DGAs with polynomial homology

In this work, we study the classification of differential graded algebras over Z (DGAs) whose homology is F-p[x], i.e. the polynomial algebra over Fp on a single generator. This classification problem was left open in work of Dwyer, Greenlees and Iyengar. For |y(2p)-2| = 2p - 2, we show that there is a unique non formal DGA with homology F-p[y2p-2] and a non-formal 2p -2 Postnikov section. Among a classification result, this provides the first example of a non-formal DGA with homology F-p[x]. By duality, this also shows that there is a non-formal DGA whose homology is an exterior algebra over F-p with a generator in degree -(2p - 1). Considering the classification of the ring spectra corresponding to these DGAs, we show that every E-2 DGA with homology Fp[x] (with no restrictions on |x|) is topologically equivalent to the formal DGA with homology F-p[x], i.e. they are topologically formal. This follows by a theorem of Hopkins and Mahowald. (C) 2021 Elsevier Inc. All rights reserved.