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\begin{document}

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\section*{reduced cohomology}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - 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{brown_representability}{Brown representability}\dotfill \pageref*{brown_representability} \linebreak
\noindent\hyperlink{relation_to_unreduced_cohomology}{Relation to unreduced cohomology}\dotfill \pageref*{relation_to_unreduced_cohomology} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{ReducedGeneralizedCohomology}\hypertarget{ReducedGeneralizedCohomology}{}
A \textbf{reduced [[cohomology theory]]} is a [[functor]]

\begin{displaymath}
\tilde E^\bullet 
   \;\colon\; 
  (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}
\end{displaymath}
from the [[opposite category|opposite]] of [[pointed topological spaces]] ([[CW-complexes]]) to $\mathbb{Z}$-[[graded abelian groups]] (``[[cohomology groups]]''), in components

\begin{displaymath}
\tilde E 
    \;\colon\; 
  (X \stackrel{f}{\longrightarrow} Y)
    \mapsto
  (\tilde E^\bullet(Y) 
    \stackrel{f^\ast}{\longrightarrow}
  \tilde E^\bullet(X))
  \,,
\end{displaymath}
and equipped with a [[natural isomorphism]] of degree +1, to be called the \textbf{[[suspension isomorphism]]}, of the form

\begin{displaymath}
\sigma
    \;\colon\;
  \tilde E^{\bullet +1}(\Sigma -) 
    \overset{\simeq}{\longrightarrow} 
  \tilde E^\bullet(-)
\end{displaymath}
such that:

\begin{enumerate}%
\item \textbf{([[homotopy invariance]])} If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) [[homotopy]] $f_1 \simeq f_2$ between them, then the induced [[homomorphisms]] of abelian groups are [[equality|equal]]

\begin{displaymath}
f_1^\ast = f_2^\ast
  \,.
\end{displaymath}

\item \textbf{(exactness)} For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced [[mapping cone]], then this gives an [[exact sequence]] of graded abelian groups

\begin{displaymath}
\tilde E^\bullet(Cone(i)) 
\overset{j^\ast}{\longrightarrow} 
\tilde E^\bullet(X)
\overset{i^\ast}{\longrightarrow}
\tilde E^\bullet(A)
\,.
\end{displaymath}


\end{enumerate}
We say $\tilde E^\bullet$ is \textbf{additive} if in addition

\begin{itemize}%
\item \textbf{([[wedge axiom]])} For $\{X_i\}_{i \in I}$ any set of pointed CW-complexes, then the canonical comparison morphism

\begin{displaymath}
\tilde E^\bullet(\vee_{i \in I} X_i) 
   \longrightarrow
  \prod_{i \in I} \tilde E^\bullet(X_i)
\end{displaymath}
is an [[isomorphism]], from the functor applied to their [[wedge sum]], example \ref{WedgeSumAsCoproduct}, to the [[product]] of its values on the wedge summands, .



\end{itemize}
We say $\tilde E^\bullet$ is \textbf{ordinary} if its value on the [[0-sphere]] $S^0$ is concentrated in degree 0:

\begin{itemize}%
\item \textbf{(Dimension)} $\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

\end{itemize}
A [[homomorphism]] of reduced cohomology theories

\begin{displaymath}
\eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet
\end{displaymath}
is a [[natural transformation]] between the underlying functors which is compatible with the suspension isomorphisms in that all the following [[commuting square|squares commute]]

\begin{displaymath}
\itexarray{
    \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}&  \tilde F^\bullet(X)
    \\
    {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}}
    \\
    \tilde E^{\bullet + 1}(\Sigma X) 
    &\overset{\eta_{\Sigma X}}{\longrightarrow}&
    \tilde F^{\bullet + 1}(\Sigma X)
  }
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{AGP02}{AGP 02, def. 12.1.4})

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

\hypertarget{brown_representability}{}\subsubsection*{{Brown representability}}\label{brown_representability}

The [[Brown representability theorem]] says that for any reduced cohomology theory $\tilde E^\bullet$ there is an [[Omega-spectrum]] $E$ which [[representable functor|represents]] $\tilde E^\bullet$ on pointed connected CW-complex $X$, in that

\begin{displaymath}
\tilde E^n(X) \simeq [X,E_n]_\ast
  \,.
\end{displaymath}
\hypertarget{relation_to_unreduced_cohomology}{}\subsubsection*{{Relation to unreduced cohomology}}\label{relation_to_unreduced_cohomology}

For an unreduced [[cohomology theory]] $E^\bullet$ the induced \textbf{reduced cohomology} is

\begin{displaymath}
\tilde E^k(X,x_0) \coloneqq E^k(X,\{x_0\}) = ker(H^k(X)\to H^k(\{x_0\}))
\end{displaymath}
e.g. \hyperlink{AGP02}{AGP 02, theorem 12.1.12}

For more see at \emph{\href{generalized+Eilenberg-Steenrod+cohomology#RelationBetweenReducedAndUnreduced}{generalized cohomology -- Relation btween reduced and unreduced}}.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[reduced homology]]

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

See the references at \emph{[[generalized (Eilenberg-Steenrod) cohomology]]}.

\begin{itemize}%
\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})


\item [[Jacob Lurie]], \emph{[[A Survey of Elliptic Cohomology - cohomology theories]]}



\end{itemize}
[[!redirects reduced cohomology theory]]



\end{document}