Coincidence of maps on torus fibre bundles over the circle (Q519319)

The author considers fibrations \(p:M\to S^1\) with fibre \(S=T\), the torus. Given fibre-preserving maps \(f,g:M\to M\) over \(S^1\) one then asks whether the pair \((f,g)\) can be deformed by a fibrewise homotopy over \(S^1\) into a coincidence free pair \((f',g')\). The fibre bundle can be obtained from \(S^1\times[0,1]\) by identifying \((x,0)\) with \((\phi(x),1)\) where \(\phi:T\to T\) is a homeomorphism. One then denotes the total space by \(M(\phi)\). \noindent The author starts by considering a more general problem. Let \((F_i,B_i,B,p_i)\), \(i=1,2\), be fibre bundles and \(f,g:M_1\to M_2\) be fibre-preserving maps over \(B\). Let \(M_2\times_B M_2:=\{(x,y)\in M_2\times M_2\mid \;p_2(x)=p_2(y)\}\) and \(E_B(M_2)=\{(x,w)\in (M_2\times_B M_2\setminus\Delta)\times X^{[0,1]}\mid i(x)=w(0)\}\) and \(q(x,w)=w(1)\). Denote by \(q_{(f,g)}:E_B(f,g)\to M_1\) the fibration induced by \((f,g)\) from \(q\). The author then proves that \((f,g)\) can be deformed over \(B\) into a coincidence free pair by a fibrewise homotopy if and only if there exists a section \(\sigma:M_1\to E_B(f,g)\) for \(q_{(f,g)}\). Moreover, the author provides an explicit classification of all \(T\)-bundles over \(S^1\). Finally he gives an explicit classification of all pairs of maps \((f,g)\) which can be deformed by a fibrewise homotopy over \(S^1\) to a pair of coincidence free maps.