Jeremy Hahn, Dylan Wilson
We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map $\mathrm{K}(\mathrm{BP}\langle n \rangle)_{(p)} \to \mathrm{L}_{n+1}^{f} \mathrm{K}(\mathrm{BP}\langle n\rangle)_{(p)}$ has bounded above fiber.
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Topological Hochschild homology of truncated Brown-Peterson spectra II
本文研究了截断布朗-佩特森谱的E_3-MU代数形式的拓扑霍奇希尔同调，提出了一种新的计算工具，并证明了其在素数p=2处不等于索姆谱BP⟨n⟩。
Fix the base field Q of rational numbers and let BP<n> denote the family of motivic truncated Brown-Peterson spectra over Q. We employ a "local-to-global" philosophy in order to compute the motivic Adams spectral sequence converging to the bi-graded homotopy groups of BP<n>. Along the way, we provide a new computation of the homotopy groups of BP<n> over the 2-adic rationals, prove a motivic Hasse principle for the spectra BP<n>, and deduce several classical and recent theorems about the K-theory of particular fields.
We prove that the $p$-completed Brown-Peterson spectrum is a retract of a product of Morava $E$-theory spectra. As a consequence, we generalize results of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that the notion of good group is determined by Brown-Peterson cohomology. Furthermore, we show that rational factorizations of the Morava $E$-theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown-Peterson cohomology of such groups.
An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized trun- cated Brown-Peterson spectra of height 2 as E_\infty-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level Γ_1(3)-structure.
For every $n\ge 0$, we construct classes in the Brown-Peterson cohomology $BP\langle n \rangle$ of smooth projective complex algebraic varieties which are not in the image of the cycle map from the corresponding motivic Brown-Peterson cohomology. This generalizes the examples of Atiyah and Hirzebruch to all finite levels in the Brown-Peterson tower.
Topological Hochschild homology of truncated Brown-Peterson spectra I
本文计算了带系数的截断Brown-Peterson谱充分结构化形式的拓扑Hochschild同调，涵盖特定素数下$BP\langle n\rangle$的$E_3$形式及Hill-Lawson构造的$E_\infty$形式，提供了完整的THH计算结果。
The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown-Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us to conclude that the 2-primary Brown-Peterson spectrum does not admit the structure of an E_n-algebra for any n greater than or equal to 12, answering a question of May in the negative.
We specify exterior generators for $π_* THH(MU) = π_*(MU) \otimes E(λ'_n \mid n\ge1)$ and $π_* THH(BP) = π_*(BP) \otimes E(λ_n \mid n\ge1)$, and calculate the action of the $σ$-operator on these graded rings. In particular, $σ(λ'_n) = 0$ and $σ(λ_n) = 0$, while the actions on $π_*(MU)$ and $π_*(BP)$ are expressed in terms of the right units $η_R$ in the Hopf algebroids $(π_*(MU), π_*(MU \wedge MU))$ and $(π_*(BP), π_*(BP \wedge BP))$, respectively.
We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring spectra. We also prove the same results for truncated Brown-Peterson spectra $\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results of Lawson at the prime $2$.
Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$
本文证明在特定条件下，光滑正常$\nmathbf{F}_p$概形的Hodge-de Rham退化成立，条件涉及拟凝聚层范畴提升到截断Brown-Peterson谱及HKR谱序列在$E_2$页退化，方法为非交换版本，与Mathew的论证类似。
In this article, we study the topological Hochschild homology of $\mathbf{E}_3$-forms of truncated Brown-Peterson spectra, taken relative to certain Thom spectra $X(p^n)$ (introduced by Ravenel and used by Devinatz-Hopkins-Smith in the proof of the nilpotence theorem). We prove analogues of Bökstedt's calculations $\mathrm{THH}(\mathbf{F}_p) \simeq \mathbf{F}_p[ΩS^3]$ and $\mathrm{THH}(\mathbf{Z}_p) \simeq \mathbf{Z}_p[ΩS^3\langle{3}\rangle]$. We also construct a topological analogue of the Sen operator of Bhatt-Lurie-Drinfeld, and study a higher chromatic extension. The behavior of these "top…
Syntomic cohomology of truncated Brown--Peterson spectra
计算了所有截断Brown–Peterson谱BP⟨n⟩的E1 MU-代数形式的MU基合成上同调，解决了其代数K-理论的Lichtenbaum–Quillen、望远镜和红移问题，并首次显式计算了高度3的E1-环的代数K-理论。
Splittings of truncated motivic Brown--Peterson cooperations algebras
本文构造了截断动机Brown--Peterson谱的谱级分裂，并计算了Adams谱序列的E_1页，覆盖多个基域。
We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology for complex manifolds.
The real Brown-Peterson homology of $Ω^ρS^{ρ+ 1}$
本文计算了二阶循环群正则表示下循环空间的实Brown-Peterson同调，并发展了相关的comodule Nishida关系。
Logarithmic topological Hochschild homology of the truncated Brown--Peterson spectrum
研究截断Brown-Peterson谱BP{n}的对数拓扑Hochschild同调，通过预对数结构和谱序列分析，计算了n=2时V(2)-同伦群。
Non-torsion non-algebraic classes in the Brown-Peterson tower
本文研究布朗-彼得森塔中非 torsion 非代数类的存在性，针对素数2构建了相关例子。
We compute the Brown-Peterson cohomology of QX, the free infinite loop-space on X, when X is a space whose Morava K-theory is flat over its BP-cohomology, in particular a space whose Morava K-theory is concentrated in even degrees. Our computation is in terms of a destabilization functor for BP-cohomology. We also show that for such X, the Morava K-homology of QX is a free commutative algebra.
