A228038 Dimensions in which nonzero Arf-Kervaire invariants exist.

2, 6, 14, 30, 62, 126

Offset: 2

Author

Jonathan Sondow, Sep 01 2013

Keywords

• nonn
• fini
• full

Comments

Hill, Hopkins, and Ravenel (2009) proved that nonzero Arf-Kervaire invariants exist only in dimensions 2^n - 2 for n = 2, 3, 4, 5, 6, and possibly 7, that is, in dimensions 2, 6, 14, 30, 62 and possibly 126.

The preprint by Lin, Wang, and Xu asserts that nonzero Arf-Kervaire invariants exist in dimension 126.

Links

• J. Baez, ‘Kervaire Invariant One Problem’ Solved, The n-Category Café (blog), April 2009.
• M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, arXiv:0908.3724 [math.AT], 2009-2015.
• Erica Klarreich, Dimension 126 Contains Strangely Twisted Shapes, Mathematicians Prove, Quanta Magazine, May 2025.
• Weinan Lin, Guozhen Wang, and Zhouli Xu, On the Last Kervaire Invariant Problem, arXiv:2412.10879 [math.AT], 2024-2025.
• V. P. Snaith, A history of the Arf-Kervaire invariant problem, Notices Amer. Math. Soc., 60 (No. 8, 2013), 1040-1047.
• Wikipedia, Kervaire invariant

Crossrefs

Cf. A228689, A000918.

Formula

a(n) = 2^n - 2 for n = 2, 3, 4, 5, 6, 7.

Extensions

a(7) from David Radcliffe, May 09 2025