\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*{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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{DefinitionInTermsOfExtendedSquares}{Construction in terms of extended squares}\dotfill \pageref*{DefinitionInTermsOfExtendedSquares} \linebreak
\noindent\hyperlink{axiomatic_characterization}{Axiomatic characterization}\dotfill \pageref*{axiomatic_characterization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_bockstein_homomorphism}{Relation to Bockstein homomorphism}\dotfill \pageref*{relation_to_bockstein_homomorphism} \linebreak
\noindent\hyperlink{compatibility_with_suspension}{Compatibility with suspension}\dotfill \pageref*{compatibility_with_suspension} \linebreak
\noindent\hyperlink{relation_to_massey_products}{Relation to Massey products}\dotfill \pageref*{relation_to_massey_products} \linebreak
\noindent\hyperlink{adem_relations}{Adem relations}\dotfill \pageref*{adem_relations} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{HopfInvariant}{Hopf invariant}\dotfill \pageref*{HopfInvariant} \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}

What are called the \emph{Steenrod squares} is the system of [[cohomology operations]] on [[cohomology]] with [[coefficients]] in $\mathbb{Z}_2$ (the [[cyclic group of order 2]]) which is compatible with [[suspension]] (the ``stable cohomology operations''). They are special examples of [[power operations]].

The Steenrod squares together form the \emph{[[Steenrod algebra]]}, see there for more.

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

\hypertarget{DefinitionInTermsOfExtendedSquares}{}\subsubsection*{{Construction in terms of extended squares}}\label{DefinitionInTermsOfExtendedSquares}

We discuss the explicit construction of the Steenrod-operations in terms of [[chain maps]] of [[chain complexes]] of $\mathbb{F}_2$-[[vector spaces]] equipped with a suitable product. We follow (\hyperlink{Lurie07}{Lurie 07, lecture 2}).

Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the [[field]] with two elements.

For $V$ an $\mathbb{F}_2$-[[module]], hence an $\mathbb{F}_2$-[[vector space]], and for $n \in \mathbb{N}$, write

\begin{displaymath}
V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod
\end{displaymath}
for the [[homotopy quotient]] of the $n$-fold [[tensor product]] of $V$ with itself by the [[action]] of the [[symmetric group]]. Explicitly this is presented, up to [[quasi-isomorphism]] by the ordinary [[coinvariants]] $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a [[free resolution]] $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:

\begin{displaymath}
V^{\otimes n}_{h \Sigma_n}
  \simeq
  D_n(V)
  \coloneqq
  (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n}
  \,.
\end{displaymath}
This is called the $n$th \emph{[[extended power]]} of $V$.

For instance

\begin{displaymath}
D_2(  \mathbb{F}_2[-n])
  \simeq
  \mathbb{F}_2[-2n] \otimes C^\bullet(B \Sigma_2)
  \,,
\end{displaymath}
where on the right we have the, say, [[singular cohomology]] [[cochain complex]] of the [[homotopy quotient]] $\ast //\Sigma_2 \simeq B \Sigma_2 \simeq \mathbb{R}P^\infty$, which is the [[homotopy type]] of the [[classifying space]] for $\Sigma_2$.

A [[chain map]]

\begin{displaymath}
D_2(V) \longrightarrow V
\end{displaymath}
is called a \emph{symmetric multiplication} on $V$ (a shadow of an [[E-infinity algebra]] structure). The archetypical class of examples of these are given by the [[singular cohomology]] $V = C^\bullet(X, \mathbb{F}_2)$ of any [[topological space]] $X$, for instance of $B \Sigma_2$.

Therefore there is a canonical [[isomorphism]]

\begin{displaymath}
H^k(D_2(\mathbb{F}_2[-n]))
  \simeq
  H_{2n - k}(B \Sigma_2, \mathbb{F}_2) e_{2n}
\end{displaymath}
of the [[cochain cohomology]] of the extended square of the chain compplex concentrated on $\mathbb{F}_2$ in degree $n$ with the [[singular homology]] of this classifying space shifted by $2 n$.

Using this one gets for general $V$ and for each $i \leq n$ a map that sends an element in the $n$th [[cochain cohomology]]

\begin{displaymath}
[v] \in H^n(V)
\end{displaymath}
represented by a morphism of [[chain complexes]]

\begin{displaymath}
v \;\colon\; \mathbb{F}_2[-n] \longrightarrow V
\end{displaymath}
to the element

\begin{displaymath}
\overline{Sq}^i(v) \in H^{n+1}(D_2(V))
\end{displaymath}
represented by the [[chain map]]

\begin{displaymath}
\mathbb{F}_2[-n-i]
  \stackrel{1}{\longrightarrow}
  C^\bullet(B \Sigma_2, \mathbb{F}_2)
  \stackrel{\simeq}{\longrightarrow}
  D_2(\mathbb{F}_2[-n])
   \stackrel{D_2(v)}{\longrightarrow}
  D_2(V)
  \,.
\end{displaymath}
If moreover $V$ is equipped with a \emph{symmetric product} $D_2(V) \longrightarrow V$ as above, then one can further compose and form the element

