Editing Spectrum (topology) (section)

==The definition of a spectrum==

There are many variations of the definition: in general, a ''spectrum'' is any sequence <math>X_n</math> of pointed topological spaces or pointed simplicial sets together with the structure maps <math>S^1 \wedge X_n \to X_{n+1}</math>, where <math>\wedge</math> is the [[smash product]]. The smash product of a pointed space <math>X</math> with a circle is homeomorphic to the [[reduced suspension]] of <math>X</math>, denoted <math>\Sigma X</math>.

The following is due to [[Frank Adams]] (1974): a spectrum (or CW-spectrum) is a sequence <math>E:= \{E_n\}_{n\in \mathbb{N}} </math> of [[CW complex]]es together with inclusions <math> \Sigma E_n \to E_{n+1} </math> of the [[suspension (topology)|suspension]] <math> \Sigma E_n  </math> as a subcomplex of <math> E_{n+1} </math>.

For other definitions, see [[symmetric spectrum]] and [[simplicial spectrum]].

=== Homotopy groups of a spectrum ===
Some of the most important invariants of a spectrum are its homotopy groups. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum <math>E</math> define the homotopy group <math>\pi_n(E)</math> as the [[colimit]]<blockquote><math>\begin{align}
  \pi_n(E) &= \lim_{\to k} \pi_{n+k}(E_k) \\
           &= \lim_\to \left(\cdots \to \pi_{n+k}(E_k) \to \pi_{n+k+1}(E_{k+1}) \to \cdots\right)
\end{align}</math></blockquote>where the maps are induced from the composition of the map <math>\Sigma: \pi_{n+k}(E_n) \to \pi_{n+k+1}(\Sigma E_n)</math> (that is, <math> [S^{n+k}, E_n] \to [S^{n+k+1}, \Sigma E_n]</math> given by functoriality of <math>\Sigma</math>) and the structure map <math>\Sigma E_n \to E_{n+1}</math>. A spectrum is said to be [[connective spectrum|connective]] if its <math>\pi_k</math> are zero for negative ''k''.