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I'm trying to use truncations $\tau_{\leq n}S$ of the sphere spectrum to ``interpolate'' between $\DeclareMathOperator{\H}{H} \H\mathbb{Z}$ and $S$, and I am struggling to find references for questions that feel like they should be well-known.

Specifically I'm interested in what's known about:

  • The relationship between $X$ and $X \otimes \tau_{\leq n}S$. Does the homology theory associated to $\tau_{\leq n}S$ have a more concrete interpretation for e.g. $n = 1$?
  • Given a map $\H A \to \H B$, what conditions does this map need to satisfy to be a map of $\tau_{\leq n}S$-modules for $n > 0$?

I would really appreciate any relevant resources/papers!

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  • $\begingroup$ What is $HA$ or $HB$ ? Is it an Eilenberg-MacLane spectrum ? $\endgroup$
    – Maxime Ramzi
    Commented Feb 18, 2022 at 8:27
  • $\begingroup$ yes! sorry for not clarifying that -- both Eilenberg-Maclane spectra, although not necessarily in the same dimension. $\endgroup$
    – Colin Aitken
    Commented Feb 18, 2022 at 15:23
  • $\begingroup$ See here for a description of $\tau_{\leq 1} S$ as a symmetric monoidal groupoid. I haven't thought about the associated homology theory. $\endgroup$
    – Tim Campion
    Commented Feb 28, 2022 at 0:32

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