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.

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.

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 $BBU_{\mathbb R}$.

We describe the structure present in algebras over the little disks operads for various representations of a finite group $G$, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_2$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.

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 $\pi_{(\ast,\ast)}MGL$-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 $\pi_{\star}MU_\mathbb R$-module structure on the Real Morava $K$-theories.

We construct $C_{2^n}$-equivariant Real oriented models of Lubin--Tate spectra $E_h$ at heights $h=2^{n-1}m$. We 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 $MU_{\mathbb{R}}$.

For an equivariant commutative ring spectrum $R$, $\pi_0 R$ has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If $R$ is an $N_\infty$ ring spectrum in the category of genuine $G$-spectra, then all possible additive transfers are present and $\pi_0 R$ has the structure of an incomplete Tambara functor. However, if $R$ is an $N_\infty$ ring spectrum in a category of incomplete $G$-spectra, the situation is more subtle.
In this paper, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures.

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories are built from the categories of $\mathcal{O}$-algebras in $G$-spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties of the equivariant stable category, notably the tom Dieck splitting. Our work is motivated in part by the examples arising from the equivariant units and Picard space functors.

We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C4))}\langle 2\rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E^{hC_{12}}_4$ is 384-periodic.

This is an expository survey of Equivariant Stable Homotopy Theory, written for Miller's "Handbook of Homotopy Theory".

We compute the Picard group of the category of $K(2)$-local module spectra over the ring spectrum $E^{hC_4}$, where $E$ is a height $2$ Morava $E$-theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K(2)$-local $E$-modules in genuine $C_4$-spectra. We show that in addition to a cyclic subgroup of order $32$ generated by $E\wedge S^1$ the Picard group contains a subgroup of order 2 generated by $E\wedge S^{7+\sigma}$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO(C_4)$-graded Mackey functor homotopy fixed point spectral sequence for the $C_4$-spectrum $E$.

Building off of the work of Kervaire and Milnor, and Hill, Hopkins, and Ravenel, Xu and Wang showed that the only odd dimensions n for which $S^n$ has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which $S^n$ has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.

We introduce a computationally tractable way to describe the $\mathbb Z$-homotopy fixed points of a $C_{n}$-spectrum $E$, producing a genuine $C_{n}$ spectrum $E^{hn\mathbb Z}$ whose fixed and homotopy fixed points agree and are the $\mathbb Z$-homotopy fixed points of $E$. These form a piece of a contravariant functor from the divisor poset of $n$ to genuine $C_{n}$-spectra, and when $E$ is an $N_{\infty}$-ring spectrum, this functor lifts to a functor of $N_{\infty}$-ring spectra.
For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the $RO(G)$-graded homotopy groups of the spectrum $E^{hn\mathbb Z}$, giving the homotopy groups of the $\mathbb Z$-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple $\mathbb Z$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.

We compute the cohomology of the subalgebra $\mathcal A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $\mathcal A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $\mathbb C$-motivic and $\mathbb R$-motivic calculations as stepping stones.

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology $THH_{C_n}(−)$, and it describes the $E_2$ term of the K\"unneth spectral sequence for relative $THH$. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors.

We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.

We describe a construction of the cyclotomic structure on topological Hochschild homology ($THH$) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bokstedt coherence machinery. As a consequence, we are able to define versions of topological cyclic homology ($TC$) relative to an arbitrary commutative ring spectrum $A$. We describe spectral sequences computing this relative theory ${}_{A}TR$ in terms of $TR$ over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on $TR$ and $TC$.

In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.

For $N_\infty$ operads $\mathcal O$ and $\mathcal O'$ such that there is an inclusion of the associated indexing systems, there is a forgetful functor from incomplete Tambara functors over $\mathcal O'$ to incomplete Tambara functors over $\mathcal O$. Roughly speaking, this functor forgets the norms in $\mathcal O'$ that are not present in $\mathcal O$. The forgetful functor has both a left and a right adjoint; the left adjoint is an operadic tensor product, but the right adjoint is more mysterious. We explicitly compute the right adjoint for finite cyclic groups of prime order.

In this paper, we describe the algebraic analogue of $N_\infty$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $\pi_0$ of $N_\infty$ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of $N_\infty$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a $G$-spectrum is slice $n$-connective. In particular, we show that a $G$-spectrum is slice greater than or equal to $n$ if and only if for all subgroups $H$, the $H$-geometric fixed points are $(n/|H|-1)$-connected. We use this to determine when smashing with a virtual representation sphere $S^V$ induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary $C_p$-spectrum and show how a very small number of functors determine all of the slices for $C_{p^n}$-spectra.

In this paper, we describe Mackey functor objects in the category of augmented Tambara functors, genuine equivariant derivations, and genuine Kahler differentials. We show that these are connected just as classically. This is the foundation for future work on equivariant TAQ

We show that many of the interesting derived algebraic geometry properties (Anderson duality and the computation of the Picard group) for the $C_2$-spectrum $Tmf_1(3)$ follow from fairly straightforward equivariant considerations.

We study the slice filtration and associated spectral sequence for a family of $RO(C_{p^{n}})$-graded suspensions of the Eilenberg-MacLane spectrum for the constant Mackey functor $\underline{\mathbb Z}$. Since $H\underline{\mathbb Z}$ is the zero slice of the sphere spectrum, this begins an analysis of how one can describe the slices of a suspension in terms of the original slices.

