讲座内容：

In this talk, I would like to introduce our recent work on the “Periodicity” phenomenon of the attaching map of the suspended two-cell CW complex.

I will introduce Selick-Wu's A^{min}-theory, a theory on the homotopy functor decomposition and becoming a bridge between homotopy theory and the representation theory. Employing this theory, we deduce that the attaching map of the CW complex above manifests a“periodicity”phenomenon.

The result essentially furnishes, up to the present, one of the most effective methods for computing the 2-torsion parts of the unstable homotopy groups \pi_{i}(\Sigma(S^{n} \cup e^{n+k+1})), where 3n + 2k + 1 < i <4n + 3k + 3 and n>1, provided the requisite homotopy groups of spheres are known.

The proof requires the Hopf algbra structure of the looped homotopy fibre of the pinch map, giving a generalization of a result of    Cohen-Moore-Neisendorfer’s 1979 classic  paper.