Geodesic
The Brouwer invariance theorems in reverse mathematics
Forum of Mathematics, Sigma, Tome 8 (2020)

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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 $\mathbb {R}^3$. 
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     author = {Takayuki Kihara},
     title = {The {Brouwer} invariance theorems in reverse mathematics},
     journal = {Forum of Mathematics, Sigma},
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     year = {2020},
     doi = {10.1017/fms.2020.52},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.52/}
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Takayuki Kihara. The Brouwer invariance theorems in reverse mathematics. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.52

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