## Algebraic & Geometric Topology

### Detection of a nontrivial product in the stable homotopy groups of spheres

#### Abstract

In this paper, we prove that there exists a new family of nontrivial homotopy elements in the stable homotopy groups of spheres with dimension $q(pn+sp+2)−4$. These nontrivial homotopy elements are represented by $β̃sh0hn$ in the $E2s+2,t$–term of the Adams spectral sequence, where $p≥5$, $n>4$, $p+1, $t=q(pn+sp+2)+s−2$, $q=2(p−1)$.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 5 (2013), 3009-3029.

Dates
Revised: 27 September 2012
Accepted: 30 September 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715698

Digital Object Identifier
doi:10.2140/agt.2013.13.3009

Mathematical Reviews number (MathSciNet)
MR3116311

Zentralblatt MATH identifier
1272.55009

Subjects
Primary: 55Q45: Stable homotopy of spheres
Secondary: 55T15: Adams spectral sequences 55S10: Steenrod algebra

#### Citation

Zhong, Linan; Wang, Yuyu. Detection of a nontrivial product in the stable homotopy groups of spheres. Algebr. Geom. Topol. 13 (2013), no. 5, 3009--3029. doi:10.2140/agt.2013.13.3009. https://projecteuclid.org/euclid.agt/1513715698

#### References

• T Aikawa, $3$–dimensional cohomology of the mod $p$ Steenrod algebra, Math. Scand. 47 (1980) 91–115
• R L Cohen, Odd primary infinite families in stable homotopy theory, Mem. Amer. Math. Soc. 30 (1981) viii+92
• C-N Lee, Detection of some elements in the stable homotopy groups of spheres, Math. Z. 222 (1996) 231–245
• X-G Liu, Some infinite elements in the Adams spectral sequence for the sphere spectrum, J. Math. Kyoto Univ. 48 (2008) 617–629
• X Liu, A Toda bracket in the stable homotopy groups of spheres, Algebr. Geom. Topol. 9 (2009) 221–236
• X Liu, W Li, A product involving the $\beta$–family in stable homotopy theory, Bull. Malays. Math. Sci. Soc. 33 (2010) 411–420
• X G Liu, X J Wang, A four-filtrated May spectral sequence and its applications, Acta Math. Sin. (Engl. Ser.) 24 (2008) 1507–1524
• A Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. No. 42 (1962) 112
• M Mahowald, A new infinite family in ${}_{2}\pi_{*}{}^{s}$, Topology 16 (1977)
• H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. of Math. 106 (1977) 469–516
• D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press, Orlando, FL (1986)
• X Wang, Some notes on the Adams spectral sequence, Acta Math. Sinica 10 (1994) 4–10
• X Wang, Q Zheng, The convergence of ${\tilde{\alpha}_s^{(n)}h_0 h_k}$, Sci. China Ser. A 41 (1998) 622–628