On Differential Hopf Algebras

Abstract

Let A be an associative differential graded Hopf algebra overa field K=Z\sb p or K=Q. Letφ\sp p\colon A\otimes\cdots (p)\cdots\otimes A\rightarrow Abe given byφ\sp p(a\sb 1\otimes \cdots\otimes a\sb p)=a\sb 1\cdots a\sb p.The filtration F\sp pA=\text{Im}\,φ\sp p induces a spectralsequence of associative Hopf algebras which are primitivelygenerated. If, instead, A is taken to be co associative, thenthe appropriate dual statement is valid.\par In this basic studyof differential Hopf algebras, the structure of these spectralsequences is investigated and applications (far too numerous tomention) are given to the Bockstein spectral sequences of Hspaces, the spectral sequence of a multiplicative fibering, etc.The exposition is clear despite the abundance of technicalresults and should be comprehensible to anyone familiar with thebasic results and terminology of J. Milnor and J. C. Moore [Onthe structure of Hopf algebras, Princeton Univ., Princeton, N.J.,1959 (mimeographed) (to appear)]