Towers and pyramids I

Abstract

A ®nitely iterated pullback of path ®brations is called a tower. A pyramid is de®ned as a ®nite sequence of maps with zero-homotopies of their successive compositions, and also with a system of higher zero-homotopies for all successive compositions of lower zerohomotopies. For the category of towers and the category of pyramids , we de®ne maps P X 3 and T X 3 which connect both concepts. The map P is an enhancement of the associated chain complex of a tower, which forms the bottom sequence of the associated pyramid. The map T is de®ned using a glueing construction for higher zero-homotopies. We show for a pyramid Y of length r that the associated chain complex of TY is homotopy equivalent to the r À 1-fold looping of the chain complex of Y . This gives in particular a characterization of the admissible chain complexes over the Steenrod algebra in the sense of Maunder: a chain complex comes from a higher cohomology operation if and only if its Toda bracket vanishes.1991 Mathematics Subject Classi®cation: 55P35, 55P42, 55S20, 55S45.Brought to you by |

Universal Toda brackets of ring spectra

“…Another approach is Spanier's definition of higher Toda brackets [36] using the concept of a carrier. A related concept is Klaus' definition of a pyramid [26, 3.4], which is linked to Spanier's definition by [26,Proposition 3.6]. The perhaps most general approach to Toda brackets and other higher homotopy operations is that of Blanc and Markl [7], who define them as obstructions to realizing homotopy commutative diagrams by strictly commutative ones.…”

A constructive approach to higher homotopy operations

“…Kristensen gave a description of such operations in terms of chain complexes (cf. [Kr, KK]), which was extended by Maunder and others to n-th order cohomology operations (see [Mau,Hol,K1,K2]).…”

Higher Toda brackets and Massey products

“…Toda brackets and Massey products have played an important role in homotopy theory ever since they were first defined in [Mas] and [To1,To2]: in applications, such as [Ad2,BJM,MP], and in a more theoretical vein, as in [Ad1,Ba3,He,Kri,Mar,Sa,Sp1]. There are a number of variants (see, e.g., [Al, HKM, Mi, PS] and [Ba1,§3.6.4]), as well as higher order versions including [Kl,Kra,KM,Mau,Mo,P1,P2,Re,Sp2,W]. In recent years they have appeared in many other areas of mathematics, including symplectic geometry, representation theory, deformation theory, topological robotics, number theory, mathematical physics, and algebraic geometry (see [BT, BKS, FW, G, Ki, La, LS, Ri]).…”