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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 𝒞↦K(𝒞) is invariant under the action of S1. What’s the significance of this result? Can it be generalized to actions of Sn?
S^n is not a group, usually…
Right, sorry. Was writing that in a hurry; could this be generalized to an action of S3?
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 S1 can be thought of as the action by the one object groupoid with arrows ℤ, in Lurie’s setup. The element 1∈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 ℤ is S1). 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≥2…
@Marc - or O(n) for n≥3, or SO(n) for oriented stuff, etc etc
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