arXiv subjects

Michael A. Hill

Publications and source records attributed to Michael A. Hill.

At least 19 recordsLinked to original sources

arXiv2026

Global dimension of the category of rational incomplete Mackey functors for a finite abelian group G

In this paper, we analyse the global dimension of the category of rational incomplete Mackey functors over a finite abelian group. Incomplete Mackey functors have recently risen to prominence in algebraic topology and hence it is valuable to understand their homological algebra invariants. When working over the rational numbers, results of Greenlees--May and Thévanez--Webb show that the homological algebra of complete Mackey functors is quite simple, but the incomplete case is more complicated. In this paper we use splitting results by the first, third and fourth authors to give an upper bound on the global dimension of rational incomplete Mackey functors where the incompleteness is governed by what is known as a disk-like transfer system. We then avail ourselves of a new connection to incidence algebras over posets to calculate the global dimension of rational incomplete Mackey functors in the disk-like case when the group is abelian.

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arXiv2026

Maximal compatibility of disklike $G$-transfer systems

Transfer systems are a combinatorial model for $N_{\infty}$-operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they encode the additive transfers and multiplicative norms of a ring-type structure. In this paper, given a transfer system encoding an additive structure, we give explicit formulae for the maximal transfer system it is compatible with. Our formulae simplify for disklike transfer systems, which typically encode additive structures. Further, we prove that (maximal) compatibility is functorial with respect to the inflation map induced by a quotient of groups, letting us compute maximal compatible transfer systems as inflations of connected transfer systems.

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arXiv2026

On the hyperreal dual Steenrod algebra

We compute the dual Steenrod algebra for Bredon homology with constant coefficients $\underline{\mathbb Z}$ and $\underline{\mathbb Z}/2$ in the category of modules over $MU^{((G))}$, the norm to $G=C_{2^n}$ of $MU_{\mathbb R}$. Using this and an equivariant version of the Greenlees--Serre spectral sequence, we give a spectral sequence computing the $RO$-graded homotopy of the Eilenberg--Mac Lane spectrum $H\underline{\mathbb F}_2\otimes H\underline{\mathbb Z}$.

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arXiv2025

Real topological Hochschild homology via the norm and Real Witt vectors

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero $D_{2m}$-Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.

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arXiv2025

Periodicity and finite complexity in higher real $K$-theories

In this paper, we establish periodicity results for higher real $K$-theories at all heights and for all finite subgroups of the Morava stabilizer group at the prime 2. We further analyze the $RO(G)$-periodicity lattice of the height-$h$ Lubin--Tate theory, proving new $RO(G)$-graded periodicities and explicit finiteness results for the $RO(G)$-graded homotopy groups of $E_h$. Together, these results provide a foundation for both the structural and computational study of higher real $K$-theories.

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arXiv2025

Splitting rational incomplete Mackey functors

Inspired by equivariant homotopy theory, equivariant algebra studies generalisations of G-Mackey functors that do not have all transfer maps (also known as induction maps), for G a finite group. These incomplete Mackey functors have interesting and subtle properties that are more complicated than classical algebra. The levels of incompleteness that occur are indexed by simple combinatorial data known as transfer systems for G, which are refinements of the subgroup relation satisfying certain axioms. The aim of this paper is to generalise the Greenlees--May and Thevenaz--Webb splitting result of rational G-Mackey functors to the incomplete case. By calculating idempotents of the rational incomplete Burnside ring of G, we find the maximal splitting of the category of rational incomplete G-Mackey functors. These splittings are determined by maps of the form H to G in the transfer system. We give an intrinsic definition of the split pieces beyond the idempotent description in order to understand what is the minimal information needed to determine an arbitrary rational incomplete G-Mackey functor. We end the paper with a series of examples of possible splittings and illustrate how simpler transfer systems have fewer terms in the splitting but the split pieces are more complicated.

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arXiv2025

On higher real $K$-theories and finite spectra

We study higher chromatic height analogues $eo_h$ of the connective real $K$-theory spectrum $ko$. We show that $eo_h$ is an fp spectrum of type $h$ in the sense of Mahowald--Rezk. We use these to study an Euler characteristic for fp spectra introduced by Ishan Levy, and give a partial answer to a question of Levy regarding the algebraic $K$-theory of the category of finite type $h$ spectra. As a corollary, we prove that if the generalized Moore spectrum $\mathbb{S}/(2^{i_0},v_1^{i_1},\ldots,v_h^{i_h})$ exists, then the $2$-adic valuation of $\prod i_j$ must exceed that of the height $h$.

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arXiv2024

On MU-homology of connective models of higher Real K-theories

We use the slice filtration to study the $MU$-homology of the fixed points of connective models of Lubin--Tate theory studied by Hill--Hopkins--Ravenel and Beaudry--Hill--Shi--Zeng. We show that, unlike their periodic counterparts $EO_n$, the $MU$ homology of $BP^{((G))}\langle m\rangle^G$ usually fails to be even and torsion free. This can only happen when the height $n=m|G|/2$ is less than $3$, and in the edge case $n=2$, we show that this holds for $tmf_0(3)$ but not for $tmf_0(5)$, and we give a complete computation of the $MU_*MU$-comodule algebra $MU_*tmf_0(3)$.

