A note on 𝐻_{∞} structures

Lawson, Tyler

doi:10.1090/S0002-9939-2015-12474-4

A note on $H_\infty$ structures
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Proc. Amer. Math. Soc. 143 (2015), 3177-3181 Request permission

Abstract:

We give a source of (coconnective) examples of $H_\infty$ structures that do not lift to $E_\infty$ structures, based on Mandell’s proof of the equivalence between certain cochain algebras and spaces.

References

J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI 10.2307/1970147
José Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720–726. MR 50278, DOI 10.1073/pnas.38.8.720
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
Andrew Baker and Birgit Richter, Uniqueness of $E_\infty$ structures for connective covers, Proc. Amer. Math. Soc. 136 (2008), no. 2, 707–714. MR 2358512, DOI 10.1090/S0002-9939-07-08984-8
A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, MIT OpenCourseWare: Massachusetts Institute of Technology (http://ocw.mit.edu/), Fall 2007.
Michael A. Mandell, $E_\infty$ algebras and $p$-adic homotopy theory, Topology 40 (2001), no. 1, 43–94. MR 1791268, DOI 10.1016/S0040-9383(99)00053-1
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. P. May, $H_{\infty }$ ring spectra and their applications, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 229–243. MR 520544
Justin Noel, $H_\infty \neq E_\infty$, arXiv:0910.3566.
Charles Rezk, Lectures on power operations, available at: http://www.math.uiuc .edu/˜rezk/papers.html.
Brooke Shipley, A convenient model category for commutative ring spectra, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 473–483. MR 2066511, DOI 10.1090/conm/346/06300