arXiv subjects

Jeremy Hahn

Publications and source records attributed to Jeremy Hahn.

22 records · Page 2Linked to original sources

arXiv2020

Real Orientations of Lubin-Tate Spectra

We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.

math.AT↗

arXiv2019

Multiplicative Structure in the Stable Splitting of $ΩSL_n(\mathbb{C})$

The space of based loops in $SL_n(\mathbb{C})$, also known as the affine Grassmannian of $SL_n(\mathbb{C})$, admits an $\mathbb{E}_2$ or fusion product. Work of Mitchell and Richter proves that this based loop space stably splits as an infinite wedge sum. We prove that the Mitchell--Richter splitting is coherently multiplicative, but not $\mathbb{E}_2$. Nonetheless, we show that the splitting becomes $\mathbb{E}_2$ after base-change to complex cobordism. Our proof of the $\mathbb{A}_\infty$ splitting involves on the one hand an analysis of the multiplicative properties of Weiss calculus, and on the other a use of Beilinson--Drinfeld Grassmannians to verify a conjecture of Mahowald and Richter. Other results are obtained by explicit, obstruction-theoretic computations.

math.AT↗

arXiv2018

Quotients of even rings

We prove that if $R$ is an $\mathbb{E}_2$-ring with homotopy concentrated in even degrees, and $\{x_j\}$ is any sequence of elements in $π_{2*}(R)$, then $R/(x_1,x_2,\cdots)$ admits the structure of an $\mathbb{E}_1$-$R$-algebra. This removes an assumption, common in the literature, that $\{x_j\}$ be a regular sequence.

math.AT↗

arXiv2017

Nilpotence in E_n Algebras

Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in the homotopy of $\mathbb{E}_n$-ring spectra. For all odd primes $p$ and all chromatic heights $h$, we use the Cohen-Moore-Neisendorfer theorem to construct examples of $K(h)$-local, $\mathbb{E}_{2n-1}$-algebras with non-nilpotent $p^n$-torsion. We exploit the interaction of the Bousfield-Kuhn functor on odd spheres and Rezk's logarithm to show that our bound is sharp at height $1$, and remark on the situation at height $2$.

math.AT↗