The realization space of an unstable coalgebra

Georg Biedermann, George Raptis, Manfred Stelzer

Published 2014-09-01Version 1

Unstable coalgebras over the Steenrod algebra form the 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 describe a decomposition 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 cosimplicial spaces.