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Ideal classes and Cappell-Shaneson homotopy 4-spheres

Ideal classes and Cappell-Shaneson homotopy 4-spheres

Authors :
Kim, Min Hoon
Yamada, Shohei
Publication Year :
2017

Abstract

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 $-64\leq \textrm{trace}\leq 69$ 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.<br />Comment: 30 pages

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1707.03860
Document Type :
Working Paper