Homotopy spectral sequences and obstructions

Abstract

For a pointed cosimplicial spaceX •, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX •. In this paper,X • is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.