Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.33754

Ramification in $\mathbf{E}_{n}$-algebras

Sanath K. Devalapurkar·Ishan Levy
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study $\mathbf{E}_n$-algebras with ramification in their $π_0$ and in particular show that the existence of ramification places strong limitations on their chromatic behavior. In particular, we show that $π_0$ of nonzero $K(1)$-local $\mathbf{E}_4$-$\mathrm{KU}$-algebras cannot have a map from the ring of integers in a ramified extension of $\mathbf{Q}_p$, and we show that a $p$-complete $\mathbf{E}_2$-ring with a map from a perfectoid ring into its $π_0$ is an associative $\mathbf{Z}_p$-algebra.

Explore related subjects

math.ATmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sanath K. Devalapurkar, Ishan Levy. 2026-10-24. Ramification in $\mathbf{E}_{n}$-algebras. https://arxiv.org/abs/2610.33754

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural graded $\mathbb{Q}$-linear section of the homomorphism induced in cohomology by the symmetric diagonal. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to distributional homotopic distance. A sequential version of this invariant was recently introduced by Jauhari and Oprea; we give an equivalent formulation in terms of distributed homotopies and derive new fibration and multiplicative triangle inequalities.

math.AT↗
arXiv2026

On smooth and bundle structures on topological manifolds homotopy equivalent to $S^{4k-1}$-bundles over $S^{4k}$

We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of $S^{4k-1}$-bundles over $S^{4k}$ for $k\ge3$, in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.

math.AT↗
arXiv2026

BV rigidity in the string topology of compact Lie groups

Let $G$ be a connected compact Lie group and $R$ a commutative ring over which $H_*(G;R)$ is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra $\mathbb{H}_*(LG;R)$ of Chas and Sullivan on the homology of the free loop space is that of the product of odd spheres with the classical exponents of $G$, tensored with the group ring of the torsion of $π_1(G)$. Conversely, over an integral domain the graded algebra $\mathbb{H}_*(LG;R)$ already determines the exponents and the group ring, hence the BV algebra. The isomorphism with the product of spheres is algebraic: for $\mathrm{Sp}(n)$, $n\geq2$, and $\mathrm{SU}(n)$, $n\geq3$, no map of spaces induces it over $\mathbb{Z}$. If moreover $G$ is simply connected, then $\mathbb{H}_*(LG;R)$, as a BV algebra, is the Hochschild cohomology of $H^*(G;R)$ with the BV operator induced by Poincaré duality.

math.AT↗