The homology of homotopy inverse limits

Abstract: The homology of a homotopy inverse limit can be studied by a spectral sequence with has E2 term the derived functors of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonne 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 innite product of spaces. This paper presents a systematic method for dealing with this problem and works out a series of examples. It simplies 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 aect the homotopy theory. Homology is with Fp coecients. 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 fXg be an arbitrary set of connected nilpotent spaces and suppose for all , HX is an abelian Hopf algebra. Then there is a natural isomorphism H