\(\mathcal {E}_\infty \) ring spectra and elements of Hopf invariant 1. (English) Zbl 1401.55010
In this paper, the author constructs and studies a family of \(2\)-local \(E_{\infty}\) Thom spectra \(M_{j_1}, M_{j_2}, M_{j_3}\). They are shown to be equivalent to reduced free commutative \(S\)-algebras on certain CW-spectra built by attaching cells along Hopf invariant \(1\) elements. It is pointed out that the latter CW-spectra also appear in work of M. Behrens et al. [“On the ring of cooperations for 2-primary connective topological modular forms”, Preprint, arXiv:1501.01050]. The author’s computation of the homology of these Thom spectra as \(\mathcal A_*\)-comodule algebras motivates his conjecture that \(M{j_2}\) is a wedge of \(kO\)-module spectra and that \(M_{j_3}\) is a wedge of \(\text{tmf}\)-module spectra.
Reviewer: Steffen Sagave (Nijmegen)
MSC:
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
stable homotopy theory; Thom spectrum; \(\mathcal {E}_\infty \) ring spectrum; comodule algebraReferences:
| [1] | Adams, J.F.: Stable Homotopy and Generalised Homology. University of Chicago Press, Chicago (1974) · Zbl 0309.55016 |
| [2] | Adams, J.F., Priddy, S.B.: Uniqueness of \[B\] BSO. Math. Proc. Camb. Philos. Soc. 80, 475-509 (1976) · Zbl 0338.55011 · doi:10.1017/S0305004100053111 |
| [3] | Ando, M., Hopkins, M., Rezk, C.: Multiplicative orientations of \[KO\] KO-theory and of the spectrum of topological modular forms. http://www.math.uiuc.edu/mando/papers/koandtmf.pdf (2010) · Zbl 1055.55007 |
| [4] | Bahri, A.P., Mahowald, M.E.: A direct summand in \[H^*(M{\rm O}\langle 8\rangle,\mathbb{Z}_2)H\]∗(MO⟨8⟩,Z2). Proc. Am. Math. Soc. 78, 295-298 (1980) · Zbl 0436.57014 |
| [5] | Bailey, S.M.: On the spectrum \[b{\rm o}\mathit{\wedge \rm tmf}\] bo∧tmf. J. Pure Appl. Algebra 214(4), 392-401 (2010) · Zbl 1188.55005 · doi:10.1016/j.jpaa.2009.06.005 |
| [6] | Baker, A.: Husemoller-Witt decompositions and actions of the Steenrod algebra. Proc. Edinb. Math. Soc. (2) 28, 271-288 (1985) · Zbl 0558.55012 |
| [7] | Baker, A.: Calculating with topological André-Quillen theory, I: homotopical properties of universal derivations and free commutative \[SS\]-algebras (2012). arXiv:1208.1868 (v5+) · Zbl 1179.55006 |
| [8] | Baker, A.: BP: close encounters of the \[{\cal{E}}_{\infty }\] E∞ kind. J. Homotopy Relat. Struct. 92, 257-282 (2014) · Zbl 1314.55001 |
| [9] | Baker, A.: Power operations and coactions in highly commutative homology theories. Publ. Res. Inst. Math. Sci. Kyoto Univ. 51, 237-272 (2015) · Zbl 1351.55014 |
| [10] | Baker, A., Clarke, F.W., Ray, N., Schwartz, L.: On the Kummer congruences and the stable homotopy of BU. Trans. Am. Math. Soc. 316, 385-432 (1989) · Zbl 0709.55012 |
| [11] | Baker, A., Gilmour, H., Reinhard, P.: Topological André-Quillen homology for cellular commutative \[SS\]-algebras. Abh. Math. Semin. Univ. Hambg. 78(1), 27-50 (2008) · Zbl 1179.55006 · doi:10.1007/s12188-008-0005-9 |
| [12] | Baker, A.J., May, J.P.: Minimal atomic complexes. Topology 43(2), 645-665 (2004) · Zbl 1055.55007 · doi:10.1016/j.top.2003.09.004 |
| [13] | Behrens, M., Ormsby, K., Stapleton, N., Stojanoska, V.: On the ring of cooperations for \[22\]-primary connective topological modular forms (2015). arXiv:1501.01050 · Zbl 1444.55004 |
| [14] | Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: \[H_\infty H\]∞ ring spectra and their applications. In: Lect. Notes Math. vol. 1176 (1986) · Zbl 0585.55016 |
| [15] | Cohen, F.R., Davis, D.M., Goerss, P.G., Mahowald, M.E.: Integral Brown-Gitler spectra. Proc. Am. Math. Soc. 103, 1299-1304 (1988) · Zbl 0669.55004 · doi:10.1090/S0002-9939-1988-0955026-0 |
