arXiv subjects
Publications and source records attributed to David Jongwon Lee.
We prove the $K(n)$-local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height $1$. Along the way, we prove the existence of $K(n)$-local finite complexes with particularly regular rings of homotopy groups.
We compute the homotopy groups of $\mathrm{THH}(\mathrm{ku})$ as a $\mathrm{ku}_\ast$-module using the descent spectral sequence for the map $\mathrm{THH}(\mathrm{ku})\to\mathrm{THH}(\mathrm{ku}/\mathrm{MU})$, which is the motivic spectral sequence for $\mathrm{THH}(\mathrm{ku})$ in the sense of Hahn-Raksit-Wilson. We reduce the computation of homotopy groups to the algebra of the universal formal group law, providing a systematic way to compute THH of quotients of MU. We compute the $E_2$-page of the motivic spectral sequence computing $\mathrm{THH}(\mathrm{ku})$, and we show that it degenerates at the $E_2$-page.
We compute the mod $(p,v_1)$ and mod $(2,η,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_ζ$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_ζ \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_ζ$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$.
When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of $\mathrm{BP}\langle n\rangle$ as a module over the Steenrod algebra determines the $p$-local spectrum $\mathrm{BP}\langle n\rangle$. In particular, we prove that the $p$-local spectrum $\mathrm{BP}\langle n\rangle$ only depends on its $p$-completion $\mathrm{BP}\langle n\rangle_p^\wedge$. As a corollary, this proves that the $p$-local homotopy type of $\mathrm{BP}\langle n\rangle$ does not depend on the ideal by which we take the quotient of $\mathrm{BP}$. In the course of the argument, we show that there is a vanishing line for odd degree classes in the Adams spectral sequence for endomorphisms of $\mathrm{BP}\langle n\rangle$. We also prove that there are enough endomorphisms of $\mathrm{BP}\langle n\rangle$ in a suitable sense. When $p=2$, we obtain the results for $n\leq 3$.