Coincidences in codimension one
Let \((f_{1},f_{2})\) be a pair of maps from the \((n+1)\)-sphere \(S^{n+1}\) to a manifold \(Y\) of dimension \(n \geq 1.\) In this paper the authors study the minimum numbers \(MC(f_{1}, f_{2})\) and \(MCC(f_{1},f_{2}),\) where \[ MC(f_{1},f_{2}):= \min\{\#\mathrm{Coin}(f^{'}_{1},f^{'}_{2})\mid (f^{'}_{1},f^{'}_{2}) \simeq (f_{1},f_{2}) \} \] and \[ MCC(f_{1},f_{2}):= \min\{\#\pi_{0}(\mathrm{Coin}(f^{'}_{1},f^{'}_{2}))\mid (f^{'}_{1},f^{'}_{2}) \simeq (f_{1},f_{2}) \} \] The study of these sets is based on the so-called \(\omega\)-invariant which is defined using framed bordism. Computations of \(MC(f_{1}, f_{2})\) and \(MCC(f_{1},f_{2}),\) in the case of \(Y = \mathbb{C}P(n), \,\, n \geq 1\) or \(Y = S^{n}/G ,\) the orbit manifold of a finite group \(G\) acting freely and smoothly on the \(n\)-sphere, \(n \geq 3,\) are presented. The authors also present a type of Wecken theorem for coincidence, the proof is in terms of the \(\omega\)-invariant.
scientific article; zbMATH DE number 6905246
scientific article