Geometry & Topology

The strong Kervaire invariant problem in dimension $62$

Zhouli Xu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Using a Toda bracket computation θ4,2,σ2 due to Daniel C Isaksen, we investigate the 45–stem more thoroughly. We prove that θ42 = 0 using a 4–fold Toda bracket. By work of Barratt, Jones and Mahowald, this implies that θ5 exists and there exists a θ5 such that 2θ5 = 0. Based on θ42 = 0, we simplify significantly their 9–cell complex construction to a 4–cell complex, which leads to another proof that θ5 exists.

Article information

Geom. Topol., Volume 20, Number 3 (2016), 1611-1624.

Received: 4 November 2014
Accepted: 18 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q45: Stable homotopy of spheres

Kervaire invariant Toda brackets


Xu, Zhouli. The strong Kervaire invariant problem in dimension $62$. Geom. Topol. 20 (2016), no. 3, 1611--1624. doi:10.2140/gt.2016.20.1611.

Export citation


  • M G Barratt, J D S Jones, M E Mahowald, The Kervaire invariant problem, from: “Proceedings of the Northwestern Homotopy Theory Conference”, (H R Miller, S B Priddy, editors), Contemp. Math. 19, Amer. Math. Soc., Providence, RI (1983) 9–22
  • M G Barratt, J D S Jones, M E Mahowald, Relations amongst Toda brackets and the Kervaire invariant in dimension $62$, J. London Math. Soc. 30 (1984) 533–550
  • M G Barratt, M E Mahowald, M C Tangora, Some differentials in the Adams spectral sequence, II, Topology 9 (1970) 309–316
  • T Bauer, Computation of the homotopy of the spectrum tmf, from: “Groups, homotopy and configuration spaces”, (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13 (2008) 11–40
  • W Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. 90 (1969) 157–186
  • A Henriques, The homotopy groups of tmf and of its localizations, electronic notes (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • M A Hill, M J Hopkins, D C Ravenel, On the non-existence of elements of Kervaire invariant one, preprint (2009) To appear in Ann. of Math.
  • D C Isaksen, Classical and motivic Adams charts, preprint (2014)
  • D C Isaksen, Stable stems, preprint (2014)
  • S O Kochman, Stable homotopy groups of spheres: a computer-assisted approach, Lecture Notes in Mathematics 1423, Springer, Berlin (1990)
  • S O Kochman, Bordism, stable homotopy and Adams spectral sequences, Fields Institute Monographs 7, Amer. Math. Soc., Providence, RI (1996)
  • W-H Lin, A proof of the strong Kervaire invariant in dimension $62$, from: “First International Congress of Chinese Mathematicians”, (L Yang, S-T Yau, editors), AMS/IP Stud. Adv. Math. 20, Amer. Math. Soc., Providence, RI (2001) 351–358
  • M Mahowald, M Tangora, Some differentials in the Adams spectral sequence, Topology 6 (1967) 349–369
  • R J Milgram, Symmetries and operations in homotopy theory, from: “Algebraic topology”, (A Liulevicius, editor), Proc. Sympos. Pure Math. 22, Amer. Math. Soc., Providence, RI (1971) 203–210
  • R E Mosher, M C Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, New York (1968)
  • R M F Moss, Secondary compositions and the Adams spectral sequence, Math. Z. 115 (1970) 283–310
  • F P Peterson, N Stein, Secondary cohomology operations: two formulas, Amer. J. Math. 81 (1959) 281–305
  • M Tangora, Some extension questions in the Adams spectral sequence, from: “Proceedings of the Advanced Study Institute on Algebraic Topology, III”, Various Publ. Ser 13, Math. Inst. Aarhus Univ. (1970) 578–587