Realizing coalgebras over the Steenrod algebra

Abstract

We describe algebraic obstruction theories for realizing an abstract (co)algebra K * over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K * .

Assigning a classifying space to a fusion system up to F-isomorphism

“…The obstruction theory in terms of characteristic classes for the realizabilty of the cohomology of a space was developed by Blanc [8]. This obstruction theory does not simplify in our situation.…”

Truncated derived functors and spectral sequences

“…Note that because Y → H * (Y; R) is contravariant the category Π A -Alg resembles Π B -Alg in being a category of graded universal algebras, so the resolutions we need for the Π A -algebra Λ = H * (Y; R) will be simplicial, rather than cosimplicial, and we can use the notion of a CW resolution V • → Λ as in §3.3. However, only when R is a field do we know that any free simplicial resolution in Π A -Alg ∆ op has a CW basis (V n ) ∞ n=0 of free Π A -algebras (see [Bl3,Proposition 3.12]). For the cosimplicial resolutions of spaces, we need to dualize §3.3 as follows:…”

Higher cohomology operations and R–completion

“…Thus one can easily construct (non-functorial) CW resolutions of any Θ-algebra Γ. Moreover, by dualizing [Bl3,Proposition 3.12] (see also [Bl2,Proposition 12]) we can choose a CW basis for any free simplicial resolution in Θ R -Alg, when R is a field. However, we shall not make use of this fact.…”