\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\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*{Bockstein homomorphism}

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

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

[[!include cohomology - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - 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{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{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Bockstein homomorphism} is a [[connecting homomorphism]] induced from a [[short exact sequence]] whose injective map is given by multiplication with an [[integer]].

The archetypical examples are the Bockstein homomorphisms induced this way from the short exact sequence

\begin{displaymath}
\mathbb{Z} 
    \stackrel{\cdot 2}{\to} 
  \mathbb{Z}
    \stackrel{}{\to}
  \mathbb{Z}/2\mathbb{Z}
  \,.
\end{displaymath}
These relate notably degree-$n$ [[cohomology]] with [[coefficients]] in $\mathbb{Z}_2$ (such as [[Stiefel-Whitney classes]]) to cohomology with integral coefficients in degree $n+1$ (such as [[integral Stiefel-Whitney classes]]).

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

Let $A$ be an [[abelian group]] and $m$ be an [[integer]]. Then multiplication by $m$

\begin{displaymath}
A \stackrel{m\cdot}{\to} A
\end{displaymath}
induces a [[short exact sequence]] of abelian groups

\begin{displaymath}
0\to A/A_{m-tors} \stackrel{m\cdot}{\to} A \to A/m A\to 0,
\end{displaymath}
where $A_{m-tors}$ is the subgroup of $m$-[[torsion subgroup|torsion elements]] of $A$, and so a long [[fiber sequence]]

\begin{displaymath}
\cdots \mathbf{B}^n (A/A_{m-tors}) \to \mathbf{B}^n A \to \mathbf B^n(A/ m A) \to \mathbf{B}^{n+1} (A/A_{m-tors}) \to \cdots
\end{displaymath}
of [[∞-groupoid]]s, where $\mathbf{B}^n(-)$ denotes the $n$-fold [[delooping]] (hence $\mathbf{B}^n A$ is the [[Eilenberg-MacLane object]] on $A$ in degree $n$).

This induces, in turn, for any object $X \in \mathbf{H}$ (for $\mathbf{H}$ the ambient [[(∞,1)-topos]], such as [[Top]] $\simeq$ [[∞Grpd]]) , a long [[fiber sequence]]

\begin{displaymath}
\cdots \mathbf{H}(X,\mathbf{B}^n (A/A_{m-tors})) \to \mathbf{H}(X,\mathbf{B}^n A) \to \mathbf{H}(X,\mathbf B^n(A/ m A)) \stackrel{\beta_m}{\to} \mathbf{H}(X,\mathbf{B}^{n+1} (A/A_{m-tors})) \to \cdots
\end{displaymath}
of [[cocycle]] [[∞-groupoids]].

Here the [[connecting homomorphisms]] $\beta_m$ are called the \textbf{Bockstein homomorphisms}.

Notice that often this term is used to refer only to the image of the above in [[cohomology]], hence to the image of $\beta_m$ under [[0-truncated|0-truncation]]/[[homotopy group|0th homotopy group]] $\pi_0$:

\begin{displaymath}
\beta_m : H^n(X,A/ m A) \to H^{n+1}(X,(A/A_{m-tors}))
  \,.
\end{displaymath}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{Mod2BocksteinIntoIntegralCohomology}\hypertarget{Mod2BocksteinIntoIntegralCohomology}{}
\textbf{(mod 2 Bockstein homomorphism into [[integral cohomology]])}

The Bockstein homomorphism $\beta$ for the sequence

\begin{displaymath}
\mathbb{Z} 
    \stackrel{\cdot 2}{\longrightarrow} 
  \mathbb{Z} 
    \stackrel{mod\, 2}{\longrightarrow} 
  \mathbb{Z}/2\mathbb{Z}
\end{displaymath}
serves to define [[integral Stiefel-Whitney classes]]

\begin{displaymath}
W_{n+1} \coloneqq \beta w_n
\end{displaymath}
in degree $n+1$ from $\mathbb{Z}/2\mathbb{Z}$-valued [[Stiefel-Whitney classes]] in degree $n$.

\end{example}
\begin{example}
\label{Mod2BocksteinIntoMod2Cohomology}\hypertarget{Mod2BocksteinIntoMod2Cohomology}{}
\textbf{(first [[Steenrod square]])}

The Bockstein homomorphism for the sequence

\begin{displaymath}
\mathbb{Z}/2\mathbb{Z}
    \overset{\cdot 2}{\longrightarrow}
  \mathbb{Z}/4\mathbb{Z}
    \overset{mod\, 2}{\longrightarrow}
  \mathbb{Z}/2\mathbb{Z}
\end{displaymath}
is also called the \emph{first [[Steenrod square]]}, denoted $Sq^1$.