| [16] | Douglas, C.L., Francis, J., Henriques, A.G., Hill, M.A.: Topological modular forms. Math. Surv. Monogr. 201 (2015). (Based on the Talbot workshop, North Conway, NH, USA, March 25-31, 2007) · Zbl 1304.55002 |
| [17] | Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. Math. Surv. Monogr. 47 (1997). (With an appendix by M. Cole) · Zbl 0894.55001 |
| [18] | Goerss, P.G., Jones, J.D.S., Mahowald, M.E.: Some generalized Brown-Gitler spectra. Trans. Am. Math. Soc. 294, 113-132 (1986) · Zbl 0597.55006 · doi:10.1090/S0002-9947-1986-0819938-3 |
| [19] | Harada, M., Kono, A.: Cohomology mod \[22\] of the classifying space of \[{\rm Spin}^c(n)\] Spinc(n). Publ. Res. Inst. Math. Sci. 22, 543-549 (1986) · Zbl 0627.55011 · doi:10.2977/prims/1195177851 |
| [20] | Hill, M.A.: Cyclic comodules, the homology of \[j\] j, and \[j\] j-homology. Topol. Appl. 155, 1730-1736 (2008) · Zbl 1169.55004 · doi:10.1016/j.topol.2008.05.013 |
| [21] | Kochman, S.O.: Homology of the classical groups over the Dyer-Lashof algebra. Trans. Am. Math. Soc. 185, 83-136 (1973) · Zbl 0271.57013 · doi:10.1090/S0002-9947-1973-0331386-2 |
| [22] | Lance, T.: Steenrod and Dyer-Lashof operations on \[BU\] BU. Trans. Am. Math. Soc. 276, 497-510 (1983) · Zbl 0522.55016 |
| [23] | Lawson, T., Naumann, N.: Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime \[22\]. Int. Math. Res. Not. IMRN 10, 2773-2813 (2014) · Zbl 1419.55011 |
| [24] | Liulevicius, A.: Notes on homotopy of Thom spectra. Am. J. Math. 86, 1-16 (1964) · Zbl 0173.25802 · doi:10.2307/2373032 |
| [25] | Liulevicius, A.: Homology comodules. Trans. Am. Math. Soc. 134, 375-382 (1968) · Zbl 0169.54502 · doi:10.1090/S0002-9947-1968-0251720-X |
| [26] | Margolis, H.R.: Spectra and the Steenrod algebra. In: Modules Over the Steenrod Algebra and the Stable Homotopy Category, North-Holland (1983) · Zbl 0552.55002 |
| [27] | May, J.P.: A general algebraic approach to Steenrod operations. Lect. Notes Math. 168, 153-231 (1970) · Zbl 0242.55023 · doi:10.1007/BFb0058524 |
| [28] | Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. (2) 81, 211-264 (1965) · Zbl 0163.28202 |
| [29] | Montgomery, S.: Hopf algebras and their actions on rings. In: CBMS Regional Conference Series in Mathematics, vol. 82 (1993) · Zbl 0793.16029 |
| [30] | Pengelley, D.J.: The \[A\] A-algebra structure of Thom spectra: \[MSO\] MSO as an example. In: CMS Conf. Proc. Current Trends in Algebraic Topology, Part 1 (London, Ont., 1981), vol. 2, pp. 511-513 (1982) · Zbl 0627.55011 |
| [31] | Pengelley, D.J.: The mod two homology of \[MSO\] MSO and \[MSU\] MSU as A comodule algebras, and the cobordism ring. J. Lond. Math. Soc. 2(25), 467-472 (1982) · Zbl 0458.55003 · doi:10.1112/jlms/s2-25.3.467 |
| [32] | Pengelley, D.J.: \[H^*(M{\rm O}\langle 8\rangle; \,\mathbb{Z}/2)H\]∗(MO⟨8⟩;Z/2) is an extended \[A^*_2\] A2∗-coalgebra. Proc. Am. Math. Soc. 87, 355-356 (1983) · Zbl 0517.55012 |
| [33] | Ravenel, D.C.: Complex cobordism and stable homotopy groups of spheres. In: Pure and Applied Mathematics, vol. 121. Academic Press, New York (1986) · Zbl 0608.55001 |
| [34] | Stong, R.E.: Determination of \[H^*(BO(k,\ldots ), Z_2)H\]∗(BO(k,…),Z2) and \[H^*(BU(k,\ldots ), Z_2)H\]∗(BU(k,…),Z2). Trans. Am. Math. Soc. 107, 526-544 (1963) · Zbl 0116.14702 |
| [35] | Stong, R.E.: Notes on cobordism theory. In: Mathematical Notes. Princeton University Press, Princeton and University of Tokyo Press, Tokyo (1968) · Zbl 0181.26604 |
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