contains a [[higher dimensional Chern-Simons theory|higher Chern-Simons term]] which up to prefactors is of the form \begin{displaymath} \hat G \mapsto \exp i \int_X ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,, \end{displaymath} where \begin{itemize}% \item the cup product now is the differential [[Beilinson-Deligne cup product]] refinement of the integral cup product; \item the symbol $\exp(i \int_X (-))$ denotes [[fiber integration in ordinary differential cohomology]]. \end{itemize} Using \eqref{DifferentialQuantizationCondition} this is \begin{displaymath} \cdots = \exp i \int_X \left( \hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1 \right) \,. \end{displaymath} But by corollary \ref{DivisibilityOfCupSquare} this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this \begin{displaymath} \hat G \mapsto \exp i \int_X \frac{1}{2} ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,. \end{displaymath} In (\hyperlink{Witten}{Witten 1996}) it is discussed that the space of [[states]] of this ``fractional'' functional over a 6-dimensional $\Sigma$ is the space of [[conformal blocks]] of the [[self-dual higher gauge theory]] on the [[M5-brane]]. \end{remark} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[integral Wu structure]], [[twisted Wu structure]] \item [[Pontryagin class]], [[Stiefel-Whitney class]], [[one-loop anomaly polynomial I8]] \item [[Euler class]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The original reference is \begin{itemize}% \item [[Wen-Tsun Wu]], \emph{On Pontrjagin classes: II} Sientia Sinica 4 (1955) 455-490 \end{itemize} See also around p. 228 of \begin{itemize}% \item [[John Milnor]], [[Jim Stasheff]], \emph{Characteristic classes}, Princeton University Press (1974) \end{itemize} and section 2 of \begin{itemize}% \item Yanghyun Byun, \emph{On vanishing of characteristic numbers in Poincar\'e{} complexes}, Transactions of the AMS, vol 348, number 8 (1996) (\href{http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01495-X/S0002-9947-96-01495-X.pdf}{pdf}) \end{itemize} and \begin{itemize}% \item [[Robert Stong]], Toshio Yoshida, \emph{Wu classes} Proceedings of the American Mathematical Society Vol. 100, No. 2, (1987) (\href{http://www.jstor.org/pss/2045970}{JSTOR}) \end{itemize} Details are reviewed in appendix E of \begin{itemize}% \item [[Mike Hopkins]], [[Isadore Singer]], \emph{[[Quadratic Functions in Geometry, Topology, and M-Theory]]} \end{itemize} This is based on or motivated from observations in \begin{itemize}% \item [[Edward Witten]], \emph{Five-Brane Effective Action In M-Theory} (\href{http://arxiv.org/abs/hep-th/9610234}{arXiv:hep-th/9610234}) \end{itemize} More discussion of Wu classes in this physical context is in \begin{itemize}% \item [[Hisham Sati]], \emph{Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures} (\href{http://arxiv.org/abs/1109.4461}{arXiv:1109.4461}) \end{itemize} which also summarizes many standard properties of Wu classes. [[!redirects Wu classes]] \end{document}