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I’ve seen the recent Lurie paper, Rotation invariance in K-theory; the main theorem states that the Waldhausen K-theory $\mathcal{C}\mapsto K(\mathcal{C})$ is invariant under the action of $S^1$. What’s the significance of this result? Can it be generalized to actions of $S^n$?
S^n is not a group, usually…
Right, sorry. Was writing that in a hurry; could this be generalized to an action of $S^3$?
Such a result wouldn’t be a generalisation, but a completely different result. S^3 and S^1 are very different beasts.
What I guess I was trying to ask is, if we impose additional conditions on C, then can the Waldhausen K-theory construction be invariant under the action of S^3?
“the action of S^3” What action of S^3? In this case there is some sort of action on the input that is “naturally occurring,” I see no such naturally occurring action of S^3.
I don’t really know. It was simply a not-very-thought-over-speculation, so perhaps I should think better. What I’m more interested in is the significance of this result.
The action by $S^1$ can be thought of as the action by the one object groupoid with arrows $\mathbb{Z}$, in Lurie’s setup. The element $1\in \mathbb{Z}$ is sent to the double suspension functor. 2-periodicity of various forms of K-theory (when it holds) is the result that the double suspension functor is a self-equivalence of the relevant ring spectrum. For instance ordinary complex K-theory is 2-periodic essentially by Bott periodicity. Lurie’s result is a massive generalisation, and as he says, ’delooping’ of this (the connected delooping of $\mathbb{Z}$ is $S^1$). One application is extension of the combinatorial construction of the topological Fukaya category from taking 2-periodic dg-categories over a base field to where the field is replaced by a ring spectrum.
A sensible question is whether the result can be generalized to $U(n)$ for $n\geq 2$…
@Marc - or $O(n)$ for $n \geq 3$, or $SO(n)$ for oriented stuff, etc etc
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