**Title:** Unstable $v_1$-periodic homotopy groups through Goodwillie calculus

**Speaker:** Jens Kjaer

**Speaker Info:** University of Notre Dame

**Brief Description:**

**Special Note**:

**Abstract:**

It is a classical result that the rational homotopy groups, $\pi_*(X) \otimes Q$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on X.The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its "spectral Lie algebra". Unfortunately both input and differentials are hard to get at.

We therefore simplify the homotopy groups by taking the unstable $v_h$-periodic homotopy groups, $v_h^{-1} \pi_*( )$ (note h = 0 recovers rational homotopy groups). For h = 1 we are able to compute the K-theory based $\nu_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $\nu_1^{-1} \pi_*SU(d)$ in a very different way from the original computation by Davis.

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