Fiberwise Fuller Trace in Parameterized Maps

• Fiberwise Fuller trace is a parameterized periodic point invariant that extends the classical Fuller trace to families of maps, capturing obstructions via cyclic group actions.
• It leverages equivariant and parameterized stable homotopy theory along with categorical frameworks to reveal periodic phenomena that fiberwise Reidemeister traces alone cannot detect.
• Applications include analyzing fiberwise fixed point and coincidence problems, providing sharper obstruction criteria in deformation and intersection theories.

The fiberwise Fuller trace is a parameterized periodic point invariant that generalizes the classical Fuller trace from fixed point theory to families of maps over a base space. It is constructed using techniques from equivariant and parameterized stable homotopy theory, and its nonvanishing provides obstructions to deforming families of maps so that periodic points are eliminated on each fiber. The invariant is strictly more sensitive than the collection of fiberwise Reidemeister traces of iterates, and its behavior is governed by the global structure induced by the cycling action of the symmetric group on n factors. The fiberwise Fuller trace serves as a central tool in the paper of fiberwise periodic points, parameterized coincidence theory, and fiberwise fixed point problems.

1. Formal Definition and Construction

The fiberwise Fuller trace is defined for a family of self-maps $f : E \to E$ parameterized over a base manifold $B$ via a fiber bundle $E \to B$. For a fixed $n \geq 1$, the Fuller construction forms a $C_n$-equivariant map

$\Psi_B^n f : E^{\times_B n} \to E^{\times_B n},$

where $E^{\times_B n}$ is the $n$-fold fiber product over $B$. The map acts by

$$(x_1, x_2, \ldots, x_n) \mapsto (f(x_n), f(x_1), \ldots, f(x_{n-1})),$$

so cyclic fixed points of $\Psi_B^n f$ correspond to $n$-periodic points of $f$ within each fiber.

The fiberwise Fuller trace $R_{B, C_n}(\Psi_B^n f)$ is then extracted as the $C_n$-equivariant fiberwise Reidemeister trace of $\Psi_B^n f$:

$$R_{B, C_n}(\Psi_B^n f) : B \times S \to \Sigma_+^B{}^\infty \Lambda^B{}^{\Psi_B^n f}(E^{\times_B n}),$$

where $\Lambda^B{}^{\Psi_B^n f}$ denotes the fiberwise twisted free loop space and $\Sigma_+^B{}^\infty$ is the fiberwise suspension spectrum. The construction relies on categories of retractive $G$-spaces and parameterized orthogonal $G$-spectra.

2. Comparison to Fiberwise Reidemeister Traces

The fiberwise Fuller trace is strictly stronger than the collection of fiberwise Reidemeister traces of iterates $\{R_B(f^k) : k|n\}$ for several reasons:

• In the single map case (unparameterized), the Fuller trace reduces to the set of Reidemeister traces for the iterates. However, in the parameterized context, the cycling symmetry creates new obstructions and periodic phenomena detectable only by the Fuller trace.
• Explicit examples show that vanishing of the fiberwise Reidemeister traces for $f$ and $f^2$ does not imply vanishing of the fiberwise Fuller trace $R_{B, C_2}(\Psi_B^2 f)$ (Williams, 25 Aug 2025).

This separation confirms that periodic data encoded by the cycling action is invisible to the iterated fiberwise traces alone. The Fuller trace thereby detects global periodic effects.

3. Algebraic and Categorical Frameworks

The construction and understanding of the fiberwise Fuller trace leverages several categorical and algebraic structures:

• In closed monoidal derivators, the external trace for fiberwise dualizable objects is functorial and recovers the fiberwise Fuller trace via homotopy colimit formulas, particularly for finite EI-categories (Gallauer, 2013). For an object $A \in (I)_0$ and endomorphism $f: A \otimes S \to A \otimes T$, the trace is given by

$$\mathrm{Tr}(f) = (q_{2!} \otimes p_2^* S \xrightarrow{\mathrm{coev} \otimes 1} A^\vee \boxtimes (A \otimes S) \xrightarrow{1 \boxtimes f} A^\vee \boxtimes (A \otimes T) \xrightarrow{\mathrm{ev} \otimes 1} q_{1*} \otimes p_2^* T),$$

and in the presence of group actions, the trace of the homotopy colimit is

$$\mathrm{tr}(p_{I!} f) = \sum_{(i, h) \in \pi_0({I})} \lambda_{(i, h)} \, \mathrm{tr}(h^* \circ f_i),$$

generalizing the additivity of traces and giving explicit computation of the fiberwise Fuller trace.

