Finite Cover Resolution Complexity of Nielsen Fixed Point Spectra

Abstract

For a self-map (f\colon X\to X) of a finite connected CW complex, we introduce visible Nielsen numbers by pushing the Reidemeister trace to finite regular covers compatible with (f). Optimizing over such covers gives a visibility profile and three resolution thresholds, including the observer complexity (\operatorname{oc}(f)), the least cover degree that separates all essential Nielsen fixed-point classes. We identify finite-observer indistinguishability with twisted conjugacy in the profinite completion and determine the associated blind subgroup. We also realize nonzero Reidemeister traces that vanish for every finite observer, and show that observer complexity is unbounded on a fixed finite complex even when the induced maps on the fundamental group and homology, the Lefschetz number, and the Nielsen number are fixed. For toral endomorphisms we compute the complete visibility profile as an exact divisor staircase. For compatible maps of principal torus bundles, in the finite-Reidemeister regime, we identify the unique maximal resolving kernel and express (\operatorname{oc}(F)) in terms of the Nielsen number and the radical of the bundle characteristic class modulo the fibre map. This yields exact formulas for scalar dilations of integral Heisenberg nilmanifolds, together with observer entropy and a rational observer-complexity zeta function. Finally, we construct toral and Heisenberg systems with identical Nielsen and Lefschetz sequences, Nielsen zeta functions, differential eigenvalues, and topological entropy, but exponentially different observer-complexity sequences.