DGAs with polynomial homology

Abstract

In this work, we study the classification of differential graded algebras over Z (DGAs) whose homology is Fp[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 |y2p−2|=2p−2, we show that there is a unique non-formal DGA with homology Fp[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 Fp[x]. By duality, this also shows that there is a non-formal DGA whose homology is an exterior algebra over Fp with a generator in degree −(2p−1).Considering the classification of the ring spectra corresponding to these DGAs, we show that every E2 DGA with homology Fp[x] (with no restrictions on |x|) is topologically equivalent to the formal DGA with homology Fp[x], i.e. they are topologically formal. This follows by a theorem of Hopkins and Mahowald.