For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the Atiyah-Hirzebruch spectral sequence for G; 3. The Brown-Peterson generalized cohomology of G. It follows that these groups afford counterexamples to conjectures of C. B. Thomas, M. F. Atiyah and P. Landweber.
For an odd prime $p$, we study the image of the Thom map from Brown-Peterson cohomology of $BPU_n$ to the ordinary cohomology in dimensions $0\leq i\leq 2p+2$, where $BPU_n$ is the classifying space of the projective unitary group $PU_n$. Also we show that a family of well understood $p$-torsion cohomology classes $y_{p,k}\in H^{2p^{k+1}+2}(BPU_n;Z_{(p)})$ are in the image of the Thom map.
We provide a lower bound for the coherence of the homotopy commutativity of the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on Dyer-Lashof operations that BP cannot be a Thom spectrum associated to n-fold loop maps to BSF for n=4 at 2 and n=2p+4 at odd primes. Other examples where we obtain estimates for coherence are the Johnson-Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-K-theory.
Structured Quotients in Real Homotopy Theory
本文赋予实配边商环对合结构，以截断实Brown-Peterson谱为例，定向Lubin-Tate理论，并刻画色局部化后等价于Lubin-Tate理论形式的更高截断Brown-Peterson谱。
Generalized Thom Spectra, Conjugate Stacks, and truncated Brown-Peterson Spectra
本文针对含截断布朗-彼得森谱$\rm BP\braket{n}$的$\rm E_3$-$\rm MU$-代数类定义共轭层，研究广义托姆谱及其相对拓扑霍奇同调，证明多个谱为对应托姆谱并给出其无法实现$\rm E_\bbinfty$-细化的结果。
We characterize ring spectra morphisms from the algebraic cobordism spectrum $\QTR{Bbb}{MGL}$ (\QCITE{cite}{}{Vo1}) to an oriented spectrum $\QTR{Bbb}{E}$ (in the sense of Morel \QCITE{cite}{}{Mo}) via formal group laws on the ''topological'' subring $E^{*}=\oplus_iE^{2i,i}$ of $E^{**}$. This result is then used to construct for any prime $p$ a motivic Quillen idempotent on $\QTR{Bbb}{MGL}_{(p)}$. This defines the $BP$-spectrum associated to the prime $p$ as in Quillen's \QCITE{cite}{}{Q1} for the complex-oriented topological case.
We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by products of standard lens spaces. This considerably improves previous calculations by various authors. Our approach relies on the investigation of the submodule of the Brown-Peterson homology of $B (\mathbb{Z}/p)^n$ generated by elements coming from proper subgroups of $(\mathbb{…
Uniqueness of $p$-local truncated Brown-Peterson spectra
本文证明当p为奇素数时，BP⟨n⟩的F_p-上同调作为Steenrod代数上的模可唯一决定其p-局部谱，并由此推出其p-局部同伦型不依赖于取商所用的理想；对p=2的情形得到n≤3的结果。
These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent w…
This paper provides some explicit expressions concerning the formal group laws of the Brown-Peterson cohomology, the cohomology theory obtained from Brown-Peterson theory by killing all but one Witt symbol, the Morava $K$-theory and the Abel cohomology.
Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theor…
In previous work, the author analyzed the co-operations algebra for the second truncated Brown-Peterson spectrum at the prime $p=2$. The purpose of this paper is to carry out the necessary modifications to odd primes.
Examples of disk algebras
本文细化了若干重要谱的乘法结构，使得相应的拓扑Hochschild同调获得圆作用。
A structure theorem for bounded-below modules over the subalgebra A(1) of the mod 2 Steenrod algebra generated by Sq^1, Sq^2 is proved; this is applied to prove a classification theorem for a family of indecomposable A(1)-modules. The action of the A(1)-Picard group on this family is described, as is the behaviour of duality. The cohomology of dual Brown-Gitler spectra is identified within this family and the relation with members of the A(1)-Picard group is made explicit. Similarly, the cohomology of truncated projective spaces is considered within this classification; this leads to a concept…
We use Brown-Peterson cohomology to obtain lower bounds for the higher topological complexity, TC_k(RP^n), of real projective spaces, which are often much stronger than those implied by ordinary mod-2 cohomology.
Following Hu and Kriz, we study the $C_2$-spectra $BP\mathbb{R}\langle n \rangle$ and $E\mathbb{R}(n)$ that refine the usual truncated Brown-Peterson and the Johnson-Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases $n=1$ and $2$.
In this paper we develop techniques to compute the cooperations algebra for the second truncated Brown-Peterson spectrum $\tBP{2}$. We prove that the cooperations algebra $\tBP{2}_*\tBP{2}$ decomposes as a direct some of a $\F_2$-vector space concentrated in Adams filtration 0 and a $\F_2[v_0,v_1,v_2]$-module which is concentrated in even degrees and $v_2$-torsion free. A recursive procedure is also developed to provide an basis of the $v_2$-torsion free part.
Structured Real Snaith Equivalences
通过Wilson空间理论控制结构化实定向，给出实Snaith等价及其乘法精化的简短证明，并应用于计算THR(KU_R)和THR(MUP_R)。
The homotopy fixed points of Real spin bordism
该研究将spinⁿ配边的2-局部分裂细化为C₂-等变映射，推导出实自旋配边同伦不动点的2-局部分裂，并讨论了真实情形下的前景。
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