Direct computation of the restriction map from π0 TCR(k;2)^{φZ/2} to (π0 TC(k;2))^{Z/2}/Im(1+w)

Determine, by a direct calculation, the restriction map res^{Z/2}_e: π0 TCR(k;2)^{φ Z/2} → (π0 TC(k;2))^{Z/2}/Im(1+w) for a general field k of characteristic 2. In particular, under the identification π0 THR(k)^{φ Z/2} ≅ k ⊗_S k where S ⊆ k is the subfield generated by squares and C2 acts by swapping tensor factors, establish that res^{Z/2}_e sends the class a^{-1} ⊗ a to 1 for all units a ∈ k^×.

Background

In the proof of the Milnor-type result for real topological cyclic homology (Theorem in Section 3.3), the authors need to compare two ideals: the augmentation ideal I in the symmetric Witt group W^s(k) and the kernel K of the restriction map from π0 TCR(k;2)^{φZ/2} to (π0 TC(k;2))^{Z/2}/Im(1+w). They show the desired identification by employing a real K-theory trace map KR(k) → TCR(k;2) and its effect on π0, avoiding a direct computation of the restriction map.

Concretely, they reduce the comparison to verifying that the restriction map sends the element a^{-1} ⊗ a (the image of the rank-1 symmetric form ⟨a⟩) to 1. Lacking an explicit description of π0 TC(k;2) for general fields of characteristic 2, they state they are unable to establish this directly and instead use the trace map to deduce the necessary property indirectly. The open question is thus to determine this restriction map explicitly and verify the stated behavior on generators.