On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$

Abstract

We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$. We relate these groups to the values of the non-abelian derived functors of the functor $M \mapsto (M \otimes_{\mathbb{Z}/4} M)^{C_2}$ at the $\mathbb{Z}/4$-module $\mathbb{Z}/2$, 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 $\pi_i(\mathrm{TCR}(\mathbb{Z}/4)^{\phi \mathbb{Z}/2})$ exactly for $i \le 1$, up to an extension problem for $2 \le i \le 5$, and describe the asymptotic growth of this group for large $i$. A consequence of these computations is that there exists some $0 \le i \le 5$ such that the canonical map comparing the genuine symmetric and symmetric $L$-theory spectra of $\mathbb{Z}/4$ is not an isomorphism on degree $i$ homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees.