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The Brouwer invariance theorems in reverse mathematics

Part of: General logic Special properties Classical Topics in Algebraic Topology

Published online by Cambridge University Press:  13 November 2020

Takayuki Kihara
Takayuki Kihara*
Affiliation:
Graduate School of Informatics, Nagoya University, Nagoya464-8601, Japan; E-mail: kihara@i.nagoya-u.ac.jp

Abstract

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In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit algorithm which, whenever the weak König’s lemma is false, constructs a topological embedding of $\mathbb {R}^4$ into $\mathbb {R}^3$ .

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 54F45: Dimension theory 55M10: Dimension theory
Type
Foundations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e51
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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