Acknowledgement
The first author is partly supported by NRF young researcher program (2021R1C1C1012939) and Samsung Science and Technology Foundation Grant.
References
- I. R. Aitchison and J. H. Rubinstein, Fibered knots and involutions on homotopy spheres, In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., 1-74. Amer. Math. Soc., Providence, RI (1984). DOI:10.1090/conm/035/780575.
- S. Akbulut, A fake 4-manifold, In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 75-141. Amer. Math. Soc., Providence, RI (1984). DOI:10.1090/conm/035/780576.
-
S. Akbulut, On fake
$S^3{\tilde{\times}}S^1{\sharp}S^2{\times}S^2$ , In Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), volume 44 of Contemp. Math., 281-286. Amer. Math. Soc., Providence, RI (1985). DOI:10.1090/conm/044/813119. - S. Akbulut, Constructing a fake 4-manifold by Gluck construction to a standard 4-manifold, Topology, 27(2)(1988), 239-243. DOI:10.1016/0040-9383(88)90041-9.
- S. Akbulut, Scharlemann's manifold is standard, Ann. of Math. (2), 149(2)(1999), 497-510. DOI:10.2307/120972.
- S. Akbulut, Cappell-Shaneson's 4-dimensional s-cobordism, Geom. Topol., 6(2002), 425-494. DOI:10.2140/gt.2002.6.425.
- S. Akbulut, Cappell-Shaneson homotopy spheres are standard, Ann. of Math. (2), 171(3)(2010), 2171-2175. DOI:10.4007/annals.2010.171.2171.
- S. Akbulut, The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture, Comment. Math. Helv., 87(1)(2012), 187-241. DOI:10.4171/CMH/252.
- S. Akbulut, 4-manifolds, volume 25 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford (2016). ISBN 978-0-19-878486-9. DOI:10.1093/acprof:oso/9780198784869.001.0001.
- S. Akbulut and R. C. Kirby, An exotic involution of S4, Topology, 18(1)(1979), 75-81. DOI:10.1016/0040-9383(79)90015-6.
- S. Akbulut and R. C. Kirby, A potential smooth counterexample in dimension 4 to the Poincar'e conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology, 24(4)(1985), 375-390. DOI:10.1016/0040-9383(85)90010-2.
- M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969).
- S. E. Cappell and J. L. Shaneson, Some new four-manifolds, Ann. of Math. (2), 104(1)(1976), 61-72. https://doi.org/10.2307/1971056
- S. E. Cappell and J. L. Shaneson, There exist inequivalent knots with the same complement, Ann. of Math. (2), 103(2)(1976), 349-353. https://doi.org/10.2307/1970942
- H. Cohen et al, PARI/GP version 2.7.5, The PARI Group, Bordeaux (2015). Available from http://pari.math.u-bordeaux.fr.
- G. Earle, Even more Cappell-Shaneson spheres are standard, Master's thesis, The University of Texas at Austin (2014).
- M. H. Freedman, R. E. Gompf, S. Morrison and K. Walker, Man and machine thinking about the smooth 4-dimensional Poincar'e conjecture, Quantum Topol., 1(2)(2010), 171-208. https://doi.org/10.4171/qt/5
- R. E. Gompf, Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems, Topology, 30(1)(1991), 97-115. DOI:10.1016/0040-9383(91)90036-4.
- R. E. Gompf, On Cappell-Shaneson 4-spheres, Topology Appl., 38(2)(1991), 123-136. DOI:10.1016/0166-8641(91)90079-2.
- R. E. Gompf, More Cappell-Shaneson spheres are standard, Algebr. Geom. Topol., 10(3)(2010), 1665-1681. DOI:10.2140/agt.2010.10.1665.
- C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. (2), 34(2)(1933), 313-316. DOI:10.2307/1968204.
- M. Newman, Integral matrices, Academic Press, New York-London (1972). Pure and Applied Mathematics, Vol. 45.
- P. Stevenhagen, The arithmetic of number rings, In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., 209-266. Cambridge Univ. Press, Cambridge (2008).
- O. Taussky, On a theorem of Latimer and MacDuffee, Canadian J. Math., 1(1949), 300-302. https://doi.org/10.4153/CJM-1949-026-1