Title: Recent results in the fixed point theory of continuous maps
Abstract: 1 1. Introduction.Given a function/: X-+X, any question which inquires into the existence, nature and number of points xÇzX such that ƒ(#)=# is called fixed point theory.The assumptions on ƒ and X range from practically none (e.g., X is a set, ƒ a function) to quite stringent assumptions on ƒ and -X" (e.g., X is a Riemannian manifold and ƒ is an isometry).Our attention will be focused on results which require X to be a fairly reasonable space (e.g., a finite polyhedron) and ƒ a map ( = continuous function).Furthermore, we will limit our discussion to results which are not included in the expository tract [49] by Van der Walt (1967), which adequately covers the history of the subject from its beginning around 1910 to the early sixties.2. The Lefschetz theorem and local index theory.One of the most useful tools in fixed point theory is the Lefschetz Fixed Point Theorem [34], [35], [25].In its most elementary form it is simply this.Let X denote a finite polyhedron and ƒ : X-*X a map.Then, using the field of rationals Q as coefficients, ƒ induces homomorphisms.(1) fa:H k (X;Q)->H k (X;Q).The number (it turns out to be an integer)(2) L(f) = E (-1)* Trace/*, k is called the Lefschetz number of ƒ.Then a sufficient condition for ƒ to have at least one fixed point is that L(f)-^0.In short,(3) L(f) T* 0 =»ƒ(*) = x for some x G X.