This is an afterthought on this MO question, and also on Gannon's book mentioned there, about $K_3(\mathbb{Z})=\mathbb{Z}/48$. Neither the question nor the book mentions a possible connection with the third stable homotopy group of spheres, $\pi ^S_3=\mathbb{Z}/24$. Does anyone know about such a connection?
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4$\begingroup$ There is a map of spectra $\mathbb{S}\to K(\mathbb{Z})$. I'm not completely sure, but I'd expect it to be injective on $\pi_3$. $\endgroup$– Achim KrauseCommented Aug 31, 2019 at 8:40
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3$\begingroup$ @abx It's just the unit of the ring structure. I think pretty much any book on algebraic K-theory constructs it, you can see my answer here for a few references (I like a lot Mitchell's survey in that list) $\endgroup$– Denis NardinCommented Aug 31, 2019 at 9:15
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1$\begingroup$ One quick way of constructing it is as the map induced on K-theory by the exact functor of Waldhausen categories $\mathrm{Fin}_*\to\mathrm{Proj}_{\mathbb{Z}}$ sending $1_+$ to $\mathbb{Z}$ $\endgroup$– Denis NardinCommented Aug 31, 2019 at 9:23
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1$\begingroup$ @abx ah, but which is the image of index 2? Is there some cokernel with another interpretation? $\endgroup$– David Roberts ♦Commented Aug 31, 2019 at 9:56
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2$\begingroup$ The map to the cokernel of $\pi_3(S) \to K_3(\mathbb{Z})$ can be interpreted as the Hurewicz homomorphism to $H_3(K(\mathbb{Z})) \cong \mathbb{Z}/2$, or as the Bökstedt trace map to $\pi_3 THH(\mathbb{Z}) \cong \mathbb{Z}/2$. A key point in the 1976 paper by Lee and Szczarba is why the extension is nontrivial. I use $\lambda$ to denote a generator of $K_3(\mathbb{Z})$ to refer to Lee (and Szczarba). $\endgroup$– John RognesCommented Aug 31, 2019 at 14:04
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1 Answer
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We have $\pi_3(\mathbb{S}) \cong \mathbb{Z}/24\{ \nu\}$ and $\pi_3K(\mathbb{Z}) \cong \mathbb{Z}/48\{ \lambda \}$. As Achim suggested, the unit map $\mathbb{S} \to K(\mathbb{Z})$ induces on $\pi_3$ the injection sending $\nu$ to $2\lambda$.
See the first paragraph of Section 2 of 'Divisibility of the Dirac magnetic monopole as a two-vector bundle over the three-sphere' by Ausoni, Dundas, and Rognes.
answered Aug 31, 2019 at 11:54
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4$\begingroup$ Thank you for the reference. Actually, this is already in the original paper of Lee and Szczarba where they prove $K_3(\mathbb{Z})=\mathbb{Z}/48$ (Annals of Math. 104 (1976), 31-60. $\endgroup$– abxCommented Aug 31, 2019 at 13:31