Prenilpotent Spaces

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Introduction. Nilpotent spaces, as introduced in [3], have proven to be as easy to handle as simply connected spaces. Being much more general than the latter, they seem to provide the correct conceptual framework which was earlier assigned to simply connected spaces. along with many other people, have investigated the relation between the homology groups, and operation thereon, and the homotopy type. For nilpotent spaces this relation is expressed by the existence of a stable and unstable Adams spectral sequence [2]. However, for general nonnilpotent spaces very little is known about the relation between homotopy and homology. The problems here are not only computational, since there are many different homotopy types which have the same homology groups and operations thereon. We have indicated a general direction of analysis in the work on acyclic spaces and homology spheres [4,5]. In the course of the attempt to generalize that work to more complicated spaces, like knot complements, it became clear that one has to solve the following general problem.