Removing coincidences of maps between manifolds of different dimensions (Q1876355)

The author considers the coincidence set of maps \(f, g\colon N^{n+m}\to M^m\) between oriented compact smooth manifolds. Let \(C\) be an isolated coincidence set. A local removability condition is obtained, namely, \(C\) is mapped by \(f\) and \(g\) into the same point in the interior of \(M\), and \(H^{k+1}(W, W-V; \pi_k(S^{n-1})) = 0\) for all \(k\geq n+1\), where \(W\) and \(V\) are two neighborhoods of \(C\) in \(N\) with \(C\subset V\subset \overline{V}\subset W\) and \(W\cap Coin(f,g) =C\). Under these conditions, \(C\) can be removed via a local homotopy of \(f\) provided the cohomology coincidence index of \(C\) is zero. The author also illustrates some cases in which the local removability condition is satisfied, including e.g. that the homotopy groups \(\pi_k(S^{n-1})\) vanish.