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Register a new userRelative Loday constructions and applications to higher THH-calculations
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We define a relative version of the Loday construction for a sequence of commutative S-algebras $A rightarrow B rightarrow C$ and a pointed simplicial subset $Y subset X$. We use this to construct several spectral sequences for the calculation of higher topological Hochschild homology and apply those for calculations in some examples that could not be treated before.
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We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $Sigma X$. We prove several structural properties about different notions of stability, corresponding to different levels of invariance required of $Xotimes A$, and establish stability in important cases, such as complex and real periodic topological K-theory, $KU$ and $KO$.
We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology.
Tensoring finite pointed simplicial sets with commutative ring spectra yields important homology theories such as (higher) topological Hochschild homology and torus homology. We prove several structural properties of these constructions relating $X otimes (-)$ to $Sigma X otimes (-)$ and we establish splitting results. This allows us, among other important examples, to determine $THH^{[n]}_*(mathbb{Z}/p^m; mathbb{Z}/p)$ for all $n geq 1$ and for all $m geq 2$.
Let $f:Gto mathrm{Pic}(R)$ be a map of $E_infty$-groups, where $mathrm{Pic}(R)$ denotes the Picard space of an $E_infty$-ring spectrum $R$. We determine the tensor $Xotimes_R Mf$ of the Thom $E_infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is the circle, the tensor with $X$ is topological Hochschild homology over $R$. We use the theory of localizations of $infty$-categories as a technical tool: we contribute to this theory an $infty$-categorical analogue of Days reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization $L$ of the $infty$-category of presentable $infty$-categories, the free $L$-local presentable $infty$-category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$-localization of the $infty$-category of spaces. If $X$ is a pointed space, a map $g: Ato B$ of $E_infty$-ring spectra satisfies $X$-base change if $Xotimes B$ is the pushout of $Ato Xotimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is etale then it satisfies $X$-base change provided $X$ is connected. We also prove that $g$ satisfies $X$-base change provided the multiplication map of $B$ is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of $S^0$-base change.
We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra.
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