The Brouwer invariance theorems in reverse mathematics

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...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2020, Vol.8, Article e51
Main Author: Kihara, Takayuki
Format: Article
Language:English
Subjects:
03B30
54F45
55M10
Algorithms
Foundations
Invariance
Mathematical analysis
Theorems
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Summary: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 $\mathbb {R}^3$ .
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2020.52