Related papers: The Chromatic Splitting Conjecture at n=p=2

We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1L_{K(2)}S^0$…

Ravenel proved the remarkable fact that the $K$-theoretic localization $L_K S^0$ of the sphere spectrum has $\mathbb{Q}/\mathbb{Z}$ as homotopy group in dimension -2. Mike Hopkins' chromatic splitting conjecture implies more generally that…

We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case.

We formulate a version of Hopkins' chromatic splitting conjecture for an arbitrary structured ring spectrum $R$, and prove it whenever $\pi_*R$ is Noetherian. As an application, these results provide a new local-to-global principle in the…

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4…

Fix a prime $p$ and a chromatic height $h$. We prove that the homotopy $(k,1)$-category of $L_h$-local spectra $\mathrm{h}_k\big(\mathrm{Sp}_{p,h}\big)$ is algebraic as a symmetric monoidal category when $p > O(h^2+kh)$. To achieve this, we…

In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These…

We provide a lower bound for the coherence of the homotopy commutativity of the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on Dyer-Lashof…

As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the $K(n)$-local category coincides with the algebraic one when $2p-2 > n^{2}+n$.

Let V(0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F_4 defined by the Weierstrass equation y^2+y=x^3. Let F_C be its formal group law and E_C be the spectrum classifying the deformations of F_C. The…

Let $K_p$ be a complete graph of order $p\geq 2$. A $K_p$-free $k$-coloring of a graph $H$ is a partition of $V(H)$ into $V_1, V_2\ldots,V_k$ such that $H[V_i]$ does not contain $K_p$ for each $i\leq k $. In 1977 Borodin and Kostochka…

In this article we investigate the question of chromatic purity of L-theory. To do so, we utilize the theory of additive GW and L-theory in the language of Poincar\'e categories as laid out in the series of papers by Calm\`es et al. We…

We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local…

Given a class $\mathcal{H}$ of $m$ hypergraphs ${H}_1, {H}_2, \ldots, {H}_m$ with the same vertex set $V$, a cooperative coloring of them is a partition $\{I_1, I_2, \ldots, I_m\}$ of $V$ in such a way that each $I_i$ is an independent set…

Haj\'os conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $\Omega(s^2/\log s)$.…

Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that if p > 5,…

In an article [3] published recently in this journal, it was shown that when k >= 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to…

We study a certain monoid of endofunctors of the stable homotopy category that includes localizations with respect to finite unions of Morava $K$-theories. We work in an axiomatic framework that can also be applied to analogous questions in…

We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for…

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the $\mathbf{E}_{2}$-topological Hochschild cohomology of certain Thom spectra (denoted $A$,…