Suspension Spectra are Harmonic In memory

The notion of a harmonic spectrum in stable homotopy theory was introduced by the second author in [Rav84]. Roughly speaking, a spectrum is harmonic if its homotopy is computable with the chromatic spectral sequence. It was shown in [Rav84, Theorem 4.4] that a large class of connective spectra (including all finite spectra) are harmonic. On the other hand, the EilenbergMacLane spectrum for a torsion group was shown to be dissonant (the opposite of harmonic), i.e., chromatic methods give no information at all about its homotopy type. The purpose of this paper is to show that all suspension spectra are harmonic (Theorem 8), thereby giving an affirmative answer to a question raised in [Rav84]. This result has some interesting corollaries. One is a new proof of the Whitehead theorem for Morava K-theory, Theorem 2. (This was originally proved, although not explicitly stated by Bousfield in [Bou82].) Another (Theorem 12) is that a suspension spectrum is uniquely determined by any of its connective covers. It is a pleasure to thank Joe Neisendorfer and Fred Cohen for helpful conversations, and Nick Kuhn for pointing out the relevance of Bousfield’s paper and that Theorem 2 below is a consequence of Theorem 8.