Maps into the torus and minimal coincidence sets for homotopies (Q2773379)

The authors deal with the ``minimal coincidence problem'': Let \(X,Y\) be compact manifolds of the same dimension. If \((f,g)\) is a pair of mappings from \(X\) to \(Y\) one denotes by \(\text{MC}[f,g]\) the minimal number of coincidence points among all pairs \((f',g')\) which are homotopic (as pairs) to \((f,g)\). Let then \((f_1,g_1)\) and \((f_2,g_2)\) be homotopic pairs such that the number of coincidence points of both of them equals \(\text{MC}[f_1,g_1]\). The minimal coincidence problem asks whether there is a pair \((H,G)\) of homotopies from \((f_1,g_1)\) to \((f_2,g_2)\) such that the number of coincidence points of \(H(\cdot,t),G(\cdot,t)\) equals \(\text{MC}[f_1,g_1]\) for each \(t\). The authors here consider the case where \(X=Y\) equals \(S^1\) or the two-dimensional torus \(T\). In this context, they obtain the following results: Let \((f,g):T\to T\) be a coincidence free pair of maps. Then there is a sequence \((f_n,g_n)\) of coincidence free pairs of maps homotopic to \((f,g)\) such that any homotopy between two distinct pairs \((f_n,g_n)\) and \((f_m,g_m)\) has coincidence points. On the other hand, if \((f_1,g_1):T\to T\) are homotopic pairs such that \(|\Lambda(f_1,g_1)|=:n\not=0\) (where \(\Lambda\) denotes the Lefschetz coincidence number) and \((f_i,g_i)\) have \(n\) coincidences for \(i=1,2\) then there is a pair of homotopies \((H,G)\) between \((f_1,g_1)\) and \((f_2,g_2)\) that has \(n\) coincidences at each level. A similar result holds if \(T\) is replaced by \(S^1\).