On differential Hopf algebras

Abstract

Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit