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Register a new userTowards topological Hochschild homology of Johnson-Wilson spectra
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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 $K(i)_*THH(E(n))$.
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We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $operatorname{THH}_*(operatorname{taf}^D;M)$ for $Min { Hmathbb{Z}_{(3)},k(1),k(2)}$ where $operatorname{taf}^D$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Hill--Lawson. We compute $operatorname{THH}_*(operatorname{tmf}_1(3);M)$ when $Min { Hmathbb{Z}_{(2)},k(2)}$ where $operatorname{tmf}_1(3)$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Lawson--Naumann. We also compute $operatorname{THH}_*(Blangle nrangle;M)$ for $M=Hmathbb{Z}_{(p)}$ and certain $E_3$ forms $Blangle nrangle$ of $BPlangle nrangle$. For example at $p=2$, this result applies to the $E_3$ forms of $BPlangle nrangle$ constructed by Hahn--Wilson.
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{mathbb{R}}$. In particular, we construct an equivariant version of the Bokstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson.
We show that an important classical fixed point invariant, the Reidemeister trace, arises as a topological Hochschild homology transfer. This generalizes a corresponding classical result for the Euler characteristic and is a first step in showing the Reidemeister trace is in the image of the cyclotomic trace. The main result follows from developing the relationship between shadows, topological Hochschild homology, and Morita invariance in bicategorical generality.
We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.
In this short note we study the topological Hoschschild homology of Eilenberg-MacLane spectra for finite cyclic groups. In particular, we show that the Eilenberg-MacLane spectrum H(Z/p^k) is a Thom spectrum for any prime p (except, possibly, when p=k=2) and we also compute its topological Hoschshild homology. This yields a short proof of the results obtained by Brun, and by Pirashvili except for the anomalous case p=k=2.
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