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\begin{document}

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\section*{logarithmic cohomology operation}

\begin{quote}%
under construction


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

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

[[!include higher algebra - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - 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{ActionOnCohomologyGroups}{Action on cohomology groups}\dotfill \pageref*{ActionOnCohomologyGroups} \linebreak
\noindent\hyperlink{explicit_formula_in_terms_of_power_operations}{Explicit formula in terms of power operations}\dotfill \pageref*{explicit_formula_in_terms_of_power_operations} \linebreak
\noindent\hyperlink{relation_to_the_stringorientation_of_}{Relation to the string-orientation of $tmf$}\dotfill \pageref*{relation_to_the_stringorientation_of_} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

In analogy to how in ordinary [[algebra]] the natural [[logarithm]] of [[positive number|positive]] [[rational numbers]] is a [[group]] [[homomorphism]] from the [[group of units]] to the completion of the rationals by the (additive) [[real numbers]]

\begin{displaymath}
log \;\colon\; \mathbb{Q}^\times_{\gt 0}\longrightarrow \mathbb{R}
\end{displaymath}
so in [[higher algebra]] for $E$ an [[E-∞ ring]] there is a natural homomorphism

\begin{displaymath}
\ell_{n,p} \;\colon\; gl_1(E) \longrightarrow L_{K(n)} E
\end{displaymath}
from the [[∞-group of units]] of $E$ to the [[K(n)-local spectrum]] obtained from $E$ (see \hyperlink{Rezk06}{Rezk 06, section 1.7}).

On the [[cohomology theory]] [[Brown representability theorem|represented]] by $E$ this induces a [[cohomology operation]] called, therefore, the ``logarithmic cohomology operation''.

More in detail, for $X$ the [[homotopy type]] of a [[topological space]], then the [[cohomology]] [[Brown representability theorem|represented]] by $gl_1(E)$ in degree 0 is the ordinary [[group of units]] in the [[cohomology ring]] of $E$:

\begin{displaymath}
H^0(X, gl_1(E))  \simeq (E^0(X))^\times
  \,.
\end{displaymath}
In positive degree the canonical map of pointed homotopy types $GL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E$ is in fact an [[isomorphism]] on all [[homotopy groups]]

\begin{displaymath}
\pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E
  \,.
\end{displaymath}
On cohomology elements this map

\begin{displaymath}
\pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq  (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times
\end{displaymath}
is [[logarithm]]-like, in that it sends $1 + x \mapsto x$.

But there is not a homomorphism of [[spectra]] of this form. This only exists after [[K(n)-local stable homotopy theory|K(n)-localization]], and that is the logarithmic cohomology operation.

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

By the [[Bousfield-Kuhn construction]] there is an [[equivalence]] of spectra

\begin{displaymath}
L_{K(n)} gl_1(E) \simeq L_{K(n)}E
\end{displaymath}
between the [[K(n)-local spectrum]] induced by the [[abelian ∞-group|abelian]] [[∞-group of units]] of $E$ (regarded as a [[connective spectrum]]) with that induced by $E$ itself. The logarithm on $E$ is the [[composition|composite]] of that with the [[Bousfield localization of spectra|localization map]]

\begin{displaymath}
\ell_{n,p}
  \;\colon\; 
 gl_1(E) \stackrel{}{\longrightarrow} L_{K(n)}gl_1(E) \stackrel{\simeq}{\to}  L_{K(n)} E
  \,.
\end{displaymath}
(see \hyperlink{Rezk06}{Rezk 06, section 3}).

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

\hypertarget{ActionOnCohomologyGroups}{}\subsubsection*{{Action on cohomology groups}}\label{ActionOnCohomologyGroups}

For every [[E-∞ ring]] $E$ and spaces $X$, [[prime number]] $p$ and [[natural number]] $n$, the logarith induces a homomorphism of [[cohomology groups]] of the form

\begin{displaymath}
\ell_{n,p}
   \;\colon\;
  (E^0(X))^\times
    \longrightarrow
  (L_{K(n)}E)^0(X)
  \,.
\end{displaymath}
\hypertarget{explicit_formula_in_terms_of_power_operations}{}\subsubsection*{{Explicit formula in terms of power operations}}\label{explicit_formula_in_terms_of_power_operations}

Under some conditions there is an explicit formula of the logarithmic cohomology operation by a [[series]] of [[power operations]].

Let $E$ be a [[K(n)-local stable homotopy theory|K(1)-local]] [[E-∞ ring]] such that

\begin{itemize}%
\item the [[kernel]] of $\pi_0 L_{K(1)}\mathbb{S} \longrightarrow \pi_0 E$ contains the [[torsion subgroup]] of $\pi_0 L_{K(1)}\mathbb{S}$.

\end{itemize}
(This is for instance the case for $L_{K(1)}$[[tmf]]).

Then on a [[finite CW complex]] $X$ the logarithmic cohomology operation from \hyperlink{ActionOnCohomologyGroups}{above}

\begin{displaymath}
\ell_{1,p}\;\colon\; (E^0(X))^\times \longrightarrow E^0(X)
\end{displaymath}
is given by the [[series]]

\begin{displaymath}
\begin{aligned}
    \ell_{1,p} \colon x 
     & \mapsto 
     \left(
       1 - \frac{1}{p}\psi
     \right)
     log(x)
     \\
     & = \frac{1}{p} log \frac{x^p}{\psi(x)}
     \\
     & = \sum_{k=1}^\infty (-1)^k \frac{p^{k-1}}{k}\left( \frac{\theta(x)}{x^p}\right)^k
     \\
   \end{aligned}
\end{displaymath}
which [[convergence of a sequence|converges]] [[p-adic numbers|p-adically]].

(\hyperlink{Rezk06}{Rezk 06, theorem 1.9}, see also \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 4.5})

Here $\theta$ \ldots{}.

In the special case that $x = 1 + \epsilon$ with $\epsilon^2 = 0$ then this reduces to

\begin{displaymath}
\ell_{1,p}(1+ \epsilon)= \epsilon - \frac{1}{p}\psi(\epsilon)
   \,.
\end{displaymath}
\hypertarget{relation_to_the_stringorientation_of_}{}\subsubsection*{{Relation to the string-orientation of $tmf$}}\label{relation_to_the_stringorientation_of_}

The above expression in terms of power operations may be used to establish the [[string orientation of tmf]] (\hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10}).

\ldots{}

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

The logarithmic operation for $p$-complete [[K-theory]] was first described in

\begin{itemize}%
\item [[Tammo tom Dieck]], \emph{The Artin-Hasse logarithm for $\lambda$-rings}, Algebraic topology (Arcata, CA, 1986), 409--415, Lecture Notes in Math., 1370, Springer, Berlin, 1989.

\end{itemize}
The formulation in terms of the [[Bousfield-Kuhn functor]] and the expression in terms of [[power operations]] is due to

\begin{itemize}%
\item [[Charles Rezk]], \emph{The units of a ring spectrum and a logarithmic cohomology operation}, J. Amer. Math. Soc. 19 (2006), 969-1014 (\href{http://arxiv.org/abs/math/0407022}{arXiv:math/0407022})

\end{itemize}
The application of this to the [[string orientation of tmf]] is due to

\begin{itemize}%
\item [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf})

\end{itemize}
[[!redirects logarithmic cohomology operations]]



\end{document}