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

Thomas Read

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 $π_i(\mathrm{TCR}(\mathbb{Z}/4)^{φ\mathbb{Z}/2})$ 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.

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 . We relate these groups to the values of the non-abelian derived functors of the functor at the -module , which we precisely calculate with computer assistance up to degree , and calculate in general up to slight remaining ambiguity. Using these results we compute exactly for , up to an extension problem for , and describe the asymptotic growth of this group for large . A consequence of these computations is that there exists some such that the canonical map comparing the genuine symmetric and symmetric -theory spectra of is not an isomorphism on degree homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees.

Paper Structure

This paper contains 12 sections, 44 theorems, 182 equations.

Table of Contents

  1. Preliminaries and conventions
  2. Point-set models and model structures
  3. Dold-Kan equivalence
  4. Real topological Hochschild homology
  5. Low degree homotopy of TCR(Z/4){phi Z/2}
  6. Decomposition of the equaliser formula for TCR(Z/4){phi Z/2}
  7. Homotopy groups of THR(Z/4){phi Z/2} and (THR(Z/4){phi Z/4}){C_2}
  8. Action of the maps f and r on homotopy
  9. Resolving the extension problem in degree 1
  10. Asymptotic growth estimates
  11. Non-abelian derived functor computations
  12. Application to size estimates

Key Result

Lemma 1

Let $F_{R} : \mathrm{Mod}_{R} \to \mathrm{Ab}$ defined by $F_{R}(M) = (M \otimes_{R} M)^{C_2}$. Then where $L_i^{(n)}F_{R} : \mathrm{Mod}_{R} \to \text{Ab}$ is the $i$th non-abelian derived functor of type $n$ of $F_{R}$.

Theorems & Definitions (96)

  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • proof
  • Definition 1.1
  • Lemma 1.2
  • proof
  • ...and 86 more