Towards topological Hochschild homology of Johnson-Wilson spectra

The ER(2)-cohomology of X^nCP^\infty and BU(n)

We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory for products of infinite complex projective spaces and for the classifying spaces for the unitary groups.

Infinitesimal thickenings of Morava K-theories

A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well.

Smashing Localizations in Equivariant Stable Homotopy

We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations i…

On fibrations related to real spectra

We consider real spectra, collections of Z/(2)-spaces indexed over Z oplus Z alpha with compatibility conditions. We produce fibrations connecting the homotopy fixed points and the spaces in these spectra. We also evaluate the map which is the analogue of the forgetful functor from complex to reals composed with complexification. Our first fibration is used to connect the real 2^{n+2}(2^n-1)-periodic Johnson--Wilson spectrum ER(n) to the usual 2(2^n-1)-periodic Johnson--Wilson spectrum, E(n). Our main result is the fibration Sigma^{lambda(n)} ER(n) --> ER(n) --> E(n)$, where lambda(n) = 2^{2n+…

Duality in BP<n> (co)homology

Let E=BP<n> denote the Johnson-Wilson spectrum, localized at p. It is proved that if E_*(X) is locally finite, then there is an isomorphism of right E_*-modules E^*(X) = (E_*(Sigma^{D+n+1}X))^V, where D=Sum |v_i| and M^V=Hom(M,Q/Z) is the Pontryagin dual. This result was motivated by work of the author and W.S.Wilson regarding the 2-local ku-homology and -cohomology of the Eilenberg-MacLane space K(Z/2,2).

连通复数K理论的积分拓扑学Hochschild同调

Integral topological Hochschild homology of connective complex K-theory

本文通过motivic谱序列计算连通复数K理论的THH，并展示其在E_2页退化。

Unstable $K$-cohomology algebra is filtered lambda-ring

Boardman, Johnson, and Wilson gave a precise formulation for an unstable algebra over a generalized cohomology theory. Modifying their definition slightly in the case of complex K-theory by taking into account its periodicity, we prove that an unstable algebra for complex $K$-theory is precisely a filtered $λ$-ring, and vice versa.

色差、Wood定理与高阶实K理论

Chromatic defect, Wood's theorem, and higher real $K$-theories

本文利用Ravenel的Thom谱X(n)引入色差概念以衡量谱距离复可定向的程度，计算了多种重要谱的色差，发展了确定色差的阻碍理论，并证明了经典Wood等价类似物的广泛存在性以构造Z指标的Adams--Novikov塔。

等变非阿贝尔庞加莱对偶性与Thom谱的等变因子同调

Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra

该研究利用参数化高范畴论构造等变Thom谱，证明等变非阿贝尔庞加莱对偶定理，刻画Thom谱的等变因子同调并完成多类谱与球面的相关同调计算。

The $\mathbb Z$-homotopy fixed points of $C_{n}$ spectra with applications to norms of $MU_{\mathbb R}$

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 pro…

On the Algebraic Classification of Module Spectra

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $π_*R$ has graded global homological dimension 2 and is concentrated in degrees divisible by some natural number $N \geq 4$, we prove that the homotopy category of $R$-modules is equivalent to the derived category of the homotopy ring $π_*R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of $R$-modules. T…

高阶半加法代数K理论与红移

Higher Semiadditive Algebraic K-Theory and Redshift

本文定义高阶半加法代数K理论，证明其满足红移猜想：高度≤n的环谱其半加法K理论高度≤n+1，并在Lubin-Tate谱等条件下达到恰好n+1，且与T(n+1)-局部化K理论一致。

关于莫拉瓦K理论的积分代数K理论

On the integral algebraic K-theory of Morava K-theory

该研究计算连通莫拉瓦K理论\(k(n)\)在特定度数下积分代数K理论群的基数，通过轨道过滤等方法，结合对特定形式DGA的分析及动机过滤，得出相关基数结果，还计算了\(W(\overline{\mathbb{F}}_p)/p^n\)代数K理论群基数，其偶数度数下消失。