In this first of a series of three papers, we compare various notions of equivariant smash powers. This generalizes work of Shipley and shows that the norm provides both conceptual and homotopical control in a way similar to that of the Bokstedt model.

We completely describe the slice spectral sequence computing the homotopy groups of the spectrum of topological modular forms with a $\Gamma_0(5)$-structure, $TMF_0(5)$. We describe the slice spectral sequence of a 32-periodic $C_4$-spectrum $K_H$ related to the $C_4$ norm $N_{C_{2}}^{C_4}MU_R$ of the real cobordism spectrum $MU_R$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor $\pi_\ast K_H$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real $K$-theory spectrum $K_R$ was first analyzed by Dugger. The $C_8$ analog of $K_H$ is 256-periodic and detects the Kervaire invariant classes $\theta_j$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that $\theta_j$ does not exist for $j\geq 7$.

In this paper, we prove that there are no smooth Kervaire invariant one manifolds of dimension larger than 126 using equivariant homotopy theory.

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. These are generalizations of the usual $E_\infty$ operads, but have a priori less structure than classical $G-E_\infty$-operads. Algebras over our operads, called $N_\infty$ operads, have norms maps which, in the case of spectra, are the Hill-Hopkins-Ravenel norms, and in the case of spaces, are the transfer.

We show that the Goerss-Hopkins-Miller sheaf of $E_\infty$ ring spectra $\mathcal O^{top}$ on the moduli stack of elliptic curves extends over the log-etale site. This allows us to functorially produce models for so-called ``elliptic curves with level structure''.

This is the long-awaited proceedings for the 2007 Talbot conference on topological modular forms. The book contains chapters covering all of the background materials, together with Behrens' description of the construction of the $tmf$-sheaf, Hopkins and Miller's original paper on $tmf$, Hopkins' paper on $K(1)$-local $E_\infty$-ring spectra, and Hopkins and Mahowald's paper on elliptic curves and stable homotopy. The PDF linked here is done with permission from the AMS. If you would like to purchase a physical copy of the book, you may do so at the AMS Bookstore.

This paper describes an issue that arises when inverting elements of the homotopy groups of an equivariant commutative ring.

We compute using cyclotomic trace methods the algebraic K-theory of truncated polynomial algebras. We also describe a convenient category for cyclotimic things and show how this simplifies several kinds of computations.

In this paper, I give an overview of the equivariant slice filtration. I produce a large family of slices, and then I determine the slice tower associated to Eilenberg-MacLane spectra.

This is a detailed sketch of our arguments written for the Current Developments in Mathematics conference at Harvard. It contains several distinct arguments, and is largely self-contained.

This is a write-up of the historical talk given by Ravenel at the CDM conference at Harvard.

This paper analyzes Poincare duality in the context of modules over the Steenrod algebra

This demonstrates a motivic Bockstein spectral sequence used to compute motivic Ext over finite subalgebras of the Motivic Steenrod algebra.

We compute several rings of automorphic forms and apply this to compute the homotopy groups of several spectra produced by Lurie's Artin Representability.

This paper simultaneously runs two pairs of Bockstein spectral sequences to compute topological Hochschild homology.

Using Hochschild homology, we show that ko and ku are not Thom spectra.

This computes the String Bordism groups of $BE_8$ and $BE_8\times BE_8$ through dimension 14. This computation amounts to computing the tmf-homology of $K(\mathbb Z,3)$.

This proves the existence of a $v_2^{32}$-self map on the generalized Smith-Toda complex $M(1,4)$.

A Bockstein and Adams-Novikov spectral sequence computation for the homotopy ring of the conjectural spectrum $eo_4$.

An elementary presentation of the cohomology of the connective $j$-theory spectrum.

This gives a form of the Adams spectral sequence for tmf-homology and applies it to compute the tmf-homology of $B\Sigma_3$.

For a motivic spectrum $E\in \mathcal{SH}(k)$, let $\Gamma(E)$ 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 $\Gamma(E)$. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of $\Gamma(E)$ 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.

In this paper, we study equivariant quotients of the multiplicative norm $MU^{((C_{2^n}))}$ of the Real bordism spectrum by permutation summands, a concept defined here. These quotients are interesting because of their relationship to the so-called "higher real K-theories".

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.

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 $THR(\underline{\mathbb Z})$ for m odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.

This paper introduces the notion of a "$G$-symmetric monoidal category". Loosely speaking, this is a symmetric monoidal category in which exponential by a finite $G$-set is compatible defined. Using this, we give a general treatment of when Bousfield localization preserves multiplicative structures on a ring spectrum. We also describe how the category of modules over a monoid in a $G$-symmetric monoidal category inherits a $G$-symmetric monoidal structure. We close looking at closely related variants for compact Lie groups and in motivic homotopy.

This paper shows that $\eta^3$ is zero in all of the norms of $MU_{\mathbb R}$ considered in the Kervaire paper. This implies also similar vanishing results in the Hopkins-Miller higher real $K$-theory spectra. The proof uses that the slice spectral sequence is a spectral sequence of Mackey functors.

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.

This is the first part in a series of papers establishing an equivariant analogue of Steve Wilson's theory of even spaces, including the fact that the spaces in the loop spectrum for complex cobordism are even.