A survey of unstable $v_n$-periodic homotopy

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Robert Thompson, Hunter College, CUNY
Fine Hall 110

In this talk I will survey some constructions and techniques designed to study unstable chromatic homotopy theory.  If $W$ is a finite complex with a periodic self map $v:\Sigma^k W \to W$, and one takes homotopy groups of a space or spectrum $X$ with coefficients in $W$, then the resulting localized homotopy groups $v^{-1}\pi(X;W)$ are called the $v$-periodic homotopy groups. Computing periodic homotopy groups is a fundamental problem in homotopy theory.

I will begin by discussing the situation stably, where we will immediately come up against the telescope conjecture, which has not yet been settled.  Then I will turn to the unstable case and survey some methods and known results involving the unstable Adams spectral, the Goodwill tower, and the Bousfield-Kunn Phi functor. I will discuss unstable analogs of the telescope conjecture and various possible relationships between the above mentioned constructions.  I will probably state more conjectures than theorems.