A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles

We give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups.

💡 Research Summary

The paper addresses the computation of the coincidence Reidemeister trace for self‑maps on a bouquet of circles, a classical object in fixed‑point and coincidence theory. A bouquet of n circles, denoted ∨ⁿS¹, is a K(Fₙ,1) space whose fundamental group is the free group Fₙ on n generators. Given two continuous self‑maps f,g : X → X, the coincidence set Coin(f,g) = {x ∈ X | f(x)=g(x)} decomposes into coincidence classes, each corresponding to a doubly‑twisted conjugacy class in π₁(X). The Reidemeister trace R(f,g) is defined as the algebraic sum of the indices of these classes; it provides a lower bound for the Nielsen coincidence number and encodes subtle homotopy‑theoretic information.

The authors’ main contribution is an explicit formula for R(f,g) expressed entirely in terms of Fox calculus. Each map induces an endomorphism of the free group, φ_f, φ_g : Fₙ → Fₙ. Using Fox derivatives ∂/∂a_i, one forms the Jacobian‑like matrices J_f = (∂φ_f(a_j)/∂a_i) and J_g = (∂φ_g(a_j)/∂a_i) with entries in the group ring ℤ