A fixed point theorem for o‐minimal structures

Abstract

We prove a definable analogue to Brouwer's Fixed Point Theorem for o-minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o-minimal functions, with an application of Sperner's Lemma. This paper operates in the definable context of o-minimal structures as developed in the series of papers of van den Dries [6], Pillay and Steinhorn [18], Knight et al. [14], Wilkie [21], and others. We prove a definable analogue to Brouwer's Fixed Point Theorem for o-minimal structures of real closed field expansions: Every continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete 1) . Our proof applies a triangulation technique for o-minimal functions (cf. van den Dries [7]), with an application of Sperner's Lemma 2) .Definable analogues to Brouwer's Theorem, and its generalization, the Lefschetz Fixed Point Theorem, have been recently obtained by Edmundo [8,9] and Berarducci and Otero [1]. These papers use o-minimal homology developed by Woerheide [22] and others. In contrast, our proof is direct and elementary and does not use o-minimal homology. We begin by reviewing some notations about o-minimal structures 3) . Recall that for a structure M = (M, . . . ), a set X M n is parametrically definable if there is a first order formula ϕ(x 1 , . . . , x n ) ∈ L(M) (the first order predicate language 4) of M together with names for the elements of M ), such that X is the set of all b ∈ M n satisfying ϕ( · ) in M, i. e. X = {b ∈ M n : M ϕ[b ]}. For brevity, we drop mention of parameters, and refer simply to "definable". A function isAn o-minimal structure is an ordered structure M = (M, <, . . . ) in which every definable (with parameters) subset of M is a finite union of intervals 5) .Our results apply to those structures M that are expansions of real closed ordered fields 6) , so M = (M, <, +, · , 0, 1, . . . ) 7) . As usual, we endow M with the interval topology, i. e. intervals (a, b) form a basis; and M n is given the product topology. We use to denote the closed unit simplex in M n , i. e.,

Computing o-minimal topological invariants using differential topology

“…In the o-minimal setting this theorem has already been proved by various methods (e.g., see [20], [2], [21]), however, we think that the proof presented here is a good illustration of the usefulness of our notion of degree.…”