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These days I've been trying to wrap my head around the current proposed approaches to algebraic geometry over the elusive "field with one element", one of whose main objects of interest is the arithmetic curve $\mathrm{Spec}(\mathbb{Z})\times_{\mathrm{Spec}(\mathbb{F}_{1})}\mathrm{Spec}(\mathbb{Z})\cong\mathrm{Spec}(\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z})$. (Incidentally, Martin Brandenburg showed here that in the binoid approach to $\mathbb{F}_{1}$-geometry, one has $\mathrm{Spec}(\mathbb{Z}\otimes_{\mathbb{F}_{1}}\mathbb{Z})\cong\mathcal{P}(\mathbb{P}\times\mathbb{P})$ as sets.)

Coming to this from a different point of view, I've heard people compare algebraic geometry over the field with one element a bit with spectral algebraic geometry, such as in a recent question here on MathOverflow. From what I understand, the analogue of the arithmetic curve $\mathrm{Spec}(\mathbb{Z})\times_{\mathrm{Spec}(\mathbb{F}_{1})}\mathrm{Spec}(\mathbb{Z})$ in SAG is the spectral Deligne--Mumford stack $\mathrm{Spét}(\mathbb{Z})\times_{\mathrm{Spét}(\mathbb{S})}\mathrm{Spét}(\mathbb{Z})\cong\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$.

For this reason, I asked on MSE a question on how to picture $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$. There, Jack Davies gave a really awesome answer, but mentioned at the end of their answer that they would also be interested if other people had more to add.

So I thought it might be a good idea to re-ask this question here.

What do we know about the homotopy groups of $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and its associated spectral Deligne--Mumford stack $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$?

Emily
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    Crossposted: MSE. – Emily Aug 14 '21 at 19:08
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    Regarding the homotopy groups of $H\mathbb Z \wedge H \mathbb Z$, see here. – Tim Campion Aug 14 '21 at 19:11
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    Let me $p$-localize. Then $\mathbf{Z}{(p)}$ is the Thom spectrum of a map $\Omega^2(S^3\langle 3\rangle) \to \mathrm{BGL}_1(S^0{(p)})$, and the Thom isomorphism tells us that $\mathbf{Z}{(p)} \otimes{S^0} \mathbf{Z}{(p)} \simeq \mathbf{Z}{(p)}[\Omega^2 S^3\langle 3\rangle]$. You can calculate its homotopy, i.e., $H_\ast(\Omega^2 S^3\langle 3\rangle; \mathbf{Z}_{(p)})$, using the Serre spectral sequence for the fibration $\Omega^2 S^3\langle 3\rangle \to \Omega^2 S^3 \to S^1$. – skd Aug 17 '21 at 23:35
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    The integral homology of $\Omega^2 S^3$ is not pretty, but it can be computed (for example) using the Eilenberg-Moore spectral sequence. In particular, $H_\ast(\Omega^2 S^3; \mathbf{Z}{(p)})$ has $p$-torsion of order $p$, and you can calculate exactly what this is using the fact that $H\ast(\Omega^2 S^3; \mathbf{F}_p)$ is isomorphic to the dual Steenrod algebra. (see section 10.26 of https://people.math.rochester.edu/faculty/jnei/exalgmethod.pdf) – skd Aug 17 '21 at 23:35

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