Malkiewich–Ponto's fiberwise Fuller trace obstruction conjecture

Let $B$ be a finite dimensional cell complex, let $E\to B$ be a manifold bundle with fibers of dimension $3+\dim(B)$, and let $f\colon E\to E$ be a family of endomorphisms over $B$. For $n\geq 1$, write $R_{B,C_n}(\Psi_B^n f)$ for the fiberwise Fuller trace of the $n$-fold equivariant construction.

Malkiewich–Ponto's conjecture. The fiberwise Fuller trace

$$R_{B,C_n}(\Psi_B^n f)$$

is a complete obstruction to removing all $n$-periodic points from the family $f$.

This conjecture proposes that, under the stated dimensional hypotheses, vanishing of the fiberwise Fuller trace is equivalent to deforming the family to one with no $n$-periodic points. The paper shows that the fiberwise Fuller trace can be strictly stronger than the collection of fiberwise Reidemeister traces of iterates, but the claimed completeness remains a conjecture.