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\begin{document}

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\section*{multiplicative cohomology theory}

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

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

[[!include cohomology - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RingAndModuleStructureOnCohomologyGroups}{Ring and module structure on cohomology groups}\dotfill \pageref*{RingAndModuleStructureOnCohomologyGroups} \linebreak
\noindent\hyperlink{multiplicative_atiyahhirzebruch_spectral_sequences}{Multiplicative Atiyah-Hirzebruch spectral sequences}\dotfill \pageref*{multiplicative_atiyahhirzebruch_spectral_sequences} \linebreak
\noindent\hyperlink{BrownRepresentabilityByRingSpectra}{Brown representability by ring spectra}\dotfill \pageref*{BrownRepresentabilityByRingSpectra} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[cohomology theory]] $E$ is called \emph{multiplicative} if each [[graded module|graded]] [[abelian group|abelian]] $E$-[[cohomology group]] $E^\bullet(X)$ is compatibly equippd with the structure of a [[graded ring]].

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

\begin{defn}
\label{PairingOfUnreducedCohomologyTheories}\hypertarget{PairingOfUnreducedCohomologyTheories}{}
Let $E_1, E_2, E_3$ be three unreduced [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology theories]] (\href{generalized+cohomology+theory#GeneralizedCohomologyTheory}{def.}). A \textbf{pairing of cohomology theories}

\begin{displaymath}
\mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3
\end{displaymath}
is a [[natural transformation]] (of functors on $(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op}$) of the form

\begin{displaymath}
\mu_{n_1,n_2}
   \;\colon\;
  E_1^{n_1}(X,A)
    \otimes
  E_2^{n_2}(Y,B)
    \longrightarrow
  E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)
\end{displaymath}
such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are [[commuting squares]]

\begin{displaymath}
\itexarray{
    E_1^{n_1}(A)
      \otimes 
    E_2^{n_2}(Y,B)
      &\overset{\delta_1 \otimes id_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y  \cup  X \times B , X \times B)}
     {E_3^{n_1 + n_2}(A \times Y, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
\end{displaymath}
and

\begin{displaymath}
\itexarray{
    E_1^{n_1}(X,A)
      \otimes 
    E_2^{n_2}(B)
      &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}&
    E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow 
      && 
    \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}}
    \\
   \underoverset
     {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)}
     {E_3^{n_1 + n_2}(X \times B, A \times B)}
     {\simeq}
     &\overset{\delta_3}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X \times Y, A \times B)
  }
  \,,
\end{displaymath}
where the isomorphisms in the bottom left are the \href{generalized+cohomology+theory##excision}{excision isomorphisms}.

\end{defn}
\begin{defn}
\label{MultiplicativeCohomologyTheory}\hypertarget{MultiplicativeCohomologyTheory}{}
An (unreduced) \textbf{multiplicative cohomology theory} is an unreduced [[generalized cohomology theory]] theory $E$ (\href{generalized+cohomology+theory#GeneralizedCohomologyTheory}{def.}) equipped with

\begin{enumerate}%
\item (external multiplication) a pairing (def. \ref{PairingOfUnreducedCohomologyTheories}) of the form $\mu \;\colon\; E \Box E  \longrightarrow E$;


\item (unit) an element $1 \in E^0(\ast)$



\end{enumerate}
such that

\begin{enumerate}%
\item ([[associativity]]) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;


\item ([[unitality]]) $\mu(1\otimes x) = \mu(x \otimes 1) = x$ for all $x \in E^n(X,A)$.



