arXiv subjects
Publications and source records attributed to Hana Jia Kong.
We give explicit product formulas for the mod 2 motivic Steenrod algebra in the conjugated motivic Milnor basis. This provides an efficient tool for computing the E2-page of the motivic Adams spectral sequence
This paper is divided into three parts. In the first part, we give the general abstract axiomatic theory and treat the classical examples of infinite loop space machines with structured spaces or $G$-spaces as input and spectra or $G$-spectra as output. The new prequel paper gives a logically compelling and more general but less useful analog that does not involve group completion. In the second part, we develop a general context of composite adjunctions that feeds into the first. It specializes to give infinite loop space machines that take either orbital presheaves or algebras over categories of operators as input. The new sequel focuses on new constructions and applications when the starting category is that of orbital presheaves of spaces and the output is $G$-spectra. In the brief third part, we show how the multiplicative theory fits into the axiomatic frameworks of the first and second parts.
We give an axiomatic homotopical analog of the classical categorical Beck monadicity theorem. It often holds when classical monadicity fails. This grew out of an understanding of a general context for recognition principles in iterated loop space theory, as treated in the logical sequel ArXiv 2402.03649, but the present result applies differently and more generally. An example gives a new perspective on the old equivalence between simplicial sets and topological spaces: both are equivalent to simplicial topological spaces, and the equivalence implies a curiously close relationship between realizations of simplicial spaces and realizations of their underlying simplicial sets, viewed as discrete simplicial spaces.
Let $G$ be a finite group. Using a new kind of operad, we axiomatize and explore an infinite loop space machine that constructs (genuine) $G$-spectra from suitably structured functors on orbital presheaves, which are just contravariant functors from the orbit category of $G$ to based spaces. The theory leads unexpectedly to a new operadic description of Mackey functors and hence to a definition of ``topological Mackey functors" and a construction of their associated $G$-spectra. It also leads to Picard $G$-spectra, Azumaya ring $G$-spectra, and Brauer $G$-spectra. These constructions raise many unanswered questions.
We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt--Morrow--Scholze, Morin, Bhatt--Lurie, and Hahn--Raksit--Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.
We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.
We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the $E_2$-page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar $v_n$-periodicities, and it displays interesting number-theoretic properties. Our techniques involve the $\mathbb{C}$-motivic Adams spectral sequence, and we obtain analogous periodic structure in $\mathbb{C}$-motivic stable homotopy.
We study a particular family of elements in the cohomology of the $\mathbb{C}$-motivic Steenrod algebra, also known as the $\mathbb{C}$-motivic Adams $E_2$-page. This family exhibits unusual periodicity properties, and it is related both to $h_1$-localization and to the algebraic Hurewicz image of the motivic modular forms spectrum $\mathrm{mmf}$.
Moss' theorem, which relates Massey products in the $E_r$-page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric monoidal stable topological model category, we prove a general version of Moss' theorem which applies to spectral sequences that arise from filtrations compatible with the monoidal structure. The theorem has broad applications, e.g. to the computation of the motivic slice and motivic Adams spectral sequences.
We consider $G=Q_8,SD_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height $2$ Morava $E$-theory $\mathbf{E}_2$ at the prime $2$. We completely compute the $G$-homotopy fixed point spectral sequences of $\mathbf{E}_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(*-σ_i)$-graded $Q_8$- and $SD_{16}$-homotopy fixed point spectral sequences, where $σ_i$ is a non-trivial one-dimensional representation of $Q_8$.
We compute the slice spectral sequence for the motivic stable homotopy groups of $L$, a motivic analogue of the connective $K(1)$-local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic $K(1)$-local sphere over arbitrary base fields of characteristic not two. To compute the slice spectral sequence, we prove several results which may be of independent interest. We describe the $d_1$-differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, R{ö}ndigs, and Østvær for the very effective cover of Hermitian K-theory. We also explicitly describe the coefficients of certain motivic Eilenberg--MacLane spectra and compute the slice spectral sequence for the very effective cover of Hermitian K-theory over prime fields.
We analyze the $\mathbb{C}$-motivic (and classical) Adams-Novikov spectral sequence for the $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$ (and for the classical topological modular forms spectrum $\mathit{tmf}$). We primarily use purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of $\mathit{tmf}$.
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.
We compute the $v_1$-periodic $\mathbb{R}$-motivic stable homotopy groups. The main tool is the effective slice spectral sequence. Along the way, we also analyze $\mathbb{C}$-motivic and $η$-periodic $v_1$-periodic homotopy from the same perspective.
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson.
We define the Chow $t$-structure on the $\infty$-category of motivic spectra $SH(k)$ over an arbitrary base field $k$. We identify the heart of this $t$-structure $SH(k)^{c\heartsuit}$ when the exponential characteristic of $k$ is inverted. Restricting to the cellular subcategory, we identify the Chow heart $SH(k)^{cell, c\heartsuit}$ as the category of even graded $MU_{2*}MU$-comodules. Furthermore, we show that the $\infty$-category of modules over the Chow truncated sphere spectrum is algebraic. Our results generalize the ones in Gheorghe--Wang--Xu in three aspects: To integral results; To all base fields other than just $C$; To the entire $\infty$-category of motivic spectra $SH(k)$, rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field $k$ using the Postnikov tower associated to the Chow $t$-structure and the motivic Adams spectral sequences over $k$.
For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the $\mathbb{R}$-motivic effective slice spectral sequence is completely determined by the $ρ$-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.
We construct a $C_2$-equivariant spectral sequence for RO$(C_2)$-graded homotopy groups. The construction is by using the motivic effective slice filtration and the $C_2$-equivariant Betti realization. We apply the spectral sequence to compute the RO$(C_2)$-graded homotopy groups of the completed $C_2$-equivariant connective real $K$-theory spectrum. The computation reproves the $C_2$-equivariant Adams spectral sequence results by Guillou, Hill, Isaksen and Ravenel.