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arXiv2024

On the slice spectral sequence for quotients of norms of Real bordism

In this paper, we investigate equivariant quotients of the Real bordism spectrum's multiplicative norm $MU^{((C_{2^n}))}$ by permutation summands. These quotients are of interest because of their close relationship with higher real $K$-theories. We introduce new techniques for computing the equivariant homotopy groups of such quotients. As a new example, we examine the theories $BP^{((C_{2^n}))}\langle m,m\rangle$. These spectra serve as natural equivariant generalizations of connective integral Morava $K$-theories. We provide a complete computation of the $a_σ$-localized slice spectral sequence of $i^*_{C_{2^{n-1}}}BP^{((C_{2^n}))}\langle m,m\rangle$, where $σ$ is the real sign representation of $C_{2^{n-1}}$. To achieve this computation, we establish a correspondence between this localized slice spectral sequence and the $H\mathbb{F}_2$-based Adams spectral sequence in the category of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$-modules. Furthermore, we provide a full computation of the $a_λ$-localized slice spectral sequence of the height-4 theory $BP^{((C_{4}))}\langle 2,2\rangle$. The $C_4$-slice spectral sequence can be entirely recovered from this computation.

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arXiv2023

The homological slice spectral sequence in motivic and Real bordism

For a motivic spectrum $E\in \mathcal{SH}(k)$, let $Γ(E)$ denote the global sections spectrum, where $E$ is viewed as a sheaf of spectra on $\mathrm{Sm}_k$. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of $Γ(E)$. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of $Γ(E)$ and study the case $E=BPGL\langle m\rangle$ for $k=\mathbb R$ in detail. We show that this spectral sequence contains the $\mathcal{A}_*$-comodule algebra $(\mathcal{A}//\mathcal{A}(m))^*$ as permanent cycles, and we determine a family of differentials interpolating between $(\mathcal{A}//\mathcal{A}(0))^*$ and $(\mathcal{A}//\mathcal{A}(m))^*$. Using this, we compute the spectral sequence completely for $m\le 3$. In the height 2 case, the Betti realization of $BPGL\langle 2\rangle$ is the $C_2$-spectrum $BP_{\mathbb R}\langle 2\rangle$, a form of which was shown by Hill and Meier to be an equivariant model for $\mathrm{tmf}_1(3)$. Our spectral sequence therefore gives a computation of the comodule algebra $H_*\mathrm{tmf}_0(3)$. As a consequence, we deduce a new ($2$-local) Wood-type splitting \[\mathrm{tmf}\wedge X\simeq \mathrm{tmf}_0(3)\] of $\mathrm{tmf}$-modules predicted by Davis and Mahowald, for $X$ a certain 10-cell complex.

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arXiv2022

Counting compatible indexing systems for $C_{p^n}$

We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count this two different ways: showing how the conditions of compatibility give natural recursive formulas for the number of admissible sets and using an enumeration of ways to extend indexing systems by conceptually simpler pieces.

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arXiv2022

Norms for compact Lie groups in equivariant stable homotopy theory

We propose a construction of an analogue of the Hill-Hopkins-Ravenel relative norm $N_{H}^{G}$ in the context of a positive dimensional compact Lie group $G$ and closed subgroup $H$. We explore expected properties of the construction. We show that in the case when $G$ is the circle group (the unit complex numbers), the proposed construction here agrees with the relative norm constructed by Angeltveit, Gerhardt, Lawson, and the authors using the cyclic bar construction. Our construction is based on a new perspective on equivariant factorization homology, using framings to convert from actions of one group to another.

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arXiv2022

Hochschild homology of mod-$p$ motivic cohomology over algebraically closed fields

We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.

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arXiv2022

Freeness and equivariant stable homotopy

We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, the $RO$-graded Bredon homology is extremely easy to compute, and if these spaces have additional structure, then this can also be easily determined. In particular, the homology of a space with this property naturally has the structure of a co-Tambara functor (and compatibly with any additional product structure). We work this out in the example of $BU_{\mathbb R}$ and coinduced versions of this. We finish by describing a readily computable bar and twisted bar spectra sequence, giving Bredon homology for various $E_{\infty}$ pushouts, and we apply this to describe the homology of $BBU_{\mathbb R}$.

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arXiv2021

Free incomplete Tambara functors are almost never flat

Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors.

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arXiv2021

Models of Lubin-Tate spectra via Real bordism theory

We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{\mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{\mathbb{R}}$.

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arXiv2021

$E_k$-pushouts and $E_{k+1}$-tensors

We prove a general result that relates certain pushouts of $E_k$-algebras to relative tensors over $E_{k+1}$-algebras. Specializations include a number of established results on classifying spaces, resolutions of modules, and (co)homology theories for ring spectra. The main results apply when the category in question has centralizers. Among our applications, we show that certain quotients of the dual Steenrod algebra are realized as associative algebras over $HF_p \wedge HF_p$ by attaching single $E_1$-algebra relation, generalizing previous work at the prime $2$. We also construct a filtered $E_2$-algebra structure on the sphere spectrum, and the resulting spectral sequence for the stable homotopy groups of spheres has $E_1$-term isomorphic to a regrading of the $E_1$-term of the May spectral sequence.

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arXiv2021

Transchromatic extensions in motivic and Real bordism

We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{\mathbb R}$. These brackets are "red-shifting" in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $π_{(\ast,\ast)}MGL$-module structure of the motivic Morava $K$-theories, including non-trivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $π_{\star}MU_\mathbb R$-module structure on the Real Morava $K$-theories.

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