Interior fixed points of unit-sphere-preserving Euclidean maps (Q385679)

A smooth map \(\phi:S^{n-1}\to S^{n-1}\) is \textit{sparse} if it is transversely fixed and it has \(N_{\pitchfork}(\phi)\) fixed points, where \(N_{\pitchfork}(\phi)\) denotes the \textit{transverse Nielsen number}. Schirmer proved that if \(n\geq 2\), \(\phi:S^{n-1}\to S^{n-1}\) is a sparse map of degree \(d\) and \(f:(B^n,S^{n-1})\to ((B^n,S^{n-1}))\) is a smooth map extending \(\phi\) with \((-1)^nd\geq 2\), then \(f\) must have a fixed point in int(\(B^n\)). In this work the authors generalize Schirmer's result by proving that if \(n\geq 2\), \(\phi:S^{n-1}\to S^{n-1}\) is a sparse map of degree \(d\) and \(f:(\mathbb{R}^n,S^{n-1})\to (\mathbb{R}^n,S^{n-1})\) is a smooth map extending \(\phi\) with \((-1)^nd\geq 2\), then \(f\) must have a fixed point in int(\(B^n\)).