The realization space of an unstable coalgebra

Abstract

Unstable coalgebras over the Steenrod algebra form a natural target category for singular homology with prime field coefficients. The realization problem asks whether an unstable coalgebra is isomorphic to the homology of a topological space. We study the moduli space of such realizations and give a description of this in terms of cohomological invariants of the unstable coalgebra. This is accomplished by a thorough comparative study of the homotopy theories of cosimplicial unstable coalgebras and of cosimplicial spaces.

118 pages. Final version ; accepted for publication in Astérisque. The presentation of the material is reorganized and generally improved. More details are added in some proofs and some corrections are made, especially in Section 7 and Appendix A. Some issues concerning basepoints are resolved and the module actions in the spiral exact sequence are explained in detail