This is often equivalently denoted $\beta$, as in example \ref{Mod2BocksteinIntoIntegralCohomology}. The difference between the two is just the mod-2 reduction in their codomain:

\begin{equation}
\itexarray{
    \mathbb{Z}
      &\overset{\cdot 2}{\longrightarrow}&
    \mathbb{Z}
      &\overset{mod\, 2}{\longrightarrow}&
    \mathbb{Z}/2\mathbb{Z}
      &\simeq&
    \mathbb{Z}/2\mathbb{Z}
      &\overset{\beta}{\longrightarrow}&
    B \mathbb{Z}
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{mod\, 4}}
      &&
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{ id }
      &&
    \downarrow^{\mathrlap{B(mod\, 2)}}
    \\
    \mathbb{Z}/2\mathbb{Z}
      &\underset{\cdot 2 }{\longrightarrow}&
    \mathbb{Z}/4\mathbb{Z}
      &\underset{mod\, 2}{\longrightarrow}&
     \mathbb{Z}/2\mathbb{Z}
      &\simeq&
     (\mathbb{Z}/4\mathbb{Z})/(\mathbb{Z}/2\mathbb{Z})
       &\underset{Sq^1}{\longrightarrow}&
     B (\mathbb{Z}/2\mathbb{Z})
  }
\label{Mod2BocksteinSequences}\end{equation}
More generally, for $p$ any [[prime number]] the multiplication by $p$ on $\mathbb{Z}_{p^2}$ induces the short exact sequence $\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/{p^2}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$. The corresponding Bockstein homomorphism $\beta_p$ appears as one of the generators of the mod $p$ [[Steenrod algebra]].

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

For [[odd natural numbers|odd]] $2n + 1 \in \mathbb{N}$ defines the [[integral Steenrod squares]] to be

\begin{displaymath}
Sq^{2n + 1}_{\mathbb{Z}}
  \;\coloneqq\;
  \beta \circ Sq^{2n}
  \,.
\end{displaymath}
By example \ref{Mod2BocksteinIntoMod2Cohomology} and by the first [[Adem relation]] $Sq^1 \circ Sq^{2n} = Sq^{2n+1}$ (\href{Steenrod+square#CompositionWithSq1}{this example}) these indeed are lifts of the odd [[Steenrod squares]]:

\begin{displaymath}
(mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}}
  \;=\;
  Sq^{2n+1}
  \,,
\end{displaymath}
because, by \eqref{Mod2BocksteinSequences} we have

\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{example}
When $A=\mathbb{Z}$, the equivalence $\vert \mathbf{B}^{n+1}\mathbb{Z} \vert \cong \vert \mathbf{B}^n U(1)\vert$ (which holds in ambient contexts such as $\mathbf{H} =$ [[ETop∞Grpd]] or [[Smooth∞Grpd]] under [[geometric realization]] $\vert - \vert : ETop \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top$) identifies the morphisms $\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n+1}\mathbb{Z}$ with the morphisms $\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n} U(1)$ induced by the inclusion of the subgroup of $m$-th roots of unity into $U(1)$. This identifies the Bockstein homomorphism $\beta_m: H^n(X;\mathbb{Z}_m)\to H^{n+1}(X;\mathbb{Z})$ with the natural homomorphism $H^n(X;\mathbb{Z}_m)\to H^{n}(X;U(1))$.

More in detail:

\begin{example}
\label{Mod2BocksteinAndExponentialExactSequence}\hypertarget{Mod2BocksteinAndExponentialExactSequence}{}
\textbf{(mod 2 [[Bockstein homomorphism]] and the [[exponential exact sequence]])}

Let

\begin{enumerate}%
\item $\beta \;\colon\; \mathbb{Z}/2\mathbb{Z} \longrightarrow B \mathbb{Z}$ be the ordinare Bockstein homomorphism


\item $\iota\coloneqq (\cdot \pi) \;\colon\; \mathbb{Z}/2\mathbb{Z} \hookrightarrow U(1)$ the canonical inclusion;


\item $\delta \;\colon\; U(1) \longrightarrow B\mathbb{Z}$ the classifying map.



\end{enumerate}
Then

\begin{displaymath}
\beta
  \;=\;
  \delta \circ \iota
  \,.
\end{displaymath}
Because

\begin{displaymath}
\itexarray{
    \mathbb{Z}
      &\overset{\cdot 2}{\longrightarrow}&
    \mathbb{Z}
      &\overset{mod\, 2}{\longrightarrow}&
    \mathbb{Z}/2\mathbb{Z}
      &\simeq&
    \mathbb{Z}/2\mathbb{Z}
      &\overset{\beta}{\longrightarrow}&
    B \mathbb{Z}
    \\
    \downarrow^{\mathrlap{id}}
      &&
    \downarrow^{\mathrlap{\cdot \pi}}
      &&
    \downarrow^{\mathrlap{\cdot \pi}}
      &&
    \downarrow^{\iota \coloneqq \mathrlap{\cdot \pi}}
      &&
    \downarrow^{\mathrlap{id}}
    \\
    \mathbb{Z}
      &\underset{\cdot 2 \pi}{\longrightarrow}&
    \mathbb{R}
      &\underset{mod\, 2\pi}{\longrightarrow}&
     U(1)
      &\simeq&
     \mathbb{R}/2\pi\mathbb{Z}
       &\underset{\delta}{\longrightarrow}&
     B \mathbb{Z}
  }
\end{displaymath}
\end{example}
\begin{prop}
\label{SteenrodSquaresAndDBCupProductOnOddClasses}\hypertarget{SteenrodSquaresAndDBCupProductOnOddClasses}{}
\textbf{([[Deligne-Beilinson cup product]] on odd-degree [[ordinary differential cohomology]])}

