Highly Structured Multiplication & The Miller Spectral Sequence

Abstract

We examine the Miller spectral sequence for determining the mod-p homology of a connective spectrum X from the mod-p homology of its associated infinite loop space, Ω∞X, considered as an algebra over the mod-p Dyer-Lashof algebra. For each prime p, we give a Koszul complex for computing the E2 page of this spectral sequence, recovering a result of Miller (at p = 2) [35] and Kraines and Lada (at odd primes) [22]. As applications, we determine H∗(HZ; Fp) and H∗(HFp; Fp) at all primes, recovering well-known results. As an original application of the Miller spectral sequence, we study the relationship between H∗(Ω∞X; Fp) and H∗(X; Fp) when X is an E∞-ring spectrum. We show that the Miller spectral sequence can be used to detect nonzero “multiplicative” k-invariants of X at all primes. We also prove that for any integer n ≥ 1, the underlying spectrum of a commutative HFp-algebra R is equivalent to its strict unit spectrum, sl1(R), in a range that is wider than the stable range: [n, pn − 1]. This is a special case of a conjecture by Mathew and Stojanoska [29].