\end{enumerate}
The mulitplicative cohomology theory is called \textbf{commutative} (often considered by default) if in addition

\begin{itemize}%
\item \textbf{(graded commutativity)}

\begin{displaymath}
\itexarray{
    E^{n_1}(X,A) \otimes E^{n_2}(Y,B)
    &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}& 
    E^{n_2}(Y,B) \otimes E^{n_1}(X,A)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}}
    \\
    E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B)
    &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}&
    E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A)
  }
  \,.
\end{displaymath}


\end{itemize}
 Given a multiplicative cohomology theory $(E, \mu, 1)$, its \textbf{[[cup product]]} is the composite of the above external multiplication with pullback along the [[diagonal]] maps $\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A)$;

\begin{displaymath}
(-) \cup (-)
   \;\colon\;
  E^{n_1}(X,A)
   \otimes
  E^{n_2}(X,A)
    \overset{\mu_{n_1,n_2}}{\longrightarrow}
  E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A)
    \overset{\Delta^\ast_{(X,A)}}{\longrightarrow}
  E^{n_1 + n_2}(X, \; A \cup B)
  \,.
\end{displaymath}
\end{defn}
e.g. (\hyperlink{TamakiKono06}{Tamaki-Kono 06, II.6})

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

\hypertarget{RingAndModuleStructureOnCohomologyGroups}{}\subsubsection*{{Ring and module structure on cohomology groups}}\label{RingAndModuleStructureOnCohomologyGroups}

\begin{prop}
\label{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}\hypertarget{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}{}
Let $(E,\mu,1)$ be a multiplicative cohomology theory, def. \ref{MultiplicativeCohomologyTheory}. Then

\begin{enumerate}%
\item For every space $X$ the \hyperlink{InternalMultiplicationOfMultiplicativeCohomologyTheory}{cup product} gives $E^\bullet(X)$ the structure of a $\mathbb{Z}$-[[graded ring]], which is graded-commutative if $(E,\mu,1)$ is commutative.


\item For every pair $(X,A)$ the external multiplication $\mu$ gives $E^\bullet(X,A)$ the structure of a left and right [[module]] over the graded ring $E^\bullet(\ast)$.


\item All pullback morphisms respect the left and right action of $E^\bullet(\ast)$ and the connecting homomorphisms respect the right action and the left action up to multiplication by $(-1)^{n_1}$



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the third point:

For pullback maps this is the [[natural transformation|naturality]] of the external product: let $f \colon (X,A) \longrightarrow (Y,B)$ be a morphism in $Top_{CW}^{\hookrightarrow}$ then naturality says that the following square commutes:

\begin{displaymath}
\itexarray{
    E^{n_1}(\ast) \otimes E^{n_2}(Y,B)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y, B)
    \\
    {}^{\mathllap{(id,f^\ast)}}\downarrow && \downarrow^{\mathrlap{f^\ast}}
    \\
    E^{n_1}(\ast) \otimes E^{n_2}(X,A)
    &\overset{\mu_{n_1,n_2}}{\longrightarrow}&
    E^{n_1 + n_2}(Y,B)
  }
  \,.
\end{displaymath}
For connecting homomorphisms this is the (graded) commutativity of the squares in def. \ref{MultiplicativeCohomologyTheory}:

\begin{displaymath}
\itexarray{
    E^{n_1}(\ast)\otimes E^{n_2}(A)
      &\overset{(-1)^{n_1} (id, \delta)}{\longrightarrow}&
    E^{n_1}(\ast) \otimes E^{n_2 + 2}(X)
    \\
    {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}}
    \\
    E^{n_1 + n_2}(A)     
      &\overset{\delta}{\longrightarrow}&
     E_3^{n_1 + n_2+ 1}(X,B)
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{multiplicative_atiyahhirzebruch_spectral_sequences}{}\subsubsection*{{Multiplicative Atiyah-Hirzebruch spectral sequences}}\label{multiplicative_atiyahhirzebruch_spectral_sequences}

\begin{prop}
\label{AHSSForMultiplicativeCohomologyTheoryIsMultiplicative}\hypertarget{AHSSForMultiplicativeCohomologyTheoryIsMultiplicative}{}
Given a multiplicative cohomology theory $(A,\mu,1)$ (def. \ref{MultiplicativeCohomologyTheory}), then for every [[Serre fibration]] $X \to B$ the corresponding [[Atiyah-Hirzebruch spectral sequence]] inherits the structure of a [[multiplicative spectral sequence]].

