Towers and pyramids. I (Q2753159)

The author works in the category \({\mathcal C}{\mathcal G}_0\) of pointed compactly generated spaces. A tower \(E\) of height \(r\) is a finitely iterated pullback \(E= \{E^s\to E^{s-1}\to X^{s-1}\}^r_{s=1}\) of path fibrations, and a map \(E^r\to X^r\). The concept of a tower appears in the Postnikov tower, the definition of higher cohomology operations, and elsewhere. A chain complex of length \(r\) is a sequence of maps \(Y= \{Y^{s-1}\to Y^s\}^r_{s=1}\) such that each composite of adjacent maps is zero, which defines a morphism from \(Y^0\to Y^r\) of a category whose objects are those of \({\mathcal C}{\mathcal G}_0\). This is called the category \({\mathcal K}\) of chain complexes. A composition of two chain complexes \(Y\) and \(Z\) is defined by deleting the connecting space \(Z^s= Y^0\). The associated chain complex \(K(E)\) of a tower \(E\) is the sequence \(\{X^{s-1}\to X^s\}^r_{s=1}\). In a similar sense, the morphisms of the category \({\mathcal T}\) of towers are towers whose composition is defined such that \(K:{\mathcal T}\to {\mathcal K}\) is an \(\Omega\)-twisted functor from the category of towers to \({\mathcal K}\). In other words, \(K(FE)= \Omega^{r-1}(K(F)K(E))\). The author defines a pyramid as a chain complex with additional conditions, and gives a similar category \({\mathcal P}\) of pyramids. Then he defines the \(\Omega\)-twisted functors \(P:{\mathcal T}\to{\mathcal P}\) and \(T:{\mathcal P}\to{\mathcal T}\). It is shown that \(KT(Y)= \Omega^{r-1}\text{chain}(Y)\), while \(PT(Y)= \Omega^{r-1}Y\) is unknown. Since the existence of a pyramid is closely related to a Toda bracket, it follows that a chain complex comes from a higher cohomology operation if and only if its Toda bracket vanishes.