arXiv subjects

Vitaly Lorman

Publications and source records attributed to Vitaly Lorman.

6 recordsLinked to original sources

arXiv2022

Tate blueshift and vanishing for Real oriented cohomology

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 of Greenlees--Sadofsky. In the course of proving these results, we develop some ideas in equivariant chromatic homotopy theory (e.g., completions of module spectra over Real cobordism, $C_2$-equivariant chromatic Bousfield localizations) and apply the parametrized Tate construction.

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arXiv2018

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.

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arXiv2017

Landweber flat real pairs and ER(n)-cohomology

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).

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arXiv2017

Multiplicative structure on Real Johnson-Wilson theory

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.

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arXiv2017

The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$

The $ER(2)$-cohomology of $B\mathbb{Z}/(2^q)$ and $\mathbb{C}P^n$ are computed along with the Atiyah-Hirzebruch spectral sequence for $ER(2)^*(\mathbb{C}P^\infty)$. This, along with other papers in this series, gives us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces. Since $ER(2)$ is $TMF_0(3)$ after a suitable completion, these computations also take care of that theory.

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arXiv2016

The Real Johnson-Wilson Cohomology of $\mathbb{C}P^\infty$

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 $ER(n)^*(\mathbb{C}P^\infty)$

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