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\begin{document}

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\section*{suspension isomorphism}

\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{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[cohomology]] of [[topological spaces]]/[[homotopy types]], the \emph{suspension isomorphism} identifies the degree-$n$ [[reduced cohomology]] of a [[pointed topological space|pointed]] space $X$ with the degree-$(n+k)$ cohomology of its $k$-fold [[suspension]], hence of its [[smash product]] with the $k$-[[sphere]]

\begin{displaymath}
\tilde H^n(X,E) \simeq \tilde H^{n+k}(S^k \wedge X, E)
  \,.
\end{displaymath}
In particular this serves to express cohomology in negative degree in terms of cohomology in non-negative degree of suspended spaces.

Requiring this to hold in [[equivariant cohomology theory]] not just for [[integer]] grading and spheres but also for [[RO(G)-grading]] and [[representation spheres]] leads to the concept of \emph{genuine} equivariant cohomology represented by [[genuine G-spectra]].

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

\begin{itemize}%
\item [[Freudenthal suspension theorem]]


\item [[Steenrod square]]


\item [[cohomology theory]]



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

Textbook accounts include

\begin{itemize}%
\item Akira Kono, Dai Tamaki, p. 19 in \emph{Generalized cohomology}, AMS 2002, esp. chapter 2 ([[GeneralizedCohomology.pdf:file]])

\end{itemize}
In [[equivariant homotopy theory]]:

\begin{itemize}%
\item [[Peter May]], chapter XI of \emph{Equivariant homotopy and cohomology theory}, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf}{pdf})

\end{itemize}
[[!redirects suspension isomorphisms]]



\end{document}