arXiv subjects

Hood Chatham

Publications and source records attributed to Hood Chatham.

6 recordsLinked to original sources

arXiv2026

High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories

Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.

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arXiv2024

Enumerating stably trivial vector bundles with higher real $K$-theory

This paper explores periodic phenomena in the group $\operatorname{Vect}_r^0(\mathbb{CP}^{r+c})$ of stably trivial, complex rank $r$ topological vector bundles on $\mathbb{CP}^{r+c}$. For $1 \leq c < r$ and $c\leq 2p-3$, we give a complete computation of the $p$-torsion in $\operatorname{Vect}_r^0(\mathbb{CP}^{r+c})$, and we relate these $p$-torsion bundles to vector bundles on spheres. We also compute $\operatorname{Vect}_r^0(\mathbb{CP}^{r+c})$ in full when $c=3$, extending computations of the second-named author when $c=1$ and $c=2$. Finally, for a fixed corank $c$ which is larger relative to the prime $p$, we show there are families of $p$-torsion in $\operatorname{Vect}_r^0(\mathbb{CP}^{r+c})$ that are periodic in $r$ with period $p$. We detect these using Weiss calculus and certain higher real $K$-theory homology groups.

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arXiv2024

Wilson Spaces, Snaith Constructions, and Elliptic Orientations

We construct a canonical family of even periodic $\mathbb{E}_{\infty}$-ring spectra, with exactly one member of the family for every prime $p$ and chromatic height $n$. At height $1$ our construction is due to Snaith, who built complex $K$-theory from $\mathbb{CP}^{\infty}$. At height $2$ we replace $\mathbb{CP}^{\infty}$ with a $p$-local retract of $\mathrm{BU} \langle 6 \rangle$, producing a new theory that orients elliptic, but not generic, height $2$ Morava $E$-theories. In general our construction exhibits a kind of redshift, whereby $\mathrm{BP}\langle n-1 \rangle$ is used to produce a height $n$ theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the $K(n)$-localization of our height $n$ ring to work of Peterson and Westerland building $E_n^{hS\mathbb{G}^{\pm}}$ from $\mathrm{K}(\mathbb{Z},n+1)$.

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arXiv2021

On the $\mathrm{EO}$-orientability of vector bundles

We study the orientability of vector bundles with respect to a family of cohomology theories called $\mathrm{EO}$-theories. The $\mathrm{EO}$-theories are higher height analogues of real $\mathrm{K}$-theory $\mathrm{KO}$. For each $\mathrm{EO}$-theory, we prove that the direct sum of $i$ copies of any vector bundle is $\mathrm{EO}$-orientable for some specific integer $i$. Using a splitting principal, we reduce to the case of the canonical line bundle over $\mathbb{CP}^{\infty}$. Our method involves understanding the action of an order $p$ subgroup of the Morava stabilizer group on the Morava $\mathrm{E}$-theory of $\mathbb{CP}^{\infty}$. Our calculations have another application: We determine the homotopy type of the $\mathrm{S}^{1}$-Tate spectrum associated to the trivial action of $\mathrm{S}^{1}$ on all $\mathrm{EO}$-theories.

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arXiv2019

An Orientation Map for Height p-1 Real E Theory

Let $p$ be an odd prime and let $\mathit{EO} = E_{p-1}^{hC_p}$ be the $C_p$ fixed points of height $p-1$ Morava $E$ theory. We say that a spectrum $X$ has algebraic $\mathit{EO}$ theory if the splitting of $K_*(X)$ as an $K_*[C_p]$-module lifts to a topological splitting of $\mathit{EO} \wedge X$. We develop criteria to show that a spectrum has algebraic $\mathit{EO}$ theory, in particular showing that any connective spectrum with mod $p$ homology concentrated in degrees $2k(p - 1)$ has algebraic $\mathit{EO}$ theory. As an application, we answer a question posed by Hovey and Ravenel by producing a unital orientation $\mathit{MY}_{4p-4}\to \mathit{EO}$ analogous to the $\mathit{MSU}$ orientation of $\mathit{KO}$ at $p=2$.

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arXiv2019

Thom Complexes and the Spectrum tmf

Many interesting spectra can be constructed as Thom spectra of easily constructed bundles. Mahowald showed that $\mathit{bu}$ and $\mathit{bo}$ cannot be realized as $E_1$ Thom spectra. We use related techniques to show that $\mathit{tmf}_{(2)}$ also cannot be realized as an $E_1$ Thom spectrum.

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