Completeness of the fiberwise Fuller trace obstruction

Establish that the fiberwise Fuller trace R_{B,C_n}(Ψ_B^n f) is a complete obstruction to removing n-periodic points from a family of endomorphisms f: E → E over a base space B, under the hypotheses that B is a finite-dimensional cell complex and E → B is a manifold bundle with fiber dimension 3 + dim(B).

Background

The paper compares periodic point invariants in the fiberwise setting, showing that, unlike the single-map case, the fiberwise Fuller trace is strictly more sensitive than the collection of fiberwise Reidemeister traces of iterates. This settles a prior conjecture of Malkiewich and Ponto concerning sensitivity.

Building on this, the authors highlight a further conjecture originating in Malkiewich–Ponto [mp2, Conjecture 1.8], proposing that the fiberwise Fuller trace serves as a complete obstruction to the removal of n-periodic points for families f: E → E over B, when the base and fibers satisfy specific dimension conditions. This conjecture seeks a fiberwise analogue of classical completeness results for fixed/periodic point invariants in high-dimensional manifold settings.

References

In future work, joint with Cary Malkiewich and Kate Ponto, we plan to follow up the present paper by investigating the following conjecture. The fiberwise Fuller trace, $R_{B,C_n}(\Psi_Bn f)$, is a complete obstruction to the removal of $n$-periodic points from a family of endomorphisms $f\colon E\to E$ over $B$ when $B$ is a finite dimensional cell complex and $E\to B$ is a manifold bundle with fibers of dimension $3+\dim(B)$.

— Comparing Periodic Point Invariants for Parameterized Families of Maps  (2508.18339 - Williams, 25 Aug 2025) in Conjecture, Section 1 (Introduction)