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Charles Rezk

Publications and source records attributed to Charles Rezk.

At least 19 recordsLinked to original sources
arXiv2026

The Witt filtration of Lubin-Tate deformation rings

This note is a meditation on a generalization $\mathbb{W}_E$ of the classical p-typical Witt vectors $\mathbb{W}_p$, which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava $E$-theory. For formal groups of height $1$ we have $\mathbb{W}_E=\mathbb{W}_p$, but the $\mathbb{W}_E$ are richer when height is $\geq 2$. We show that $\mathbb{W}_p$ splits naturally from $\mathbb{W}_E$. A key property of $\mathbb{W}_E$ is the isomorphism $\pi_0E\approx \mathbb{W}_E(\pi_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $\pi_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $\pi_0E/(p)$. We use this to give a new proof of cofreeness.

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arXiv2026

Cofreeness of the Lubin-Tate deformation ring

We give a proof of the cofreeness of the Lubin-Tate deformation ring, by generalizing earlier results by Matt Ando and Yifei Zhu about $\mathsf{H}_\infty$-orientations to the context of power operations for Morava $E$-theory.

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arXiv2019

The Bousfield-Kuhn functor and Topological Andre-Quillen cohomology

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the case where the space is a sphere. This results in a method for computing unstable v_n-periodic homotopy groups of spheres from their Morava E-cohomology (as modules over the Dyer-Lashof algebra of Morava E-theory). We relate the resulting algebraic computations to the algebraic geometry of isogenies between Lubin-Tate formal groups.

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arXiv2018

Comparison of models for $(\infty, n)$-categories, II

In this paper we complete a chain of explicit Quillen equivalences between the model category for $Θ_{n+1}$-spaces and the model category of small categories enriched in $Θ_n$-spaces. The Quillen equivalences given here connect Segal category objects in $Θ_n$-spaces, complete Segal objects in $Θ_n$-spaces, and $Θ_{n+1}$-spaces.

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arXiv2017

Looijenga line bundles in complex analytic elliptic cohomology

We present a calculation, which shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of $U(1)$-bundles on a torus. Furthermore, we show how the analogous calculation, applied to a moduli space of principal bundles for a $K(\mathbb{Z},2)$ central extension of $U(1)^d$ give rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.

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arXiv2017

Classifying spaces for 1-truncated compact Lie groups

A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.

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arXiv2017

Spectral algebra models of unstable v_n-periodic homotopy theory

We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological André-Quillen cohomology, Koszul duality, and Goodwillie calculus.

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arXiv2014

On Hopkins' Picard groups for the prime 3 and chromatic level 2

We give a calculation of Picard groups of K(2)-local invertible spectra and of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup of invertible spectra with the same Morava module as a sphere.

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arXiv2014

An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology

We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum $R$, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space $X$ to the infinity category of free rank one $R$-modules, which we show is a model for $BGL_1 R$; we show that $BGL_1 R$ classifies homotopy sheaves of rank one $R$-modules, which we call $R$-line bundles. We use our $R$-module Thom spectrum to define the twisted $R$-homology and cohomology of an $R$-line bundle over a space $X$, classified by a map from $X$ to $BGL_1 R$, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.

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arXiv2014

Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.

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arXiv2013

Frobenius Pairs and Atiyah Duality

We define a notion of "Frobenius pair", which is a mild generalization of the notion of Frobenius object in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain collection of structure obtained from a manifold forms a commutative Frobenius pair in the stable homotopy category of spectra.

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arXiv2012

Reedy categories and the $Θ$-construction

We use the notion of multi-Reedy category to prove that, if $\mathcal C$ is a Reedy category, then $Θ\mathcal C$ is also a Reedy category. This result gives a new proof that the categories $Θ_n$ are Reedy categories. We then define elegant Reedy categories, for which we prove that the Reedy and injective model structures coincide.

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arXiv2012

Comparison of models for $(\infty, n)$-categories, I

While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty, n)$-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $(\infty,n)$-categories. Furthermore, we establish Quillen equivalences between them.

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arXiv2012

Modular Isogeny Complexes

We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.

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arXiv2010

A cartesian presentation of weak n-categories

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.

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