A fixed point theorem for smooth extension maps

Nirattaya Khamsemanan,Sompong Dhompongsa,Robert F Brown,Catherine Lee

doi:10.1186/1687-1812-2014-97

Abstract

Let X be a compact smooth n-manifold, with or without boundary, and let A be an -dimensional smooth submanifold of the interior of X. Let be a smooth map and be a smooth map whose restriction to A is ϕ. If is an isolated fixed point of f that is a transversal fixed point of ϕ, that is, the linear transformation is nonsingular, then the fixed point index of f at p satisfies the inequality . It follows that if ϕ has k fixed points, all transverse, and the Lefschetz number , then there is at least one fixed point of f in . Examples demonstrate that these results do not hold if the maps are not smooth. MSC:55M20, 54C20.

Highlights

1 Introduction It has been known at least since the work of Shub and Sullivan in [ ] that the values of the fixed point index of smooth maps are more restricted than they are for continuous functions in general
We obtain a condition on the Lefschetz number L(f ) of f that implies the existence of fixed points of f in X \ A
We demonstrate by an example that the same Lefschetz number condition is not sufficient to imply the existence of fixed points in X \ A for maps f : (X, A) → (X, A) in general

Summary

Introduction

Introduction It has been known at least since the work of Shub and Sullivan in [ ] that the values of the fixed point index of smooth maps are more restricted than they are for continuous functions in general. A consequence of this result is that, under appropriate hypotheses on a smooth map f , it must have fixed points on X \ ∂X, the interior of the manifold X.

Objectives

Results

Conclusion