Higher order derived functors and the Adams spectral sequence
The famous Adams spectral sequence [\textit{J. F. Adams}, Comment. Math. Helv. 32, 180--214 (1958; Zbl 0083.17802)] was first used to obtain partial results on the Hopf Invariant One Problem. While that problem was completely resolved by Adams later, the sequence remained a powerful technique to calculate homotopy groups of spheres. The sequence begins with an Ext group arising from the Steenrod algebra, but the higher terms involve more. One may think of this as involving the homotopy groups of a mapping space.NEWLINENEWLINEIn this fascinating paper, the authors consider the point of view of the ``Eilenberg MacLane'' Map\((X,K(F_p,n))\), on which Map\((K(F_p,n),K(F_p,n))\) acts. They use a generalization of singular homology based on cubical balls, made up of cubes glued to form a ball; this gives a combinatorial structure and also permits products since a product of cubes is a cube. They also generalize chain complexes to consider higher null homotopies. Their first major theorem states that higher order resolutions exist for an algebra of left cubical balls. Before getting to this result (Theorem 14.5), there is considerable algebra. A higher order chain complex is defined as a functor from an elaborate indexing category. This then leads to higher order Ext groups.NEWLINENEWLINEFrom the point of view of the Adams sequence, this gives a new and interesting way to look at the higher order terms in the spectral sequence. For specific calculations, one might look at [\textit{H.-J. Baues} and \textit{M. Jibladze}, J. K-Theory 7, No. 2, 203--347 (2011; Zbl 1231.18009)]. There are also interesting asides relating to the WK construction and ordinary complexes equivalent to higher order complexes. The reviewer is not happy with the word ``strictification'' but cannot think of an improvement.
scientific article; zbMATH DE number 6339636
scientific article