Kyungpook Mathematical Journal

  • 제63권3호
  • /
  • Pages.373-411
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  • 2023
  • /
  • 1225-6951(pISSN)
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  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
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Ideal Classes and Cappell-Shaneson Homotopy 4-Spheres

  • Min Hoon Kim (Department of Mathematics Education, Kyungpook National University) ;
  • Shohei Yamada (College of Natural Sciences Department of Mathematics Kyungpook National University)
  • 투고 : 2023.01.18
  • 심사 : 2023.07.15
  • 발행 : 2023.09.30
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초록

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5 - n Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace n case is equivalent to the trace 5 - n case. We confirm Gompf conjecture for the special cases that -64 ≤ trace ≤ 69 and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.

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과제정보

The first author is partly supported by NRF young researcher program (2021R1C1C1012939) and Samsung Science and Technology Foundation Grant.

참고문헌

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