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\begin{document}

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\section*{suspension spectrum}

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

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{completion_to_an_spectrum}{Completion to an $\Omega$-spectrum}\dotfill \pageref*{completion_to_an_spectrum} \linebreak
\noindent\hyperlink{relation_to_looping_and_stabilization}{Relation to looping and stabilization}\dotfill \pageref*{relation_to_looping_and_stabilization} \linebreak
\noindent\hyperlink{StrongMonoidalness}{Strong monoidalness}\dotfill \pageref*{StrongMonoidalness} \linebreak
\noindent\hyperlink{recognition_and_diagonals}{Recognition and diagonals}\dotfill \pageref*{recognition_and_diagonals} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $X$ a [[pointed object|pointed]] [[topological space]], its \textbf{suspension spectrum} $\Sigma^\infty X$ is the [[spectrum]] given by the pre-spectrum whose degree-$n$ space is the $n$-fold [[reduced suspension]] of $X$:

\begin{displaymath}
(\Sigma^\infty X)_n = \Sigma^n X
  \,.
\end{displaymath}
(e.g. \hyperlink{ElmendorfKrizMay}{Elmendorf-Kriz-May, example 1.1})

As a [[symmetric spectrum]]: (\hyperlink{Schwede12}{Schwede 12, example I.2.6})

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

\hypertarget{completion_to_an_spectrum}{}\subsubsection*{{Completion to an $\Omega$-spectrum}}\label{completion_to_an_spectrum}

See at \href{Omega-spectrum#CompletionOfSuspensionSpectra}{Omega spectrum -- Completion of a suspension spectrum}.

\hypertarget{relation_to_looping_and_stabilization}{}\subsubsection*{{Relation to looping and stabilization}}\label{relation_to_looping_and_stabilization}

As an [[infinity-functor]] $\Sigma^\infty\colon Top_* \to Spec$ the suspension spectrum functor exhibits the [[stabilization]] of [[Top]].

\begin{displaymath}
(\Sigma^\infty \dashv \Omega^\infty)\colon 
  Top_* \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}}
  Spec
\end{displaymath}
\hypertarget{StrongMonoidalness}{}\subsubsection*{{Strong monoidalness}}\label{StrongMonoidalness}

The suspension spectrum functor is [[strong monoidal functor|strong monoidal]].

On the one hand, this is the case for its incarnation as a [[1-functor]] with values in [[structured spectra]] (\href{Introduction+to+Stable+homotopy+theory+--+1-2#SmashProductOfFreeSpectra}{this Prop.}) Via the corresponding [[symmetric monoidal smash product of spectra|symetric monoidal]] [[Model categories of diagram spectra|model structure on structured spectra]] this exhibits strong monoidalness also as an [[(infinity,1)-functor]].

More abstractly this follows from general properties of [[stabilization]] when regarding [[stable homotopy theory]] as the result of inverting [[smash product]] with the [[circle]], via \href{stabilization#Robalo12}{Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1)}. For emphasis see also \href{stabilization#Hoyois15}{Hoyois 15, section 6.1}, specifically \href{stabilization#Hoyois15}{Hoyois 15, Def. 6.1}.

\hypertarget{recognition_and_diagonals}{}\subsubsection*{{Recognition and diagonals}}\label{recognition_and_diagonals}

(\ldots{})

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

\begin{itemize}%
\item [[free structured spectrum]]


\item [[equivariant suspension spectrum]]



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

\begin{itemize}%
\item [[Frank Adams]], part III, section 2 of \emph{[[Stable homotopy and generalised homology]]}, 1974


\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], example 1.1 of \emph{[[Modern foundations for stable homotopy theory]]}, in [[Ioan Mackenzie James]], \emph{[[Handbook of Algebraic Topology]]}, Amsterdam: North-Holland (1995) pp. 213--253, (\href{http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf}{pdf})


\item Nicholas J. Kuhn, \emph{Suspension spectra and homology equivalences}, Trans. Amer. Math. Soc. 283, 303--313 (1984) (\href{http://www.jstor.org/stable/2000005}{JSTOR})


\item [[John Klein]], \emph{Moduli of suspension spectra} (\href{http://arxiv.org/abs/math/0210258}{arXiv:math/0210258}, \href{http://mathoverflow.net/a/84686/381}{MO})


\item [[Stefan Schwede]], Example I.2.6 in \emph{Symmetric spectra}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec.pdf}{pdf})



\end{itemize}
Suspension spectra of [[infinite loop spaces]] are discussed (in a context of [[Goodwillie calculus]] and [[chromatic homotopy theory]]) in

\begin{itemize}%
\item Nicholas J. Kuhn, section 6.2 of \emph{Goodwillie towers and chromatic homotopy: An overview} (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/KuhnKinosaki.pdf}{pdf})

\end{itemize}
[[!redirects suspension spectrum]] [[!redirects suspension spectra]]



\end{document}