not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements. Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of Massey products in cohomology. Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements. Computational methods If is any finite simplicial complex with finite fundamental group, in particular if is a sphere of dimension at least 2, then its homotopy groups are all finitely generated abelian groups. To compute these groups, they are often factored into their -components for each prime , and calculating each of these -groups separately. The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on spectral sequences. This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates. "The method of killing homotopy groups", due to involves repeatedly using the Hurewicz theorem to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an Eilenberg–MacLane space. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group. The Serre spectral sequence was used by Serre to prove some of the results mentioned previously. He used the fact that taking the loop space of a well behaved space shifts all the homotopy groups down by 1, so the th homotopy group of a space is the first homotopy group of its ()-fold repeated loop space, which is equal to the first homology group of the ()-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of to the calculation of homology groups of its repeated loop spaces. The Serre spectral