Po Hu
Answering a question of W. S. Wilson, I introduce a Z/2-equivariant Atiyah-Real analogue of Johnson-Wilson cohomology theory BP<n>, whose coefficient ring is the =< n-chromatic part of Landweber's Real cobordism ring.
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We completely compute the Real Johnson-Wilson cohomology of $\mathbb{C}P^\infty$. Applying techniques from equivariant stable homotopy theory to the Bockstein spectral sequence, we produce permanent cycles and solve extension problems to give an explicit description of the ring $ER(n)^*(\mathbb{C}P^\infty)$
We prove that the Real Johnson-Wilson theories ER(n) are homotopy associative and commutative ring spectra up to phantom maps. We further show that ER(n) represents an associatively and commutatively multiplicative cohomology theory on the category of (possibly non-compact) spaces.
We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_\infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) \rightarrow THH(E(2))\rightarrow \overline{THH}(E(2))$ and describe $\overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 \leq i \leq n$. In particular, we compute $K(i)_*THH(E(n))$.
Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)^*(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2^{n+1}] is ER(n)-orientable (but not ER(n+1)-orientable), where MO[2^{n+1}] is defined as the Thom spectrum for the self map of BO given by multiplication by 2^{n+1}.
We show that the Hopf elements, the Kervaire classes, and the $\barκ$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-Wilson theory $E\mathbb{R}(n)$, depending on $n$.
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.
Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson-Wilson spectrum E(2) to deduce certain non-immmersion results for real projective spaces. ER(n) is a $2^{n+2}(2^n-1)$-periodic spectrum. The key result to use is the existence of a stable cofibration $Σ^{λ(n)}ER(n) \rightarrow ER(n) \rightarrow E(n)$ connecting the real Johnson-Wilson spectrum with the classical one. The value of $λ(n)$ is $2^{2n+1}-2^{n+2}+1$. We extend Kitchloo-Wilson's results on non-immersions of real projective spaces by computing the second real Johnson-Wilson…
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.
Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups
该研究引入柄体群的两类Torelli子群，借助辛表示与Johnson同态定义其阿贝尔商，并通过特殊线性群表示理论刻画对应第一有理上同调类的上积，推进柄体Torelli群上同调研究。
On the structures of the Johnson cokernels of the basis-conjugating automorphism groups of free groups
本文研究自由群的基共轭自同构群的Johnson同构的结构，通过构造障碍条件确定其余核直至四阶，并进一步探讨更高阶的性质。应用中解决了四阶Andreadakis问题，证明了第一上同调群的杯积映射是满射，并计算了自由群的辫子-置换自同构群的扭曲第一上同调群。
The Johnson filtration of the automorphism group of a free group is composed of those automorphisms which act trivially on nilpotent quotients of the free group. We compute cohomology classes as follows: (i) we analyze analogous classes for a subgroup of the pure symmetric automorphism group of a free group, and (ii) we analyze features of these classes which are preserved by the Johnson homomorphism. One consequence is that the ranks of the cohomology groups in any fixed dimension between 1 and n-1 increase without bound for terms deep in the Johnson filtraton.
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…
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.
We study transchromatic phenomena for the Tate construction of Real oriented cohomology theories. First, we show that after suitable completion, the Tate construction with respect to a trivial $\mathbb{Z}/2$-action on height $n$ Real Johnson--Wilson theory splits into a wedge of height $n-1$ Real Johnson--Wilson theories. This is the first example of Tate blueshift at all chromatic heights outside of the complex oriented setting. Second, we prove that the Tate construction with respect to a trivial finite group action on Real Morava K-theory vanishes, refining a classical Tate vanishing result…
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+…
We investigate Gamma-cohomology of some commutative cooperation algebras E_*E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Gamma-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E infinity structures. As a consequence we obtain an E infinity structure for the connective Adams summand. For the Johnson-Wilson spectrum E(n) with n > 0 we establish the existence of a unique E infinit…
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).
We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to ER(n)^*(pt) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E(n), the cohomology ring ER(n)^*(Z) can be obtained from E(n)^*(Z) by base change. In particular, our results allow us to compute the Real Johnson-Wilson cohomology of the Eilenberg-MacLane spaces Z = K(Z, 2m+1), K(Z/2^q, 2m), K(Z/2, m) for all natural numbers $m$ and $q$, as well as connective covers of BO: BO, BSO, BSpin, and BO<8> (the last for n<3 only).
Bounded cohomology of classifying spaces for families of subgroups
本文引入子群族的有界 Bredon 上同调，推广有界上同调，并给出相对可遗性与相对双曲性的上同调刻画。
We study natural subalgebras Ch_E(G) of group cohomology defined in terms of infinite loop spaces E and give representation theoretic descriptions of those based on QS^0 and the Johnson-Wilson theories E(n). We describe the subalgebras arising from the Brown-Peterson spectra BP and as a result give a simple reproof of Yagita's theorem that the image of BP^*(BG) in H^*(BG;F_p) is F-isomorphic to the whole cohomology ring; the same result is shown to hold with BP replaced by any complex oriented theory E with a map of ring spectra from E to HF_p which is non-trivial in homotopy. We also extend t…
Real syntomic cohomology
本文提出实数 syntonic 上同调理论，扩展并细化了现有理论，并计算了实数拓扑 K-理论和拓扑模形式的上同调。
Perfect even modules and the even filtration
本文受Hahn-Raksit-Wilson工作启发，在$\mathbf{E}_{1}$-环及其模上定义了偶滤过的新变体，证明其满足平坦下降并与已知滤过一致，进而用于研究连通$\mathbf{E}_{1}$-环的偶上同调，得到消失性、基变换公式及低权重显式计算。
Algebraic K-theory of real topological K-theory
通过棱镜和合成上同调，确定了连通实K理论谱ko的拓扑循环同调的A(1)-同伦，得到秩52的自由F_2[v_2^4]-模，生成元在茎-1≤*≤30中。
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