On the action of \(\beta_1\) in the stable homotopy of spheres at the prime 3
This paper investigates multiplicative relations for various \(\beta\)-elements in the stable homotopy groups of spheres \(\pi_*S^0\) localized at the prime~\(3\). In particular, for the family \(\beta_{9t+i}\) with \(i=0,1,2,5,6\) and \(t\geq 0\) and Ravenel's element \(\beta_{6/3}\), it shows that \[ \begin{aligned} \beta_{9t+i}\beta_1^i\neq 0&\Longleftrightarrow i<5, \\ \beta_{9t+i}\beta_2\beta_1^j\neq 0&\Longleftrightarrow j<2,\\ \beta_{9t+i}\beta_{6/3}\beta_1^k\neq 0&\Longleftrightarrow k<4. \end{aligned} \] The methods used are calculational and include the traditional ones involving the Smith-Toda complex \(V(1)\), a self map \(B:\Sigma^{144}V(1)\rightarrow V(1)\) inducing multiplication by \(v_2^9\) in \(BP\)-homology, and the chromatic approach to understanding the \(\text{E}_2\)-term of the Adams-Novikov spectral sequence.
scientific article; zbMATH DE number 1539268