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\begin{document}

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\section*{Adams conjecture}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - 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{statements}{Statements}\dotfill \pageref*{statements} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesInEquivariantCohomology}{In equivariant cohomology}\dotfill \pageref*{ReferencesInEquivariantCohomology} \linebreak


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

The \emph{Adams conjecture} is a statement about triviality of [[spherical fibrations]] associated to certain formal differences of [[vector bundles]] ([[K-theory]] classes) via the [[J-homomorphism]]. The conjecture was stated in (\hyperlink{Adams63}{Adams 63, conjecture 1.2}), for vector bundles of rank up to two over a [[finite CW-complex]], which was proven in (\hyperlink{Adams63}{Adams 63, theorem 1.4}). A general proof was then given in (\href{Quillen71}{Quillen 71}).

The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the [[image of the J-homomorphism]].

\hypertarget{statements}{}\subsection*{{Statements}}\label{statements}

Let $X$ be (the [[homotopy type]] of) a [[topological space]]. For $V \;\colon\; X \longrightarrow B O$ classifying a real [[vector bundle]] on $X$, the corresponding [[spherical fibration]] is classified by the composite

\begin{displaymath}
J(V)
   \;\colon\;
  X \stackrel{V}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S})
\end{displaymath}
with the delooped [[J-homomorphism]]. This descends to a map from [[topological K-theory]] to [[spherical fibrations]].

Now for $L$ a [[line bundle]] on some $X$ and for non-vanishing $k \in \mathbb{Z}$, [[John Adams]] observed that the [[spherical fibration]] associated with the difference $L^{\otimes k} - L \in K O(X)$ has the property that some $k$-fold multiple of it has trivial spherical fibration, hence that there is $N \in \mathbb{N}$ for which

\begin{displaymath}
J\left(
    \oplus^{k^N} (L^{\otimes k} - L)
  \right)
  = 0
  \,.
\end{displaymath}
Noticing that $L \mapsto L^{\otimes^k} = \Psi^k(L)$ is the $k$th [[Adams operation]] on [[K-theory]] applied to the [[line bundle]] $L$, [[John Adams]] then conjectured that the above is true for all [[vector bundles]] $V$ in the form

\begin{displaymath}
J\left(
    \oplus^{k^N} (\Psi^k(V) - V)
  \right)
  = 0
  \,.
\end{displaymath}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[J-homomorphism]]


\item [[Schur index]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The conjecture originates in

\begin{itemize}%
\item [[John Adams]], \emph{On the groups $J(X)$ I: Topology}, 2 (1963) pp. 181--195

\end{itemize}
A quick review is for instance in

\begin{itemize}%
\item [[Akhil Mathew]], \emph{The Adams conjecture I} (\href{http://amathew.wordpress.com/2013/01/23/the-adams-conjecture-i/}{web})

\end{itemize}
The [[proof]] of the Adams conjecture is originally due to

\begin{itemize}%
\item [[Daniel Quillen]], \emph{The Adams conjecture}, Topology, vol 10, 1971 (\href{http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf}{pdf})

\end{itemize}
The proof using [[algebraic geometry]] is due to

\begin{itemize}%
\item [[Dennis Sullivan]], \emph{Genetics of homotopy theory and the Adams conjecture}, The Annals of Mathematics, Second Series, Vol. 100, No. 1 (Jul., 1974), pp. 1-79 (\href{http://www.jstor.org/stable/1970841}{JSTOR}, \href{http://math1.unice.fr/~cazanave/Gdt/ImJ/Sullivan.pdf}{pdf})

\end{itemize}
Yet another proof via [[Becker-Gottlieb transfer]] is due to

\begin{itemize}%
\item J. Becker, D. Gottlieb, \emph{The transfer map and fiber bundles} Topology , 14 (1975)

\end{itemize}
\hypertarget{ReferencesInEquivariantCohomology}{}\subsubsection*{{In equivariant cohomology}}\label{ReferencesInEquivariantCohomology}

The generalization to [[equivariant cohomology]] ([[equivariant K-theory]]) is discussed in

\begin{itemize}%
\item [[Tammo tom Dieck]], theorem 11.3.8 in \emph{[[Transformation Groups and Representation Theory]]} Lecture Notes in Mathematics 766 Springer 1979


\item Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of \emph{Equivariant algebraic K-theory}, \emph{Equivariant algebraic K-theory}, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (\href{http://math.uchicago.edu/~may/PAPERS/40.pdf}{pdf})


\item Henning Hauschild, [[Stefan Waner]], theorem 0.1 of \emph{The equivariant Dold theorem mod $k$ and the Adams conjecture}, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (\href{https://projecteuclid.org/euclid.ijm/1256065410}{euclid:1256065410})


\item Kuzuhisa Shimakawa, \emph{Note on the equivariant $K$-theory spectrum}, Publ. RIMS, Kyoto Univ. \textbf{29} (1993), 449-453 (\href{http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5}{pdf}, \href{https://doi.org/10.2977/prims/1195167052}{doi})


\item Christopher French, theorem 2.4 in \emph{The equivariant $J$–homomorphism for finite groups at certain primes}, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (\href{https://projecteuclid.org/euclid.agt/1513797069}{euclid:1513797069})



\end{itemize}
[[!redirects Adams conjectures]]

[[!redirects Adams' conjecture]] [[!redirects Adams' conjectures]]

[[!redirects equivariant Adams conjecture]] [[!redirects equivariant Adams conjectures]]



\end{document}