The Homology of Homotopy Inverse Limitsby

The homology of a homotopy inverse limit can be studied by a spectral sequence with has E 2 term the derived functors of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonn e modules if one has a diagram of connected abelian Hopf algebras. One of the standard problems in homotopy theory is to calculate the homology of a given type of inverse limit. For example, one might want to know the homology of the inverse limit of a tower of brations, or of the pull-back of a bration, or of the homotopy xed point set of a group action, or even of an innnite product of spaces. This paper presents a systematic method for dealing with this problem and works out a series of examples. It simpliies the foundational questions present when dealing with inverse limits to work with simplicial sets rather than topological spaces. So this paper is written simpli-cially; that is, a space is a simplicial set. As usual this doesn't aaect the homotopy theory. Homology is with F p coeecients. Here are some simple examples of the type of result we obtain. An abelian Hopf algebra is one for which both diagonal and multiplication are commutative. Theorem A. Let fX g be an arbitrary set of connected nilpotent spaces and suppose for all , H X is an abelian Hopf algebra. Then there is a natural isomorphism This is a companion to a result of Bousseld's 3, 4.4] where slightly diierent hypotheses yield a slightly diierent result. The product Q H X is in the category of graded connected coalgebras. If the set is nite it is simply the graded tensor product. If the set is innnite something more sophisticated is required. See section 1 for limits of coalgebras.