Editing Spectrum (topology) (section)

=== 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''.