\end{prop}
A proof of prop. \ref{AHSSForMultiplicativeCohomologyTheoryIsMultiplicative} via [[Cartan-Eilenberg systems]] is given at \emph{[[multiplicative spectral sequence]]}. A proof arguing via [[Brown representability theorem|representing]] [[ring spectra]] is in (\hyperlink{Kochman96}{Kochman 96, prop. 4.2.9}).

\begin{prop}
\label{LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory}\hypertarget{LinearityOfDifferentialsInSerreAHSSForMultiplicativeCohomologyTheory}{}
Given a multiplicative cohomology theory $(A,\mu,1)$ (def. \ref{MultiplicativeCohomologyTheory}), then for every [[Serre fibration]] $X \to B$ all the differentials in the corresponding [[Atiyah-Hirzebruch spectral sequence]]

\begin{displaymath}
H^\bullet(B,A^\bullet(F)) 
  \;\Rightarrow\;
  A^\bullet(X)
\end{displaymath}
are linear over $A^\bullet(\ast)$.

\end{prop}
\begin{proof}
By construction (\href{Atiyah&#8211;Hirzebruch+spectral+sequence#ConstructionByFilteringTheBaseSpace}{here}) the differentials are those induced by the [[exact couple]]

\begin{displaymath}
\itexarray{
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    &&
      \stackrel{}{\longrightarrow} 
    &&
    \underset{s,t}{\prod} A^{s+t}(X_{s})
    \\
    & \nwarrow && \swarrow
    \\
    && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1})
  }
  \;\;\;\;\;\;\;
  \left(
      \itexarray{
        A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1})
        \\
        \uparrow && \downarrow_{\mathrlap{\delta}}
        \\
        A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1})
      }
  \right)
  \,.
\end{displaymath}
consisting of the pullback homomorphisms and the connecting homomorphisms of $A$.

By the nature of spectral sequences induced from exact couples (\href{exact+couple#CohomologicalSpectralSequenceOfAnExactCouple}{this prop.}) its differentials on page $r$ are the composites of one pullback homomorphism, the preimage of $(r-1)$ pullback homomorphisms, and one connecting homomorphism of $A$. Hence the statement follows with prop. \ref{RingAndModuleStructureOnCohomologyGroupsOfMultiplicativeCohomplogyTheory}.

\end{proof}
\hypertarget{BrownRepresentabilityByRingSpectra}{}\subsubsection*{{Brown representability by ring spectra}}\label{BrownRepresentabilityByRingSpectra}

A \emph{multiplicative structure} on a [[generalized (Eilenberg-Steenrod) cohomology]] theory is the structure of a [[ring spectrum]] on the [[spectrum]] that [[Brown representability theorem|represents]] it.

e.g. (\hyperlink{TamakiKono06}{Tamaki-Kono 06, appendix C}, \hyperlink{Lurie10}{Lurie 10, lecture 4})

In particular every [[E-∞ ring]] is a [[ring spectrum]], hence represents a multiplicative cohomology theory, but the converse is in general false.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[ordinary cohomology]] with [[coefficients]] in a [[ring]], in particular [[integral cohomology]]


\item [[topological K-theory]], [[K-theory]]


\item [[cobordism cohomology theory]]


\item [[tmf]]


\item etc\ldots{}.



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Kronecker pairing]]


\item [[universal coefficient theorem]], [[module spectrum|module cohomology theories]], \ldots{}



\end{itemize}
See also

\begin{itemize}%
\item [[complex oriented cohomology theory]], [[chromatic homotopy theory]]


\item [[Thom isomorphism]]


\item [[orientation in generalized cohomology]]


\item [[fiber integration]]


\item [[K-theory of a bipermutative category]]


\item [[multiplicative spectral sequence]]



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

\begin{itemize}%
\item [[Dai Tamaki]], [[Akira Kono]], chapter 2.6 of \emph{Generalized Cohomology}, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([[GeneralizedCohomology.pdf:file]])


\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology - cohomology theories]]}


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]} 2010, Lecture 4 \emph{[[complex oriented cohomology theory|Complex-oriented cohomology theories]]} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item [[Stanley Kochman]], \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996

\end{itemize}
[[!redirects multiplicative cohomology theories]]

[[!redirects multiplicative cohomology]]



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