Coincidence of pairs of maps on torus fibre bundles over the circle (Q2307313)

Denote by \(M(\Phi )\) a fibre bundle over the circle \(S^1\), where the fiber is the torus \(T\). This space is obtained from \(T\times [0,1]\) by identifying the points \((x,0)\) with the points \((\Phi(x),1)\), where \(\Phi:T\to T\) is a homeomorphism. The authors consider fibre-preserving maps \(f,g:M(\Phi_1)\to M(\Phi_2)\) over \(S^1\). The purpose of the paper is to classify the pairs of maps \((f,g)\) which can be deformed by fibre-preserving homotopy to a coincidence-free pair \((f',g')\). This problem is equivalent to finding solutions of some equation in the free group \(\pi _2(T,T-1)\). This equation is carefully investigated. The special case with \(\Phi_1=\Phi_2\) was considered by the first author in [Topol. Methods Nonlinear Anal. 46, No. 2, 507--548 (2015; Zbl 1366.55002)].