Stable homotopy groups of suspension spectra

Example 4.4.24. For a pointed anima \(X\), the negative homotopy groups of its suspension spectrum \(\Sigma ^{\infty }X\) vanish. For \(k \geq 0\), Remark 4.3.27 gives \[ \pi _k(\Sigma ^{\infty }(X)) \; \cong \; \pi _k(\Omega ^{\infty } \Sigma ^{\infty }(X)) \; \cong \; \colim _{n \geq 0} \pi _k(\Omega ^n \Sigma ^nX) = \colim _n \pi _{n+ k}(\Sigma ^n X). \] These groups are known as the stable homotopy groups of \(X\). By the classical Freudenthal suspension theorem [Freudenthal (1937)] the maps \(\pi _{n+k}(\Sigma ^nX) \to \pi _{n+k+1}(\Sigma ^{n+1}X)\) are isomorphisms for \(n \geq k+2\).