\begin{displaymath}
{Sq}^i(v) \in H^{n+1}(V)
\end{displaymath}
represented by the [[chain map]]

\begin{displaymath}
\mathbb{F}_2[-n-i]
  \stackrel{1}{\longrightarrow}
  C^\bullet(B \Sigma_2, \mathbb{F}_2)
  \stackrel{\simeq}{\longrightarrow}
  D_2(\mathbb{F}_2[-n])
   \stackrel{D_2(v)}{\longrightarrow}
  D_2(V)
   \longrightarrow
  V
  \,.
\end{displaymath}
This [[linear map]]

\begin{displaymath}
Sq^i \;\colon\; H^\bullet(V) \longrightarrow H^{\bullet + i}(V)
\end{displaymath}
is called the $i$th \emph{Steenrod operation} or the $i$th \emph{Steenrod square} on $V$. By default this is understood for $V = C^\bullet(X,\mathbb{F}_2)$ the $\mathbb{F}_2$-[[singular homology|singular cochain complex]] of some [[topological space]] $X$, as in the above examples, in which case it has the form

\begin{displaymath}
Sq^i \;\colon\; H^\bullet(X, \mathbb{F}_2) \longrightarrow H^{\bullet+i}(X,\mathbb{F}_2)
  \,.
\end{displaymath}
\hypertarget{axiomatic_characterization}{}\subsubsection*{{Axiomatic characterization}}\label{axiomatic_characterization}

For $n \in \mathbb{N}$ write $B^n \mathbb{Z}_2$ for the [[classifying space]] of [[ordinary cohomology]] in degree $n$ with [[coefficients]] in the [[group of order 2]] $\mathbb{Z}_2$ (the [[Eilenberg-MacLane space]] $K(\mathbb{Z}_2,n)$), regarded as an [[object]] in the [[homotopy category]] $H$ [[model structure on topological spaces|of topological spaces]]).

Notice that for $X$ any topological space ([[CW-complex]]),

\begin{displaymath}
H^n(X, \mathbb{Z}_2) \coloneqq H(X, B^n \mathbb{Z}_2)
\end{displaymath}
is the [[ordinary cohomology]] of $X$ in degree $n$ with [[coefficients]] in $\mathbb{Z}_2$. Therefore, by the [[Yoneda lemma]], [[natural transformations]]

\begin{displaymath}
H^{k}(-, \mathbb{Z}_2) \to H^l(-, \mathbb{Z}_2)
\end{displaymath}
correspond bijectively to morphisms $B^k \mathbb{Z}_2 \to B^l \mathbb{Z}_2$.

The following characterization is due to (\hyperlink{SteenrodEpstein}{SteenrodEpstein}).

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{Steenrod squares} are a collection of [[cohomology operations]]

\begin{displaymath}
Sq^n \;\colon\; H^k(-, \mathbb{Z}_2) \longrightarrow H^{k+n}(-, \mathbb{Z}_2)
  \,,
\end{displaymath}
hence of [[morphisms]] in the [[homotopy category]]

\begin{displaymath}
Sq^n \;\colon\; B^k \mathbb{Z}_2 \longrightarrow B^{k + n} \mathbb{Z}_2
\end{displaymath}
for all $n,k \in \mathbb{N}$ satisfying the following conditions:

\begin{enumerate}%
\item for $n = 0$ it is the [[identity]];


\item if $n \gt deg(x)$ then $Sq^n(x) = 0$;


\item for $k = n$ the morphism $Sq^n : B^n \mathbb{Z}_2 \to B^{2n} \mathbb{Z}_2$ is the [[cup product]] $x \mapsto x \cup x$;


\item $Sq^n(x \cup y) = \sum_{i + j = n} (Sq^i x) \cup (Sq^j y)$;



\end{enumerate}
\end{defn}
An analogous definition works for [[coefficients]] in $\mathbb{Z}_p$ for any [[prime number]] $p \gt 2$. The corresponding operations are then usually denoted

\begin{displaymath}
P^n \;\colon\; B^k \mathbb{Z}_p \longrightarrow B^{k+n} \mathbb{Z}_{p}
  \,.
\end{displaymath}
Under [[composition]], the Steenrod squares form an [[associative algebra]] over $\mathbb{F}_2$, called the \emph{[[Steenrod algebra]]}. See there for more.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_bockstein_homomorphism}{}\subsubsection*{{Relation to Bockstein homomorphism}}\label{relation_to_bockstein_homomorphism}

$Sq^1$ is the [[Bockstein homomorphism]] of the [[short exact sequence]] $\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2$.

\hypertarget{compatibility_with_suspension}{}\subsubsection*{{Compatibility with suspension}}\label{compatibility_with_suspension}

The Steenrod squares are compatible with the [[suspension isomorphism]].

