Papers › On the geometric fixed points of the real topological cyclic homology of ℤ/4
On the geometric fixed points of the real topological cyclic homology of ℤ/4
Thomas Read
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We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of ℤ/4. We relate these groups to the values of the non-abelian derived functors of the functor M ↦(M ⊗_(ℤ/4) M)^(C₂) at the ℤ/4-module ℤ/2, which we precisely calculate with computer assistance up to degree 6, and calculate in general up to slight remaining ambiguity. Using these results we compute πᵢ(TCR(ℤ/4)^(ϕℤ/2)) exactly for i ≤1, up to an extension problem for 2 ≤i ≤5, and describe the asymptotic growth of this group for large i. A consequence of these computations is that there exists some 0 ≤i ≤5 such that the canonical map comparing the genuine symmetric and symmetric L-theory spectra of ℤ/4 is not an isomorphism on degree i homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees.
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