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\begin{document}

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\section*{integral Steenrod square}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{IntegralSteenrodSquares}\hypertarget{IntegralSteenrodSquares}{}
\textbf{([[integral Steenrod squares]])}

For [[odd natural numbers|odd]] $2n + 1 \in \mathbb{N}$ the \emph{integral Steenrod square} $Sq^{2n + 1}_{\mathbb{Z}}$ is the [[composition]] of the mod-2 [[Steenrod square]] $Sq^{2n}$ with the [[Bockstein homomorphism]] $\beta$ associated with the sequence $\mathbb{Z} \overset{\cdot 2}{\to} \mathbb{Z} \overset{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z}$:

\begin{displaymath}
Sq^{2n + 1}_{\mathbb{Z}}
  \;\coloneqq\;
  \beta \circ Sq^{2n}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{}\hypertarget{}{}
The odd-degree integral Steenrod squares from def. \ref{IntegralSteenrodSquares} are indeed integral lifts of the mod-2 Steenrod squares in that

\begin{displaymath}
(mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}}
  \;=\;
  Sq^{2n+1}
  \,,
\end{displaymath}
\end{prop}
\begin{proof}
This follows from the relation of the [[Bockstein homomorphism]] to the first [[Steenrod square]]

\begin{displaymath}
(mod, 2) \circ  \beta = Sq^1
\end{displaymath}
(\href{Bockstein+homomorphism#Mod2BocksteinIntoMod2Cohomology}{this example}) together with the first [[Adem relation]]

\begin{displaymath}
Sq^1 \circ Sq^{2n} = Sq^{2n+1}
\end{displaymath}
(\href{Steenrod+square#CompositionWithSq1}{this example}):

\begin{displaymath}
\itexarray{
    Sq^{2n+1}_{\mathbb{Z}}
    &\colon&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{Sk^{2n}}{\longrightarrow}&      
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{ \beta }{\longrightarrow}&
    B^{\bullet + 2n + 1} \mathbb{Z}
    \\
    &&
    \downarrow^{ id }  
    &&
    \downarrow^{ id }
      &&
    \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}}
    \\
    Sq^{2n+1}
    &\colon&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\underset{Sk^{2n}}{\longrightarrow}&   
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
       &\underset{  Sq^1 }{\longrightarrow}&
    B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z})
  }
\end{displaymath}
\end{proof}
\begin{prop}
\label{IntegralSteenrodSquareInTermsOfBocksteinForExponentialSequence}\hypertarget{IntegralSteenrodSquareInTermsOfBocksteinForExponentialSequence}{}
\textbf{([[integral Steenrod square]] in terms of [[Bockstein homomorphism]] for [[exponential sequence]])}

The integral Steenrod squares (def. \ref{IntegralSteenrodSquares}) may equivalently be written in terms of the [[Bockstein homomorphism]] $\delta$ of the [[exponential sequence]] $\mathbb{Z} \overset{\cdot 2\pi}{\longrightarrow} \mathbb{R} \overset{mod\, 2 \pi}{\longrightarrow} U(1)$ as

\begin{equation}
Sq^{2n+1}_{\mathbb{Z}}
  \;\colon\;
  B^{\bullet} (\mathbb{Z}/2\mathbb{Z})
    \overset{
      Sq^{2 n}
    }{\longrightarrow}
  B^{\bullet + 2n } (\mathbb{Z}/2\mathbb{Z})
  \overset{
    \iota    
  }{\longrightarrow}
  B^{\bullet + 2n} U(1)
   \overset{\delta}{\longrightarrow}
  B^{\bullet + 2n + 1} \mathbb{Z}
  \,.
\label{IntegralSteenrodSquareViaBocksteinOfExponentialSequence}\end{equation}
\end{prop}
\begin{proof}
Since $\beta = \delta \circ \iota$, by \href{Bockstein+homomorphism#Mod2BocksteinAndExponentialExactSequence}{this example}.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{IntegralSteenrodSquareRefinedToOrdinaryDifferentialCohomology}\hypertarget{IntegralSteenrodSquareRefinedToOrdinaryDifferentialCohomology}{}
\textbf{([[integral Steenrod square]] refined to [[ordinary differential cohomology]])}

Let $\hat G_{2n+2} \colon X \to \mathbf{B}^{2n+1} U(1)_{conn}$ be a [[cocycle]] in [[ordinary differential cohomology]] of degree $2n + 2$.

By inserting the bottom triangle of the ordinary [[differential cohomology hexagon]] (\href{differential+cohomology+diagram#eq:OrdinarCohomologyHexagon}{this diagram}) into the factorization in \eqref{IntegralSteenrodSquareViaBocksteinOfExponentialSequence} we obtain a canonical refinement of the [[integral Steenrod square]] $Sq^{2n+1}_{\mathbb{Z}} [G_{2n + 2}]$ to a cocycle $\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n+2}$ in [[ordinary differential cohomology]], which happens to be [[flat infinity-bundle|flat]]

\begin{displaymath}
\itexarray{
     && && \mathbf{B}^{4n+2} \flat U(1)
     &\longrightarrow&
     \mathbf{B}^{4n+2}U(1)_{conn}
     \\
     && 
     & {}^{\mathllap{ \iota \circ Sq^{2n} }}\nearrow 
     &
     &
     {}_{\mathllap{\delta}}\searrow
     &
     \downarrow^{\chi}
     \\
     X 
     &\underset{G_{2n + 2}}{\longrightarrow}&
     B^{2n+2} \mathbb{Z}
     &&
       \underset{ Sq^{2n+1}_{\mathbb{Z}} }{\longrightarrow}
     &&
     B^{4n+3} \mathbb{Z}
  }
  \,.
\end{displaymath}
If one moreover asks that the integral Steenrod square vanishes

\begin{displaymath}
[ Sq^{2n+1}_{\mathbb{Z}} G_{2n+2}]
  \;=\;
  0
  \;\in\;
  H^{4n+3}(X,\mathbb{Z})
\end{displaymath}
(as in \hyperlink{DiaconescuMooreWitten00}{Diaconescu-Moore-Witten 00, around (6.9)} for $n = 1$) then the curvature exact sequence and characteristic class exact sequence in [[ordinary differential cohomology]] (\href{ordinary+differential+cohomology#ExactSequencesForOrdDiffCohomology}{this prop.}) imply that the class of $\widehat{Sq}^{2n+1}_{\mathbb{Z}} \hat G_{2n + 2}$ is identified with a class in [[de Rham cohomology]] in degree $4n+3$:

\begin{displaymath}
H^{2n+2}_{diff}(X)|_{Sq^{2n+1}_{\mathbb{Z}} = 0}
    \overset{\widehat{Sq}_{\mathbb{Z}}^{2n+1}}{\longrightarrow}
  H^{4n+3}_{dR}(X)
  \,.
\end{displaymath}
\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[integral Stiefel-Whitney classes]]


\item [[integral Wu structure]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The third integral Steenrod square $Sq^3_{\mathbb{Z}}$ plays a central role in the discussion of the [[supergravity C-field]] in

\begin{itemize}%
\item [[Gregory Moore]], [[Edward Witten]], \emph{$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory}, Adv. Theor. Math. Phys. 6:1031-1134, 2003 (\href{https://arxiv.org/abs/hep-th/0005090}{arXiv:hep-th/0005090})

\end{itemize}
[[!redirects integral Steenrod squares]]



\end{document}