On the homology spectral sequence for topological Hochschild homology

Hunter, Thomas

doi:10.1090/S0002-9947-96-01742-4

On the homology spectral sequence for topological Hochschild homology
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by Thomas J. Hunter PDF

Trans. Amer. Math. Soc. 348 (1996), 3941-3953 Request permission

Abstract:

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of $\Bbb Z/p$ using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of $S$-algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable “lower” notation for the Dyer-Lashof algebra.

References

Marcel Bökstedt, The topological Hochschild homology of $\Bbb Z$ and $\Bbb Z/p$, unpublished.
R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132, DOI 10.1007/BFb0075405
H. E. A. Campbell, F. P. Peterson, and P. S. Selick, Self-maps of loop spaces. I, Trans. Amer. Math. Soc. 293 (1986), no. 1, 1–39. MR 814910, DOI 10.1090/S0002-9947-1986-0814910-1
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146, DOI 10.1007/BFb0080464
A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules and algebras in stable homotopy theory, Preprint.
Stanley O. Kochman, Symmetric Massey products and a Hirsch formula in homology, Trans. Amer. Math. Soc. 163 (1972), 245–260. MR 331388, DOI 10.1090/S0002-9947-1972-0331388-5
L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482, DOI 10.1007/BFb0075778
Hans Ligaard and Ib Madsen, Homology operations in the Eilenberg-Moore spectral sequence, Math. Z. 143 (1975), 45–54. MR 375316, DOI 10.1007/BF01173050
J. Peter May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970) Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR 0281196
J. McClure, R. Schwänzl, and R. Vogt, $\operatorname {THH}(R) \cong R \otimes S^1$ for $E_\infty$ ring spectra, To appear in The Journal of Pure and Applied Algebra.
Douglas C. Ravenel and W. Stephen Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691–748. MR 584466, DOI 10.2307/2374093