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Register a new userThe topological cyclic homology of the dual circle
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Authors
Cary Malkiewich
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We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.
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We introduce a new approach to computing topological cyclic homology by means of a descent spectral sequence. We carry out the computation for a p-adic local field with Fp-coefficients, including the case p=2 which was only covered by motivic methods except in the totally unramified case.
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 recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBokstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
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.
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