Let

\begin{displaymath}
\hat H
  \;\colon\;
  X \longrightarrow \mathbf{B}^{2n} U(1)_{conn})
\end{displaymath}
be a class in [[ordinary differential cohomology]] with underlying class in odd degree

\begin{displaymath}
[H]
  \;\colon\;
  X 
    \overset{\hat H}{\longrightarrow}
  \mathbf{B}^{n} U(1)_{conn}
    \overset{\chi}{\longrightarrow} 
  B^{2n+1} \mathbb{Z}
\end{displaymath}
This implies that its [[Beilinson-Deligne cup product]] with itself satisfies

\begin{displaymath}
\hat H \hat \cup \hat H = - \hat H \hat \cup \hat H
\end{displaymath}
hence

\begin{displaymath}
2 \hat H \hat \cup \hat H \;\simeq\; 0
\end{displaymath}
hence

\begin{displaymath}
2 [H] \cup [H] \;\simeq\; 0
\end{displaymath}
hence that the ordinary [[cup product]] $[H] \cup [H]$ is a 2-torsion class. Let then

\begin{displaymath}
j
  \;\colon\;
  \mathbf{B}^{4n+1} \mathbb{Z}/2\mathbb{Z}
    \overset{
      B^{4n+1} (\iota)
    }{\hookrightarrow}
  \flat \mathbf{B}^{4n+1}  U(1)
    \longrightarrow
  \mathbf{B}^{4n+1} U(1)_{conn}
\end{displaymath}
with $\iota$ from example \ref{Mod2BocksteinAndExponentialExactSequence}.

Then

\begin{displaymath}
\hat H  \hat \cup \hat H
  \;\simeq\;
  j Sq^{2n}([H]_{mod\,2})
  \,.
\end{displaymath}
This is a [[differential cohomology]]-refinement of the first [[Adem relation]] $Sq^1 \circ Sq^{2n} = Sq^{2n+1}$ on the [[Steenrod squares]] (\href{Steenrod+square#CompositionWithSq1}{this example}) in that, by example \ref{Mod2BocksteinAndExponentialExactSequence}, its image in ordinary cohomology with coefficients in $\mathbb{Z}/2\mathbb{Z}$ is

\begin{displaymath}
\itexarray{
    ([H]  \cup [H])_{mod 2}
     & \simeq & 
    \underset{
      \beta
    }{
    \underbrace{
      Sq^1
    }}
    \circ Sq^{2n}([H]_{mod\,2})
    \\
    =
    \\
    [H]_{mod\, 2} \cup [H]_{mod,2}
    \\
    =
    \\
    Sq^{2n+1}([H]_{mod\, 2})
  }
  \,.
\end{displaymath}
\end{prop}
This was first observed in (\hyperlink{Gomi08}{Gomi 08}). Streamlined proofs are given in (\hyperlink{Bunke12}{Bunke 12, propblem 3.106}, \hyperlink{GradySati16}{Grady-Sati 16, prop. 22}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Steenrod squares]], [[cohomology operation]]


\item [[Bockstein spectral sequence]]



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

Original references include

\begin{itemize}%
\item [[Meyer Bockstein]],

\emph{Universal systems of $\nabla$-homology rings}, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 243--245, MR0008701

\emph{A complete system of fields of coefficients for the $\nabla$-homological dimension} , C. R. (Doklady) Acad. Sci. URSS (N.S.) (1943), 38: 187--189, MR0009115


\item [[Meyer Bockstein]], \emph{Sur la formule des coefficients universels pour les groupes d'homologie} , Comptes Rendus de l'acad\'e{}mie des Sciences. S\'e{}rie I. Math\'e{}matique (1958), 247: 396--398, MR0103918



\end{itemize}
The relation to the [[Beilinson-Deligne cup product]] is discussed in

\begin{itemize}%
\item [[Kiyonori Gomi]], \emph{Differential characters and the Steenrod squares}, In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 297?308. Math. Soc. Japan, Tokyo, 2008


\item [[Ulrich Bunke]], problem 3.106 in \emph{Differential cohomology} (\href{https://arxiv.org/abs/1208.3961}{arXiv:1208.3961})


\item Daniel Grady, [[Hisham Sati]], prop. 22 in \emph{Primary operations in differential cohomology}, Adv. Math. 335 (2018), 519-562 (\href{https://arxiv.org/abs/1604.05988}{arXiv:1604.05988}, \href{https://www.sciencedirect.com/science/article/pii/S0001870818302676?via%3Dihub}{doi:10.1016/j.aim.2018.07.019})



\end{itemize}
[[!redirects Bockstein homomorphisms]] [[!redirects Bockstein morphism]] [[!redirects Bockstein morphisms]] [[!redirects Bockstein operation]] [[!redirects Bockstein operations]]



\end{document}