Unstable v1-periodic Homotopy Groups through Goodwillie Calculus

It is a classical result that the rational homotopy groups, $\pi_*(X) \otimes \mathbb{Q}$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on X.

 

This result can also be recovered from applying Goodwillie calculus to rational homotopy theory.

 

A different simplification of the homotopy theory, is $v_h$-periodic homotopy theory. For $h = 1$ we are able to compute the K-theory based $v_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $v_1^{-1} \pi_*SU(d)$ in a very different way from the original computation by Davis.