arXiv subjects
Publications and source records attributed to Michael A. Hill.
We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}\langle 2 \rangle$. 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 $E_4^{hC_{12}}$ is 384-periodic.
For an equivariant commutative ring spectrum $R$, $π_0 R$ 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 $π_0 R$ 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 $ξ_k$ in the category of associative algebras freely kills the higher generators $ξ_{k+n}$. 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.
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 Künneth 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.
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.
For all subgroups $H$ of a cyclic $p$-group $G$ we define norm functors that build a $G$-Mackey functor from an $H$-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.
We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. 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 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.
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.
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.
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.
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.
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.
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.
We lift to equivariant algebra three closely related classical algebraic concepts: abelian group objects in augmented commutative algebras, derivations, and Kähler differentials. We define Mackey functor objects in the category of Tambara functors augmented to a fixed Tambara functor $\underline{R}$, and we show that the usual square-zero extension gives an equivalence of categories between these Mackey functor objects and ordinary modules over $\underline{R}$. We then describe the natural generalization to Tambara functors of a derivation, building on the intuition that a Tambara functor has products twisted by arbitrary finite $G$-sets, and we connect this to square-zero extensions in the expected way. Finally, we show that there is an appropriate form of Kähler differentials which satisfy the classical relation that derivations out of $\underline{R}$ are the same as maps out of the Kähler differentials.
We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In particular, we compute their $C_2$-equivariant Picard groups and the $C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies corresponding results for the fixed point spectra $TMF_0(3)$ and $Tmf_0(3)$. Furthermore, we prove a Real Landweber exact functor theorem.
Building on structure observed in equivariant homotopy theory, we define an equivariant generalization of a symmetric monoidal category: a $G$-symmetric monoidal category. These record not only the symmetric monoidal products but also symmetric monoidal powers indexed by arbitrary finite $G$-sets. We then define $G$-commutative monoids to be the natural extension of ordinary commutative monoids to this new context. Using this machinery, we then describe when Bousfield localization in equivariant spectra preserves certain operadic algebra structures, and we explore the consequences of our definitions for categories of modules over a $G$-commutative monoid.
On the basis of experimental data and mathematical equations in the literature, we remodel the ionic dynamics of smooth muscle cells (SMCs) as an eigensystem formulation, which is valid for investigating finite variations of variables from the equilibrium like in common experimental operations. This algorithm provides an alternate viewpoint from frequency-domain analysis and enables one to probe functionalities of SMC's rhythm by means of a resonance-related mechanism. Numerical results show three types of calcium oscillations of SMCs in mesenteric arterioles: spontaneous calcium oscillation, agonist-dependent calcium oscillation, and agonist-dependent calcium spike. For simple single and double SMCs, we demonstrate properties of synchronization among complex signals related to calcium oscillations, and show different correlation relations between calcium and voltage signals for various synchronization and resonance conditions. For practical cell clusters, our analyses indicate that the rhythm of SMCs could (1) benefit enhancements of signal communications among remote cells, (2) respond to a significant calcium peaking against transient stimulations for triggering globally-oscillating modes, and (3) characterize the globally-oscillating modes via frog-leap (non-molecular-diffusion) calcium waves across inhomogeneous SMCs.