Prasit Bhattacharya (University of Virginia)

"A $2$-local type $3$ spectrum, its periodic $v_3$-self-map, and its $K(3)$-local homotopy groups."

Constructing type n spectrum and understanding its periodic $v_n$-self-map is an integral part chromatic homotopy theory, potentially can  produce infinite families in chromatic layer $n$ of the stable homotopy groups of spheres.  Recently, in a joint work with Philip Egger, we produce $2$-local type $2$ spectra $Z$ which admit $v_2^1$-self-map. In fact, $Z$ in many ways the height 2 analogue of the spectrum $Y:= M_2(1) \wedge C\eta$. In a joint work with Nicolas Ricka, we extend the family consisting of $Y$ and $Z$ to chromatic  height 3, by producing a type $3$ spectrum at the prime $2$. In this talk, I will explain multiple ways of constructing this spectrum, one of them uses the technique of Jeff Smith that is used in the proof of the famous `thick subcategory theorem’ due to Hopkins and Smith. Then I will outline the proof of  $v_3^4$-self-map and give evidences in favor of this spectrum admitting  $v_3^1$-self-map. Time permitting, I will also talk about the action of height $3$ Morava stabilizer group on Morava E-theory of this spectrum and give a  complete answer for the $E_2$-page of the descent spectral sequence which  computes $K(3)$-local homotopy groups of this spectrum.