Connected Choice and the Brouwer Fixed Point Theorem

TL;DR

This paper explores the computational complexity of the Brouwer Fixed Point Theorem within the Weihrauch lattice, establishing equivalences with connected choice and Weak K"onig's Lemma across different dimensions.

Contribution

It demonstrates the equivalence of Brouwer's Fixed Point Theorem and connected choice in various dimensions and introduces new representations for closed sets in Euclidean spaces.

Findings

Brouwer Fixed Point Theorem is equivalent to connected choice in each fixed dimension.

Connected choice is complete for dimensions ≥2, equivalent to Weak K"onig's Lemma.

The connected choice in dimension one relates to the Intermediate Value Theorem but is not idempotent.

Abstract

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.