Therefore the Steenrod squares are often also referred to as the \emph{[[stabilization|stable]] [[cohomology operations]]}

\hypertarget{relation_to_massey_products}{}\subsubsection*{{Relation to Massey products}}\label{relation_to_massey_products}

See at \emph{[[Massey product]]}, \emph{\href{Massey+product#RelationToSteenrodSquares}{Relation to Steenrod squares}}

\hypertarget{adem_relations}{}\subsubsection*{{Adem relations}}\label{adem_relations}

\begin{prop}
\label{AdemRelations}\hypertarget{AdemRelations}{}
\textbf{([[Adem relations]])}

The [[composition]] of Steenrod square operations satisfies the following relations

\begin{displaymath}
Sq^i \circ Sq^j 
   = 
  \sum_{0 \leq k \leq i/2}
  \left(
    {
      { j - k - 1 }
      \atop 
      { i - 2k }
    }
  \right)_{mod 2}
  Sq^{i + j -k} \circ Sq^k
\end{displaymath}
for all $0 \lt i \lt 2 j$.

Here $\left( a \atop b \right) \coloneqq 0$ if $a \lt b$.

\end{prop}
\begin{example}
\label{CompositionWithSq1}\hypertarget{CompositionWithSq1}{}
\textbf{([[Adem relation]] for [[postcomposition]] with the [[Bockstein homomorphism]] $Sq^1 = \beta$)}

For $j \geq 2$ and $i =1$, the [[Adem relations]] (prop. \ref{AdemRelations}) say that:

\begin{displaymath}
\begin{aligned}
    Sq^1 \circ Sq^j 
    & = 
    \underset{
      (j-1)_{mod 2}
    }{
    \underbrace{
    \left(
      {
        {j - 1 }
        \atop
        1
      }     
    \right)_{mod 2}
    }}
    Sq^{j + 1} 
    \\   
    & =
    \left\{
      \itexarray{
        Sq^{j+1} &\vert& j \, \text{even}
        \\
        0 &\vert& j \, \text{odd}
      }
    \right.
  \end{aligned}
\end{displaymath}
\end{example}
This gives rise to:

\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{CompositionWithSq1} and by \href{Bockstein+homomorphism#Mod2BocksteinIntoMod2Cohomology}{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}
in that we have

\begin{displaymath}
\itexarray{
    Sq^{2n+1}_{\mathbb{Z}}
    &\colon&
    B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z})
      &\overset{Sq^{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{Sq^{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}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{HopfInvariant}{}\subsubsection*{{Hopf invariant}}\label{HopfInvariant}

\begin{prop}
\label{}\hypertarget{}{}
For $\phi \colon S^{k+n-1} \to S^k$, a map of [[spheres]], the Steenrod square

\begin{displaymath}
Sq^n 
    \colon 
   H^k(cofib(\phi), \mathbb{F}_2) 
    \longrightarrow 
   H^{k+n}(cofib(\phi),\mathbb{F}_2)
\end{displaymath}
(on the [[homotopy cofiber]] $cofib(\phi)\simeq S^k \underset{S^{k+n-1}}{\cup} D^{k+n}$)

is non-vanishing exactly for $n \in \{1,2,4,8\}$.

\end{prop}
(\href{Hopf+invariant+one#Adams60}{Adams 60, theorem 1.1.1}).

See at \emph{[[Hopf invariant one theorem]]}.

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

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


\item [[Wu class]]


\item [[Steenrod algebra]]



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

The operations were first defined in

\begin{itemize}%
\item [[Norman Steenrod]], \emph{Products of cocycles and extensions of mappings}, Annals of mathematics (1947)

\end{itemize}
The axiomatic definition appears in

\begin{itemize}%
\item [[Norman Steenrod]], [[David Epstein]], \emph{Cohomology operations}, Annals of mathematics studies, Princeton University Press (1962)

\end{itemize}
Lecture notes on Steenrod squares and the [[Steenrod algebra]] include

\begin{itemize}%
\item [[Jacob Lurie]], 18.917 \emph{\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007}{Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007}}. (MIT OpenCourseWare: Massachusetts Institute of Technology), \emph{\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/}{Lecture notes}}

Lecture 2 \emph{Steenrod operations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf}{pdf})

Lecture 3 \emph{Basic properties of Steenrod operations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture3.pdf}{pdf})

Lecture 4 \emph{The Adem relations} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture4.pdf}{pdf})

Lecture 5 \emph{The Adem relations (cont.)} (\href{http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture5.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[Wen-Tsun Wu]], \emph{Sur les puissances de Steenrod}, Colloque de Topologie de Strasbourg, IX, La Biblioth\`e{}que Nationale et Universitaire de Strasbourg, (1952)

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

[[!redirects Steenrod operation]] [[!redirects Steenrod operations]]

[[!redirects stable cohomology operation]] [[!redirects stable cohomology operations]]



\end{document}