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We study the moduli space of A ∞ structures on a topological space as well as the moduli space of A ∞ -ring structures on a fixed module spectrum. In each case we show that the moduli space sits in a homotopy fiber sequence in which the other terms are representing spaces for Hochschild cohomology.

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Let en be the connective cover of the Morava E-theory spectrum En of height n. In this paper we compute its homology H * (en; Fp) for any prime p and n ≤ 4 up to possible multiplicative extensions when n is 3 or 4. We do this by using the Künneth spectral sequence based on BP which we prove is multiplicative. Contents 1. Introduction 1 1.1. Outline 3 1.2. Conventions 3 Acknowledgments 4 2. The multiplicativity of the Künneth spectral sequence for E 4 -algebras 4 2.1. Monoidal structure on the category of BP -modules 4 2.2. Multiplicative filtrations 6 2.3. Modules and Algebras under BP 9 2.4. Massey Products in the Künneth spectral sequence 13 3. The E 2 -page of the Künneth spectral sequence and differentials 15 3.1. The E 2 -page as an algebra 15 3.2. The collapse of the spectral sequence 19 3.3. Multiplicative extensions 20 References 23