• In indexed symmetric monoidal categories, the fiberwise trace is further refined using string diagram calculus, allowing passage between symmetric monoidal traces and bicategorical (Reidemeister-type) traces in a coherent way (Ponto et al., 2012).

4. Bordism and Obstruction Theoretic Aspects

The fiberwise Fuller trace is closely related to bordism invariants in fiberwise intersection theory:

• The bordism invariant $[c, c] \in \Omega_{p+q+k-m}(E(f, i_Q); \xi)$, constructed via homotopy pullbacks and normal bundle theory, acts as a complete obstruction to fiberwise removing intersections (Sunyeekhan, 2012).
• In the fixed point context, the invariant $L_{\mathrm{bord}}(f)$ in the appropriate bordism group encodes the fixed point data, and vanishing of these invariants implies the existence of fiberwise homotopies eliminating intersections or fixed points.
• Techniques developed for analyzing such bordism invariants inform computations of the fiberwise Fuller trace, connecting intersection theory and fixed point theory through a common algebraic-topological language.

5. Orbit Counting Methods and Essential Obstructions

For fiberwise coincidence theory, the minimum number of coincidence points and components—hence the essentiality of periodic points—depends on the orbit structure of the associated group actions:

• The Nielsen number $N_B(f_1, f_2)$ is computed by counting essential pathcomponents determined by the orbit structure of the affine $\pi_1(B)$-action on the cokernel group $G = \mathrm{Coker}(f_{1*} - f_{2*})$ (Koschorke, 2013).
• Orbits of odd order under this action play a critical role: they represent obstructions to deforming fiberwise maps apart (or eliminating periodic points), as even order orbits may be canceled via homotopies given extra degrees of freedom.
• The fiberwise Fuller trace tracks the full orbit structure, unlike classical invariants, thus capturing subtle essential periodic data.

6. Resolution of the Fuller Trace Conjecture

Previous conjectures posited that the fiberwise Fuller trace would be strictly stronger than the collection of iterated fiberwise Reidemeister traces [MP2]. This has been confirmed (Williams, 25 Aug 2025) via explicit examples:

• Construction of a family $f: E \to E$ over $B = S^1$ for which $R_B(f)$ and $R_B(f^2)$ vanish, but $R_{B, C_2}(\Psi_B^2 f)$ is nonzero. This provides a counterexample showing that the Fuller trace captures strictly more than the assembly of iterated trace invariants.
• Methodologically, the proof uses equivariant parametrized stable homotopy theory, Pontryagin–Thom isomorphism for singular $G$-manifolds over a map, and analysis of framings and local indices. The technical computation establishes the nontriviality of the invariant via stable homotopy group evaluations.

7. Applications and Broader Implications

The fiberwise Fuller trace has substantial impact in differential topology, fixed point theory, and the paper of parameterized dynamical systems:

• It provides sharper obstructions to removing periodic points across families than classical invariants.
• Its functoriality and explicit computability in terms of local traces and automorphism data make it amenable to application in parameterized gauge theory, string topology, and the paper of equivariant phenomena.
• The invariants permit systematic paper of minimal coincidence numbers in fiberwise settings, reveal the obstructions contributed by group actions, and afford refined control in deformation theory.

A plausible implication is that techniques derived from the fiberwise Fuller trace framework can be adapted to analyze periodic orbits, coincidence theory, and fixed point theory in broader parameterized and equivariant contexts, extending the reach of algebraic topology into geometric analysis of families and their symmetries.