Spectra and Stability

ΣY Σ f −→ΣX →ΣX ∪Σ f CΣY. Together, these turn a CW-structure on a space X into a CW-structure on ΣX by shifting all the cells of X up one dimension. There’s an analogous shift operation in graded algebraic contexts: the suspension of a graded group, for instance, is given by the formula (ΣG)n+1 =Gn . Much of modern homotopy theory revolves around the study of stable invariants, which are invariants that intertwine the topological and algebraic notions of suspension. Homology is one example of a stable invariant: the Eilenberg–Steenrod axioms can be used to produce the familiar identity Hn(X )∼=Hn+1(ΣX ). Homotopy, however, is not stable — but a theorem of Freudenthal says that a piece of it is.