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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: [email protected]

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

References

Beeson, M. J., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, Vol. 6 (Springer, Berlin, 1985).Google Scholar
Engelking, R., Dimension Theory (North-Holland Publishing Co., Amsterdam, 1978). Translated from the Polish and revised by the author.Google Scholar
Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2002).Google Scholar
Julian, W., Mines, R. and Richman, F., ‘Alexander duality’, Pacific J. Math., 106 (1983) 115–127.CrossRefGoogle Scholar
Kihara, T. and Pauly, A., ‘Point degree spectra of represented spaces’, Preprint, 2014, arXiv:1405.6866.Google Scholar
Miller, J. S., ‘Degrees of unsolvability of continuous functions’, J. Symb. Log., 69 (2004) 555–584.Google Scholar
Nagata, J., Modern Dimension Theory , rev. ed., Sigma Series in Pure Mathematics, 2 (Heldermann Verlag, Berlin, 1983).Google Scholar
Orevkov, V. P.,‘A constructive map of the square into itself, which moves every constructive point’,Dokl. Akad. Nauk, 152 (1963) 55–58.Google Scholar
Sakamoto, N. and Yokoyama, K., ‘The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic’, Arch. Math. Logic, 46 (2007) 465–480.CrossRefGoogle Scholar
Shioji, N. and Tanaka, K., ‘Fixed point theory in weak second-order arithmetic’,Ann. Pure Appl. Logic, 47 (1990) 167–188.Google Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic, 2nd ed., Perspectives in Logic (Association for Symbolic Logic, Poughkeepsie, 2009).Google Scholar
Stillwell, J., Reverse Mathematics: Proofs from the Inside Out (Princeton University Press, Princeton, 2018).Google Scholar
Tao, T., Hilbert’s Fifth Problem and Related Topics , Graduate Studies in Mathematics, 153 (American Mathematical Society, Providence, 2014).Google Scholar
van Dalen, D., Brouwer, L. E. J.— Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life (London, Springer, 2013).Google Scholar