A fixed point theorem for smooth extension maps

Abstract

Let X be a compact smooth n-manifold, with or without boundary, and let A be an (n - 1)-dimensional smooth submanifold of the interior of X. Let φ: A→A be a smooth map and f: (X, A)→(X, A) be a smooth map whose restriction to A is φ. If p ş̌ A is an isolated fixed point of f that is a transversal fixed point of φ, that is, the linear transformation dφp- IA: TpA→TpA is nonsingular, then the fixed point index of f at p satisfies the inequality |i(X, f , p)| ≤ 1. It follows that if φ has k fixed points, all transverse, and the Lefschetz number L(f) > k, then there is at least one fixed point of f in X \ A. Examples demonstrate that these results do not hold if the maps are not smooth. © 2014 Khamsemanan et al.; licensee Springer.