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

TL;DR

This paper investigates the homotopy groups of the geometric fixed points of the real topological cyclic homology of d/4, relating them to derived functors, with explicit calculations up to degree 6, revealing differences in symmetric L-theory spectra.

Contribution

It provides the first detailed computation of these homotopy groups for d/4, connecting them to derived functors and analyzing their growth and structural differences.

Findings

Computed d/4 homotopy groups up to degree 6 with computer assistance.

Identified non-isomorphism between symmetric and genuine L-theory spectra in certain degrees.

Described asymptotic growth of the homotopy groups for large degrees.

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.