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\begin{document}

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\section*{Kronecker pairing}

\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{UniversalCoefficientTheorem}{Universal coefficient theorem}\dotfill \pageref*{UniversalCoefficientTheorem} \linebreak
\noindent\hyperlink{pairing_on_atiyahhirzebruch_spectral_sequences}{Pairing on Atiyah-Hirzebruch spectral sequences}\dotfill \pageref*{pairing_on_atiyahhirzebruch_spectral_sequences} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[ordinary homology]]/[[ordinary cohomology]] represented as [[singular homology]]/[[singular cohomology]], then \emph{Kronecker pairing} refers to the defining pairing of a [[chain]] with a [[cochain]].

More generally, in [[generalized (Eilenberg-Steenrod) cohomology]]/[[generalized homology]] represented by a [[ring spectrum]] $E$, then the \emph{Kronecker pairing} is a canonical pairing of the $E$-[[generalized cohomology]] [[cohomology group|groups]] $E^\bullet(X)$ with the $E$-[[generalized homology]] groups $E_\bullet(X)$ of suitable spaces ([[homotopy types]]/[[spectra]]) $X$

\begin{displaymath}
\langle-,-\rangle_X 
    \;\colon\; 
   E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E)
  \,.
\end{displaymath}
The combination of the Kronecker pairing with a [[diagonal]] map yields the [[cap product]] pairing in generalized (co-)homology.

If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

\begin{displaymath}
E^0(X) \longrightarrow Hom_{\pi_\bullet(E)}(E_\bullet(X), \pi_\bullet(E))
\end{displaymath}
of the Kronecker pairing is an [[isomorphism]] and hence exhibits $E$-[[generalized cohomology]] as the $\pi_\bullet(E)$-[[linear dual]] of the $E$-[[generalized homology]] of $X$; an instance of a [[universal coefficient theorem]] for generalized (co-)homology (prop. \ref{KroneckerPairingAdjunctIsIsomorphism} below).

On [[CW-complexes]] $X$ of [[finite number|finite]] [[dimension]], the Kronecker pairing induces a pairing of the corresponding [[Atiyah-Hirzebruch spectral sequences]] (prop. \ref{KroneckerPairingOnAHSS} below).

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

Let $E$ be a [[ring spectrum]] with product denoted $\mu \colon E \wedge E \longrightarrow E$. Let $X,Y$ be any [[spectra]].

\begin{defn}
\label{KroneckerPairing}\hypertarget{KroneckerPairing}{}
Given $[f] \in E^k(X)$ with representative $f \colon X \longrightarrow \Sigma^k E$ and given $[w] \in E_{n+k}(X \wedge Y)$ with representative $w \colon S^{n+k} \longrightarrow E \wedge X \wedge Y$, then their \emph{Kronecker pairing} is the element

\begin{displaymath}
\langle f,w\rangle \in E_n(Y)
\end{displaymath}
represented by the composite

\begin{displaymath}
S^{k+n} 
    \stackrel{w}{\longrightarrow}
  E\wedge X \wedge Y
    \stackrel{id_E \wedge f \wedge id_Y}{\longrightarrow}
  E \wedge \Sigma^k E \wedge Y
   \stackrel{\Sigma^k \mu \wedge id_Y}{\longrightarrow}
  \Sigma^k E \wedge Y
  \,.
\end{displaymath}
This yields a homomorphism of [[graded abelian groups]]

\begin{displaymath}
\langle-,-\rangle
   \;\colon\;
  E^{\bullet_1}(X)
   \otimes
  E_{\bullet_2}(X \wedge Y)
   \longrightarrow
  E_{\bullet_2-\bullet_1}(Y)
  \,.
\end{displaymath}
(and similarly for $Y$ on the other side\ldots{})

For $Y = \mathbb{S}$ this is

\begin{displaymath}
\langle-,-\rangle
   \;\colon\;
  E^{\bullet_1}(X)
   \otimes
  E_{\bullet_2}(X)
   \longrightarrow
  \pi_{\bullet_2-\bullet_1}(E)
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{Kochmann96}{Kochmann 96, (4.2.1)}, \hyperlink{Schwede12}{Schwede 12, construction 6.13})

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

\hypertarget{UniversalCoefficientTheorem}{}\subsubsection*{{Universal coefficient theorem}}\label{UniversalCoefficientTheorem}

\begin{prop}
\label{KroneckerPairingAdjunctIsIsomorphism}\hypertarget{KroneckerPairingAdjunctIsIsomorphism}{}
If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

\begin{displaymath}
\pi_0[X, E \wedge Y]
  \longrightarrow
  Hom_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) )
\end{displaymath}
\begin{displaymath}
f \mapsto \langle f,-\rangle
\end{displaymath}
of the Kronecker pairing, def. \ref{KroneckerPairing}, is an [[isomorphism]].

\end{prop}
(e.g. \hyperlink{Schwede12}{Schwede 12, chapter II, prop.6.20})

\begin{proof}
By the formula for [[adjuncts]], the morphism factors through the [[free-forgetful adjunction]] for $E$-[[module spectra]]

\begin{displaymath}
\pi_0[X, E \wedge Y]
    \stackrel{\simeq}{\longrightarrow}
  \pi_0[E\wedge X, E \wedge Y]_{E Mod}
    \stackrel{\pi_\bullet}{\longrightarrow}
  Hom_{\pi_\bullet}( E_\bullet(X), E_\bullet(Y) )
  \,.
\end{displaymath}
Hence one is reduced to showing that under the given conditions the second morphis is an iso. (\ldots{})

\end{proof}
This may be regarded as a [[universal coefficient theorem]] (\hyperlink{Adams74}{Adams 74, part III, around prop. 13.5}).

For $Y = \mathbb{S}$ prop. \ref{KroneckerPairingAdjunctIsIsomorphism} gives:

\begin{example}
\label{}\hypertarget{}{}
If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

\begin{displaymath}
E^0(X)
  \longrightarrow
  Hom_{\pi_\bullet(E)}( E_\bullet(X), \pi_\bullet(E) )
\end{displaymath}
\begin{displaymath}
f \mapsto \langle f,-\rangle
\end{displaymath}
of the Kronecker pairing, def. \ref{KroneckerPairing}, is an [[isomorphism]].

\end{example}
\hypertarget{pairing_on_atiyahhirzebruch_spectral_sequences}{}\subsubsection*{{Pairing on Atiyah-Hirzebruch spectral sequences}}\label{pairing_on_atiyahhirzebruch_spectral_sequences}

\begin{prop}
\label{KroneckerPairingOnAHSS}\hypertarget{KroneckerPairingOnAHSS}{}
For $E$ a [[ring spectrum]] and $X$ a [[CW complex]] of [[finite number|finite]] [[dimension]], then the Kronecker pairing $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ of def. \ref{KroneckerPairing} passes to a page-wise pairing of the corresponding [[Atiyah-Hirzebruch spectral sequences]] for $E$-cohomology/homology

\begin{displaymath}
\langle-,-\rangle_r
  \;\colon\;
  \mathcal{E}_r^{n,-s}
  \otimes
  \mathcal{E}^r_{n,t}
  \longrightarrow
  \pi_{s+t}(E)
\end{displaymath}
such that

\begin{enumerate}%
\item on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for [[ordinary cohomology]]/[[ordinary homology]] with [[coefficients]] in $\pi_\bullet(E)$;


\item the [[differentials]] act as [[derivations]]

\begin{displaymath}
\langle d_r(-),-\rangle = \langle -, d^r(-)\rangle
  \,,
\end{displaymath}

\item The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.



\end{enumerate}
\end{prop}
(\hyperlink{Kochmann96}{Kochmann 96, prop. 4.2.10})

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

Presumeably named after [[Leopold Kronecker]].

\begin{itemize}%
\item [[Frank Adams]], Part III, section 13 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Stanley Kochman]], (4.2.1) and prop. 4.2.10 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Stefan Schwede]], chapter II, section 6 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})



\end{itemize}


\end{document}