Higher homotopy operations and the cohomology of diagrams

Algebraic topology tries to answer problems of homotopy theory by translating them into algebra, using invariants such as the homotopy or cohomology groups of a space. These invariants, even when endowed with extra algebraic structure such as Whitehead products or Steenrod operations, rarely reflect fully the original homotopical information, and the additional data needed for a full invariant is usually not purely algebraic in character: for example, the differentials of a DGA in rational homotopy, k-invariants (cohomology classes of the successive Postnikov sections), higher Massey products, action of a suitable operad, and so on. Higher homotopy and cohomology operations have a long history, starting with Toda brackets, Massey products, and Adem’s secondary cohomology operations. They have been used with great effect in various computations (e.g., [Ad, BJM, MP]), but little has been done to organize them systematically, or even define them in general (beyond Spanier’s attempts in [S1, S2]). In practice, higher operations often appear in an algebraic form: as differentials in spectral sequences (e.g., in [Ad, Ch. 2]), as Ext classes (see [Mar, Ch. 16, 3]), and so on. Such descriptions often serve as a good way to actually compute the operations. Here we consider the operation itself as the intrinsic homotopy-theoretic “fact,” which may manifest itself in different (seemingly unrelated) algebraic guises. All such higher homotopy or cohomology operations can be described geometrically as obstructions to rectifying homotopy-commutative diagrams A : Γ → hoTop for various indexing categories Γ (that is, making A strictly commute by finding a lift to  : Γ → Top). However, making this precise – including the exact conditions when the higher operations are defined in the first place – is not easy, and we know of no fully satisfactory general definition. However, the approach described in [BMa] appears sufficient for our purposes. So far, there have been two significant attempts to categorize complete collections of (secondary) operations: