CiE 2012: How the World Computes pp 56-67

# On the Computational Content of the Brouwer Fixed Point Theorem

• Vasco Brattka
• Stéphane Le Roux
• Arno Pauly
Conference paper

DOI: 10.1007/978-3-642-30870-3_7

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)
Cite this paper as:
Brattka V., Le Roux S., Pauly A. (2012) On the Computational Content of the Brouwer Fixed Point Theorem. In: Cooper S.B., Dawar A., Löwe B. (eds) How the World Computes. CiE 2012. Lecture Notes in Computer Science, vol 7318. Springer, Berlin, Heidelberg

## Abstract

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. 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. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. Another main result is that connected choice is complete for dimension greater or equal to three in the sense that it is computably equivalent to Weak Kőnig’s Lemma. In contrast to this, the connected choice operations in dimensions zero, one and two form a strictly increasing sequence of Weihrauch degrees, where connected choice of dimension one is known to be equivalent to the Intermediate Value Theorem. Whether connected choice of dimension two is strictly below connected choice of dimension three or equivalent to it is unknown, but we conjecture that the reduction is strict. As a side result we also prove that finding a connectedness component in a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma.

## Authors and Affiliations

• Vasco Brattka
• 1
• Stéphane Le Roux
• 2
• Arno Pauly
• 3
1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa