Secondary power operations and the Brown-Peterson spectrum at the prime \(2\). (English) Zbl 1431.55011

Lawson, Tyler

Ann. Math. (2) 188, No. 2, 513-576 (2018).

The Brown-Peterson Spectrum defines a cohomology theory which has the same strength as complex cobordism, but is much smaller and the characteristic features are more visible. \(E_\infty\)-structures have become increasingly useful, and it was natural to ask whether the Brown-Peterson spectrum enjoys this extra structure. In this paper, the author shows that localized at the prime 2 it does not. The prime 2 is the most difficult in these kinds of problems, so there is still hope that it might be better at larger primes.
The author has a lengthily and very clear introduction to these structures and the problems they entail, which I recommend.

Reviewer: Brayton Gray (Chicago)

MSC:

55P43

Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)

Keywords:

structured ring spectra; secondary operations; Brown-Peterson spectrum

References:

[1]	Ando, Matthew; Blumberg, Andrew J.; Gepner, David; Hopkins, Michael J.; Rezk, Charles, An {\(\infty\)}-categorical approach to {\(R\)}-line bundles, {\(R\)}-module {T}hom spectra, and twisted {\(R\)}-homology, J. Topol.. Journal of Topology, 7, 869-893, (2014) · Zbl 1312.55011 · doi:10.1112/jtopol/jtt035
[2]	Adams, J. F., On the groups {\(J(X)\)}. {IV}, Topology. Topology. An International Journal of Mathematics, 5, 21-71, (1966) · Zbl 0145.19902 · doi:10.1016/0040-9383(66)90004-8
[3]	Angeltveit, Vigleik; Lind, John A., Uniqueness of {\(BP \langle n\rangle\)}, J. Homotopy Relat. Struct.. Journal of Homotopy and Related Structures, 12, 17-30, (2017) · Zbl 1379.55007 · doi:10.1007/s40062-015-0120-0
[4]	Ando, Matthew, Isogenies of formal group laws and power operations in the cohomology theories {\(E_n\)}, Duke Math. J.. Duke Mathematical Journal, 79, 423-485, (1995) · Zbl 0862.55004 · doi:10.1215/S0012-7094-95-07911-3
[5]	Angeltveit, Vigleik, Topological {H}ochschild homology and cohomology of {\(A_\infty\)} ring spectra, Geom. Topol.. Geometry & Topology, 12, 987-1032, (2008) · Zbl 1149.55006 · doi:10.2140/gt.2008.12.987
[6]	Ausoni, Christian; Rognes, John, Algebraic {\(K\)}-theory of topological {\(K\)}-theory, Acta Math.. Acta Mathematica, 188, 1-39, (2002) · Zbl 1019.18008 · doi:10.1007/BF02392794
[7]	Baas, Nils Andreas, On bordism theory of manifolds with singularities, Math. Scand.. Mathematica Scandinavica, 33, 279-302, (1973) · Zbl 0281.57027 · doi:10.7146/math.scand.a-11491
[8]	Baker, Andrew, {\(BP\)}: close encounters of the {\(E_\infty\)} kind, J. Homotopy Relat. Struct.. Journal of Homotopy and Related Structures, 9, 553-578, (2014) · Zbl 1314.55001 · doi:10.1007/s40062-013-0051-6
[9]	Baker, Andrew, Power operations and coactions in highly commutative homology theories, Publ. Res. Inst. Math. Sci.. Publications of the Research Institute for Mathematical Sciences, 51, 237-272, (2015) · Zbl 1351.55014 · doi:10.4171/PRIMS/154
[10]	Basterra, M., Andr\'e-{Q}uillen cohomology of commutative {\(S\)}-algebras, J. Pure Appl. Algebra. Journal of Pure and Applied Algebra, 144, 111-143, (1999) · Zbl 0937.55006 · doi:10.1016/S0022-4049(98)00051-6
[11]	Barratt, M. G.; Eccles, Peter J., {\(\Gamma \sp{+}\)}-structures. {I}. {A} free group functor for stable homotopy theory, Topology. Topology. An International Journal of Mathematics, 13, 25-45, (1974) · Zbl 0292.55010 · doi:10.1016/0040-9383(74)90036-6
[12]	Berger, Clemens, Combinatorial models for real configuration spaces and {\(E_n\)}-operads. Operads: {P}roceedings of {R}enaissance {C}onferences, Contemp. Math., 202, 37-52, (1997) · Zbl 0860.18001 · doi:10.1090/conm/202/02582
[13]	Baker, Andrew; Jeanneret, Alain, Brave new {H}opf algebroids and extensions of {\(M\)}{U}-algebras, Homology Homotopy Appl.. Homology, Homotopy and Applications, 4, 163-173, (2002) · Zbl 1380.55009 · doi:10.4310/HHA.2002.v4.n1.a9
[14]	Basterra, Maria; Mandell, Michael A., The multiplication on {BP}, J. Topol.. Journal of Topology, 6, 285-310, (2013) · Zbl 1317.55005 · doi:10.1112/jtopol/jts032
[15]	Bruner, R. R.; May, J. P.; McClure, J. E.; Steinberger, M., {\(H\sb \infty \)} Ring Spectra and their Applications, Lecture Notes in Math., 1176, viii+388 pp., (1986) · Zbl 0585.55016 · doi:10.1007/BFb0075405
[16]	Brown, Jr., Edgar H.; Peterson, Franklin P., A spectrum whose {\(Z\sb{p}\)} cohomology is the algebra of reduced {\(p\sp{th}\)} powers, Topology. Topology. An International Journal of Mathematics, 5, 149-154, (1966) · Zbl 0168.44001 · doi:10.1016/0040-9383(66)90015-2
[17]	Bruner, Robert R., Extended powers of manifolds and the {A}dams spectral sequence. Homotopy Methods in Algebraic Topology, Contemp. Math., 271, 41-51, (2001) · Zbl 0983.55012 · doi:10.1090/conm/271/04349
[18]	Cohen, R. L.; Jones, J. D. S.; Segal, G. B., Floer’s infinite-dimensional {M}orse theory and homotopy theory. The {F}loer Memorial Volume, Progr. Math., 133, 297-325, (1995) · Zbl 0843.58019 · doi:10.1007/978-3-0348-9217-9_13
[19]	Cohen, Frederick R.; Lada, Thomas J.; May, J. Peter, The Homology of Iterated Loop Spaces, Lecture Notes in Math., 533, vii+490 pp., (1976) · Zbl 0334.55009 · doi:10.1007/BFb0080464
[20]	Chadwick, Steven Greg; Mandell, Michael A., {\(E_n\)} genera, Geom. Topol.. Geometry & Topology, 19, 3193-3232, (2015) · Zbl 1335.55008 · doi:10.2140/gt.2015.19.3193
[21]	Dwyer, W. G.; Kan, D. M., Calculating simplicial localizations, J. Pure Appl. Algebra. Journal of Pure and Applied Algebra, 18, 17-35, (1980) · Zbl 0485.18013 · doi:10.1016/0022-4049(80)90113-9
[22]	Dwyer, W. G.; Kan, D. M., Function complexes in homotopical algebra, Topology. Topology. An International Journal of Mathematics, 19, 427-440, (1980) · Zbl 0438.55011 · doi:10.1016/0040-9383(80)90025-7
[23]	Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, Modules, and Algebras in Stable Homotopy Theory, Math. Surveys Monogr., 47, xii+249 pp., (1997) · Zbl 0894.55001
[24]	Elmendorf, A. D.; Mandell, M. A., Rings, modules, and algebras in infinite loop space theory, Adv. Math.. Advances in Mathematics, 205, 163-228, (2006) · Zbl 1117.19001 · doi:10.1016/j.aim.2005.07.007
[25]	Goerss, P. G.; Hopkins, M. J., Moduli problems for structured ring spectra, (2020)
[26]	Goerss, P. G.; Hopkins, M. J., Moduli Spaces of Commutative Ring Spectra. Structured Ring Spectra, London Math. Soc. Lecture Note Ser., 315, 151-200, (2004) · Zbl 1086.55006 · doi:10.1017/CBO9780511529955.009
[27]	Gepner, David; Haugseng, Rune, Enriched {\(\infty\)}-categories via non-symmetric {\(\infty\)}-operads, Adv. Math.. Advances in Mathematics, 279, 575-716, (2015) · Zbl 1342.18009 · doi:10.1016/j.aim.2015.02.007
[28]	Goerss, Paul G., Hopf rings, {D}ieudonn\'e modules, and {\(E_*\Omega^2S^3\)}. Homotopy Invariant Algebraic Structures, Contemp. Math., 239, 115-174, (1999) · Zbl 0954.55006 · doi:10.1090/conm/239/03600
[29]	Harper, John R., Secondary Cohomology Operations, Grad. Stud. in Math., 49, xii+268 pp., (2002) · Zbl 1003.55001 · doi:10.1090/gsm/049
[30]	Hu, P.; Kriz, I.; May, J. P., Cores of spaces, spectra, and {\(E_\infty\)} ring spectra, Homology Homotopy Appl.. Homology, Homotopy and Applications, 3, 341-354, (2001) · Zbl 0987.55009 · doi:10.4310/HHA.2001.v3.n2.a3
[31]	Hovey, Mark; Shipley, Brooke; Smith, Jeff, Symmetric spectra, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 13, 149-208, (2000) · Zbl 0931.55006 · doi:10.1090/S0894-0347-99-00320-3
[32]	Johnson, Niles; Noel, Justin, For complex orientations preserving power operations, {\(p\)}-typicality is atypical, Topology Appl.. Topology and its Applications, 157, 2271-2288, (2010) · Zbl 1196.55011 · doi:10.1016/j.topol.2010.06.007
[33]	Kochman, Stanley O., Homology of the classical groups over the {D}yer-{L}ashof algebra, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 185, 83-136, (1973) · Zbl 0271.57013 · doi:10.2307/1996429
[34]	Kriz, I., Towers of {\(E_\infty\)}-ring spectra with an application to {\(BP\)}, (1995)
[35]	Lawson, T., Calculating obstruction groups for {E}-infinity ring spectra, (2017)
[36]	Lazarev, A., Homotopy theory of {\(A_\infty\)} ring spectra and applications to {\(M{\rm U}\)}-modules, \(K\)-Theory. \(K\)-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of \(K\)-Theory in the Mathematical Sciences, 24, 243-281, (2001) · Zbl 1008.55007 · doi:10.1023/A:1013394125552
[37]	Lewis, Jr., L. G.; May, J. P.; Steinberger, M.; McClure, J. E., Equivariant Stable Homotopy Theory, Lecture Notes in Math., 1213, x+538 pp., (1986) · Zbl 0611.55001 · doi:10.1007/BFb0075778
[38]	Lawson, Tyler; Naumann, Niko, Strictly commutative realizations of diagrams over the {S}teenrod algebra and topological modular forms at the prime 2, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 2773-2813, (2014) · Zbl 1419.55011 · doi:10.1093/imrn/rnt010
[39]	Lurie, Jacob, Higher Topos Theory, Ann. of Math. Stud., 170, xviii+925 pp., (2009) · Zbl 1175.18001 · doi:10.1515/9781400830558
[40]	Lurie, Jacob, Higher algebra, (2017) · Zbl 1175.18001
[41]	Mandell, Michael A., The smash product for derived categories in stable homotopy theory, Adv. Math.. Advances in Mathematics, 230, 1531-1556, (2012) · Zbl 1246.55010 · doi:10.1016/j.aim.2012.04.012
[42]	May, J. P., Problems in infinite loop space theory. Conference on Homotopy Theory, Notas Mat. Simpos., 1, 111-125, (1975) · Zbl 0332.55007
[43]	May, J. Peter, {\(E\sb{\infty }\)} Ring Spaces and {\(E\sb{\infty }\)} Ring Spectra, Lecture Notes in Math., 577, 268 pp., (1977) · Zbl 0345.55007 · doi:10.1007/BFb0097608
[44]	May, J. P., Idempotents and {L}andweber exactness in brave new algebra, Homology Homotopy Appl.. Homology, Homotopy and Applications, 3, 355-359, (2001) · Zbl 0986.55007 · doi:10.4310/HHA.2001.v3.n2.a4
[45]	Miller, Haynes, A spectral sequence for the homology of an infinite delooping, Pacific J. Math.. Pacific Journal of Mathematics, 79, 139-155, (1978) · Zbl 0383.55007 · doi:10.2140/pjm.1978.79.139
[46]	Mandell, M. A.; May, J. P.; Schwede, S.; Shipley, B., Model categories of diagram spectra, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 82, 441-512, (2001) · Zbl 1017.55004 · doi:10.1112/S0024611501012692
[47]	May, J. P.; Sigurdsson, J., Parametrized Homotopy Theory, Math. Surveys Monogr., 132, x+441 pp., (2006) · Zbl 1119.55001 · doi:10.1090/surv/132
[48]	McClure, James E.; Smith, Jeffrey H., Cosimplicial objects and little {\(n\)}-cubes. {I}, Amer. J. Math.. American Journal of Mathematics, 126, 1109-1153, (2004) · Zbl 1064.55008 · doi:10.1353/ajm.2004.0038
[49]	Mathew, Akhil; Stojanoska, Vesna, The {P}icard group of topological modular forms via descent theory, Geom. Topol.. Geometry & Topology, 20, 3133-3217, (2016) · Zbl 1373.14008 · doi:10.2140/gt.2016.20.3133
[50]	Priddy, Stewart B., Koszul resolutions, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 152, 39-60, (1970) · Zbl 0261.18016 · doi:10.2307/1995637
[51]	Priddy, Stewart, Dyer-{L}ashof operations for the classifying spaces of certain matrix groups, Quart. J. Math. Oxford Ser. (2). The Quarterly Journal of Mathematics. Oxford. Second Series, 26, 179-193, (1975) · Zbl 0311.55013 · doi:10.1093/qmath/26.1.179
[52]	Priddy, Stewart, A cellular construction of {BP} and other irreducible spectra, Math. Z.. Mathematische Zeitschrift, 173, 29-34, (1980) · Zbl 0417.55009 · doi:10.1007/BF01215522
[53]	Peterson, F. P.; Stein, N., Secondary cohomology operations: two formulas, Amer. J. Math.. American Journal of Mathematics, 81, 281-305, (1959) · Zbl 0088.38801 · doi:10.2307/2372745
[54]	Quillen, Daniel, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc.. Bulletin of the American Mathematical Society, 75, 1293-1298, (1969) · Zbl 0199.26705 · doi:10.1090/S0002-9904-1969-12401-8
[55]	Quillen, Daniel, Elementary proofs of some results of cobordism theory using {S}teenrod operations, Advances in Math.. Advances in Mathematics, 7, 29-56, (1971) · Zbl 0214.50502 · doi:10.1016/0001-8708(71)90041-7
[56]	Ravenel, Douglas C., Complex Cobordism and Stable Homotopy Groups of Spheres, Pure Appl. Math., 121, xx+413 pp., (1986) · Zbl 0608.55001
[57]	Richter, Birgit, A lower bound for coherences on the {B}rown-{P}eterson spectrum, Algebr. Geom. Topol.. Algebraic & Geometric Topology, 6, 287-308, (2006) · Zbl 1095.55005 · doi:10.2140/agt.2006.6.287
[58]	Robinson, Alan, Obstruction theory and the strict associativity of {M}orava {\(K\)}-theories. Advances in Homotopy Theory, London Math. Soc. Lecture Note Ser., 139, 143-152, (1989) · Zbl 0688.55008 · doi:10.1017/CBO9780511662614.014
[59]	Robinson, Alan, Gamma homology, {L}ie representations and {\(E_\infty\)} multiplications, Invent. Math.. Inventiones Mathematicae, 152, 331-348, (2003) · Zbl 1027.55010 · doi:10.1007/s00222-002-0272-5
[60]	Ravenel, Douglas C.; Wilson, W. Stephen, The {H}opf ring for complex cobordism, Bull. Amer. Math. Soc.. Bulletin of the American Mathematical Society, 80, 1185-1189, (1974) · Zbl 0294.57022 · doi:10.1090/S0002-9904-1974-13668-2
[61]	Senger, A., On the realization of truncated {B}rown–{P}eterson spectra as {\(E_\infty\)} ring spectra
[62]	Senger, A., The {B}rown-{P}eterson spectrum is not \({E}\_{2(p^2+2)}\) at odd primes, (2017)
[63]	Steenrod, N. E., Cohomology Operations. lectures by {N. E. S}teenrod written and revised by {D. B. A. E}pstein, Ann. of Math. Stud., 50, vii+139 pp., (1962) · Zbl 0102.38104
[64]	Strickland, N. P., Products on {\({\rm MU}\)}-modules, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 351, 2569-2606, (1999) · Zbl 0924.55005 · doi:10.1090/S0002-9947-99-02436-8
[65]	tom Dieck, Tammo, Steenrod-{O}perationen in {K}obordismen-{T}heorien, Math. Z.. Mathematische Zeitschrift, 107, 380-401, (1968) · Zbl 0167.51801 · doi:10.1007/BF01110069
[66]	Tilson, Sean, Power operations in the {K}{\"{u}}nneth spectral sequence and commutative {\(\text{H}\mathbb{F}_p\)}-algebras, (2016)
[67]	Turner, Paul R., Dyer-{L}ashof operations in the {H}opf ring for complex cobordism, Math. Proc. Cambridge Philos. Soc.. Mathematical Proceedings of the Cambridge Philosophical Society, 114, 453-460, (1993) · Zbl 0797.55013 · doi:10.1017/S0305004100071747
[68]	Vogt, Rainer M., Homotopy limits and colimits, Math. Z.. Mathematische Zeitschrift, 134, 11-52, (1973) · Zbl 0276.55006 · doi:10.1007/BF01219090