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Contractible 3-manifolds and the double 3-space property


Authors: Dennis J. Garity, Dušan D. Repovš and David G. Wright
Journal: Trans. Amer. Math. Soc. 370 (2018), 2039-2055
MSC (2010): Primary 54E45, 54F65; Secondary 57M30, 57N10
DOI: https://doi.org/10.1090/tran/7035
Published electronically: November 7, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $ \mathbb{R}^3$ and whose intersection is again homeomorphic to $ \mathbb{R}^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.


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  • [AS89] Fredric D. Ancel and Michael P. Starbird, The shrinkability of Bing-Whitehead decompositions, Topology 28 (1989), no. 3, 291-304. MR 1014463, https://doi.org/10.1016/0040-9383(89)90010-4
  • [AW] Kathryn B. Andrist and David G. Wright, On computing the geometric index, preprint, B.Y.U., Provo, 2000.
  • [Bro61] Morton Brown, The monotone union of open $ n$-cells is an open $ n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812-814. MR 0126835, https://doi.org/10.2307/2034881
  • [Dav07] Robert J. Daverman, Decompositions of manifolds, AMS Chelsea Publishing, Providence, RI, 2007. Reprint of the 1986 original. MR 2341468
  • [DV09] Robert J. Daverman and Gerard A. Venema, Embeddings in manifolds, Graduate Studies in Mathematics, vol. 106, American Mathematical Society, Providence, RI, 2009. MR 2561389
  • [Dol95] Albrecht Dold, Lectures on algebraic topology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1335915
  • [Gab11] David Gabai, The Whitehead manifold is a union of two Euclidean spaces, J. Topol. 4 (2011), no. 3, 529-534. MR 2832566, https://doi.org/10.1112/jtopol/jtr010
  • [GRWŽ11] Dennis Garity, Dušan Repovš, David Wright, and Matjaž Željko, Distinguishing Bing-Whitehead Cantor sets, Trans. Amer. Math. Soc. 363 (2011), no. 2, 1007-1022. MR 2728594, https://doi.org/10.1090/S0002-9947-2010-05175-X
  • [Hem04] John Hempel, $ 3$-manifolds, AMS Chelsea Publishing, Providence, RI, 2004. Reprint of the 1976 original. MR 2098385
  • [McM62] D. R. McMillan Jr., Some contractible open $ 3$-manifolds, Trans. Amer. Math. Soc. 102 (1962), 373-382. MR 0137105, https://doi.org/10.2307/1993684
  • [Mye88] Robert Myers, Contractible open $ 3$-manifolds which are not covering spaces, Topology 27 (1988), no. 1, 27-35. MR 935526, https://doi.org/10.1016/0040-9383(88)90005-5
  • [Mye99] Robert Myers, Contractible open $ 3$-manifolds which non-trivially cover only non-compact $ 3$-manifolds, Topology 38 (1999), no. 1, 85-94. MR 1644087, https://doi.org/10.1016/S0040-9383(98)00004-4
  • [Rol90] Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
  • [RS82] Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
  • [Sch53] Horst Schubert, Knoten und Vollringe, Acta Math. 90 (1953), 131-286 (German). MR 0072482, https://doi.org/10.1007/BF02392437
  • [ST80] Herbert Seifert and William Threlfall, Seifert and Threlfall: a textbook of topology, Pure and Applied Mathematics, vol. 89, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Translated from the German edition of 1934 by Michael A. Goldman; With a preface by Joan S. Birman; With ``Topology of $ 3$-dimensional fibered spaces'' by Seifert; Translated from the German by Wolfgang Heil. MR 575168
  • [Whi35] J. H. C. Whitehead, A certain open manifold whose group is unity, Quart. J. Math. 6 (1935), no. 6, 268-279.
  • [Wri89] David G. Wright, Bing-Whitehead Cantor sets, Fund. Math. 132 (1989), no. 2, 105-116. MR 1002625
  • [Wri92] David G. Wright, Contractible open manifolds which are not covering spaces, Topology 31 (1992), no. 2, 281-291. MR 1167170, https://doi.org/10.1016/0040-9383(92)90021-9

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Additional Information

Dennis J. Garity
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: garity@math.oregonstate.edu

Dušan D. Repovš
Affiliation: Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia 1000
Email: dusan.repovs@guest.arnes.si

David G. Wright
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: wright@math.byu.edu

DOI: https://doi.org/10.1090/tran/7035
Keywords: Contractible 3-manifold, open 3-manifold, Whitehead Link, defining sequence, geometric index, McMillan contractible 3-manifold, Gabai Link
Received by editor(s): December 12, 2015
Received by editor(s) in revised form: July 21, 2016
Published electronically: November 7, 2017
Additional Notes: The first and second authors were supported in part by the Slovenian Research Agency grant BI-US/15-16-029. The first author was supported in part by the National Science Foundation grant DMS0453304. The first and third authors were supported in part by the National Science Foundation grant DMS0707489. The second author was supported in part by the Slovenian Research Agency grants P1-0292, JL-7025, and J1-6721.
Article copyright: © Copyright 2017 American Mathematical Society

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