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 Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward 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 upward. 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 than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

中文翻译：

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连通选择和 Brouwer 不动点定理

我们研究了 Weihrauch 晶格中 Brouwer 不动点定理的计算内容。连通选择是在负信息给定的非空连通封闭集中找到一个点的操作。我们的主要结果之一是，对于任何固定维度，该维度的 Brouwer 不动点定理在计算上等价于相同维度的欧几里德单位立方体的连通选择。另一个主要结果是，对于大于或等于 2 的维度，连通选择是完整的，因为它在计算上等同于 Weak Kőnig 引理。虽然我们可以为维度 3 和更高维度提供两个独立的证明，它们要么基于简单的几何构造，要么基于组合论证，但维度 2 的证明基于更复杂的逆极限构造。已知维度一的连通选择操作等价于中值定理；我们证明了这个问题与二维及向上的情况相比不是幂等的。我们还证明了具有严格大于 1 的 Lipschitz 常数的 Lipschitz 连续性并不能简化寻找不动点的过程。最后，我们证明了找到任何维度大于或等于 1 的欧几里得单位立方体的闭合子集的连通性分量等价于 Weak Kőnig 引理。为了描述这些结果，我们通过有理复形树引入单位立方体的封闭子集的表示。我们还证明了具有严格大于 1 的 Lipschitz 常数的 Lipschitz 连续性并不能简化寻找不动点的过程。最后，我们证明了找到任何维度大于或等于 1 的欧几里得单位立方体的闭合子集的连通性分量等价于 Weak Kőnig 引理。为了描述这些结果，我们通过有理复形树引入单位立方体的封闭子集的表示。我们还证明了具有严格大于 1 的 Lipschitz 常数的 Lipschitz 连续性并不能简化寻找不动点的过程。最后，我们证明了找到任何维度大于或等于 1 的欧几里得单位立方体的闭合子集的连通性分量等价于 Weak Kőnig 引理。为了描述这些结果，我们通过有理复形树引入单位立方体的封闭子集的表示。