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Detecting periodic elements in higher topological Hochschild homology. (English) Zbl 1384.55007

The topological Hochschild homology spectrum of a commutative ring spectrum \(R\) can be modeled by the tensor \(R \otimes S^1\) of \(R\) with the \(1\)-sphere. The tensor \(\Lambda_X R = R \otimes X\) of \(R\) with a more general space \(X\) is also known as the Loday construction and has been studied by M.Brun, G. Carlsson and B. I. Dundas [Adv. Math. 225, No. 6, 3166-3213 (2010; Zbl 1208.19003)]. The purpose of the present work is to calculate the homotopy groups \(\Lambda_X R\) when \(R\) is the Eilenberg-Mac Lane spectrum \(H \mathbb F_p\) at a prime \(p \geq 5\), the space \(X\) is either an \(n\)-sphere \(S^n\) or an \(n\)-torus \(T^n\), and \(n\leq p\).
The relevance of these calculations comes from the hope that fixed point information of the iterated topological Hochschild homology spectrum \(\Lambda_{T^n}(R)\) should provide an approximation to the iterated algebraic \(K\)-theory \(K^{(n)}(R) = K(K(\dots K(R)))\) of \(R\), just as the topological cyclic homology spectrum built from fixed points of the topological Hochschild homology spectrum \(\Lambda_{S^1}R\) approximates the algebraic \(K\)-theory of \(R\). The chromatic redshift conjecture predicts that under suitable assumptions, algebraic \(K\)-theory should increase chromatic complexity. Thus \(\Lambda_{T^n} H \mathbb F_p\) may be among the most accessible objects that can provide evidence for chromatic redshift in high chromatic levels. In fact, the author shows that in accordance with these expectations, the image of \(v_{n-1}\) under the unit map \(\mathbb F_p[v_{n-1}] \cong \pi_* k(n-1) \to k(n-1)_*( \Lambda_{T^n} H \mathbb F_p)^{h T^n}\) is non-zero. Here \(k(n-1)\) is the connective Morava \(K\)-theory spectrum, \(k(n-1)_*\) denotes \(k(n-1)\)-homology, and \( ( \Lambda_{T^n} H \mathbb F_p)^{h T^n}\) is the \(T^n\)-homotopy fixed point spectrum of \(\Lambda_{T^n} H \mathbb F_p\).
To establish these results, the author first determines the homotopy groups of \(\Lambda_{S^n} H \mathbb F_p\). Using this, he sets up a family of bar spectral sequences and exploits a multifold Hopf algebra structure coming from the various circles in \(T^n\) in order to compute the homotopy groups of \(\Lambda_{T^n} H \mathbb F_p\). The result about the image of \(v_{n-1}\) in the \(k(n-1)\)-homology of \((\Lambda_{T^n} H \mathbb F_p)^{h T^n}\) is the consequence of an analysis of the homotopy fixed point spectral sequence.

MSC:

55P42 Stable homotopy theory, spectra
55P91 Equivariant homotopy theory in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55T99 Spectral sequences in algebraic topology

Citations:

Zbl 1208.19003
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