Nathaniel Stapleton
YOU?
Author Swipe
View article: Using the coaches voice to improve the representation and experience of females in coaching: a Gaelic games perspective
Using the coaches voice to improve the representation and experience of females in coaching: a Gaelic games perspective Open
Background Female coaches across all sports and levels are underrepresented on a global scale, existing as peripheral figures on the coaching landscape. This is evident in an Irish context, with a recent report suggesting that just 18.7% o…
View article: On Hopkins' Picard group
On Hopkins' Picard group Open
We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$…
View article: On the image of the total power operation for Burnside rings
On the image of the total power operation for Burnside rings Open
We prove that the image of the total power operation for Burnside rings $A(G) \to A(G\wrΣ_n)$ lies inside a relatively small, combinatorial subring $\mathring A(G,n) \subseteq A(G \wr Σ_n)$. As $n$ varies, the subrings $\mathring A(G,n)$ a…
View article: On the rationalization of the $K(n)$-local sphere
On the rationalization of the $K(n)$-local sphere Open
We compute the rational homotopy groups of the $K(n)$-local sphere for all heights $n$ and all primes $p$, verifying a prediction that goes back to the pioneering work of Morava in the early 1970s. More precisely, we show that the inclusio…
View article: The homotopy of the KU_G-local equivariant sphere spectrum
The homotopy of the KU_G-local equivariant sphere spectrum Open
We compute the homotopy Mackey functors of the $KU_G$-local equivariant sphere spectrum when $G$ is a finite $q$-group for an odd prime $q$, building on the degree zero case from arXiv:2204.03797.
View article: Power operations in the Stolz–Teichnerprogram
Power operations in the Stolz–Teichnerprogram Open
The Stolz--Teichner program proposes a deep connection between geometric\nfield theories and certain cohomology theories. In this paper, we extend this\nconnection by developing a theory of geometric power operations for geometric\nfield t…
View article: On the $KU_G$-local equivariant sphere
On the $KU_G$-local equivariant sphere Open
Equivariant complex $K$-theory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd $p$-group, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spec…
View article: Evaluation maps and transfers for free loop spaces II
Evaluation maps and transfers for free loop spaces II Open
In our previous paper, we constructed and studied a functorial extension of the evaluation map $S^1 \times \mathcal{L}X \to X$ to transfers along finite covers. In this paper, we show that this induces a natural evaluation map on the full …
View article: Evaluation maps and transfers for free loop spaces I
Evaluation maps and transfers for free loop spaces I Open
We construct and study a functorial extension of the evaluation map $S^1 \times \mathcal{L} X \to X$ to transfers along finite covers. For finite covers of classifying spaces of finite groups, we provide algebraic formulas for this extensi…
View article: Power operations in the Stolz--Teichner program
Power operations in the Stolz--Teichner program Open
The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theo…
View article: Additive power operations in equivariant cohomology
Additive power operations in equivariant cohomology Open
Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any $G$-space $X$ and positive integer $m$, we give an explicit description of the smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times BΣ_m)$ for which…
View article: Level structures on $p$-divisible groups from the Morava $E$-theory of abelian groups
Level structures on $p$-divisible groups from the Morava $E$-theory of abelian groups Open
The close relationship between the scheme of level structures on the universal deformation of a formal group and the Morava $E$-cohomology of finite abelian groups has played an important role in the study of power operations for Morava $E…
View article: Lubin-Tate theory, character theory, and power operations
Lubin-Tate theory, character theory, and power operations Open
This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories.In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their symme…
View article: On the ring of cooperations for 2‐primary connective topological modular forms
On the ring of cooperations for 2‐primary connective topological modular forms Open
We analyze the ring tmf_*tmf of cooperations for the connective spectrum of\ntopological modular forms (at the prime 2) through a variety of perspectives:\n(1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a\ndecompositi…
View article: A canonical lift of Frobenius in Morava $E$-theory
A canonical lift of Frobenius in Morava $E$-theory Open
We prove that the $p$th Hecke operator on the Morava $E$-cohomology of a space is congruent to the Frobenius mod $p$. This is a generalization of the fact that the $p$th Adams operation on the complex $K$-theory of a space is congruent to …
View article: Lubin-Tate theory, character theory, and power operations
Lubin-Tate theory, character theory, and power operations Open
This expository paper introduces several ideas in chromatic homotopy theory around Morava's extraordinary E-theories. In particular, we construct various moduli problems closely related to Lubin-Tate deformation theory and study their symm…
View article: Excellent rings in transchromatic homotopy theory
Excellent rings in transchromatic homotopy theory Open
The purpose of this note is to verify that several basic rings appearing in transchromatic homotopy theory are Noetherian excellent normal domains and thus amenable to standard techniques from commutative algebra. In particular, we show th…
View article: Chromatic homotopy theory is asymptotically algebraic
Chromatic homotopy theory is asymptotically algebraic Open
Inspired by the Ax--Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the…
View article: A formula for $p$-completion by way of the Segal conjecture
A formula for $p$-completion by way of the Segal conjecture Open
The Segal conjecture describes stable maps between classifying spaces in terms of (virtual) bisets for the finite groups in question. Along these lines, we give an algebraic formula for the p-completion functor applied to stable maps betwe…
Brown–Peterson cohomology from Morava -theory Open
We prove that the $p$ -completed Brown–Peterson spectrum is a retract of a product of Morava $E$ -theory spectra. As a consequence, we generalize results of Kashiwabara and of Ravenel, Wilson and Yagita from spaces to spectra and deduce th…
View article: The character of the total power operation
The character of the total power operation Open
We compute the total power operation for the Morava [math] –theory of any finite group up to torsion. Our formula is stated in terms of the [math] –action on the Drinfel’d ring of full level structures on the formal group associated to [ma…