Products on $MU$-modules

Author:
N. P. Strickland

Journal:
Trans. Amer. Math. Soc. **351** (1999), 2569-2606

MSC (1991):
Primary 55T25

DOI:
https://doi.org/10.1090/S0002-9947-99-02436-8

Published electronically:
March 1, 1999

MathSciNet review:
1641115

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of $MU$-modules such as $BP$, $K(n)$ and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over $MU[\frac {1}{2}]_*$ that are concentrated in degrees divisible by $4$; this guarantees that various obstruction groups are trivial. We extend these results to the cases where $2=0$ or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising $2$-local $MU_*$-modules as $MU$-modules.

- J. F. Adams,
*A variant of E. H. Brown’s representability theorem*, Topology**10**(1971), 185–198. MR**283788**, DOI https://doi.org/10.1016/0040-9383%2871%2990003-6 - J. F. Adams,
*Stable homotopy and generalised homology*, University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics. MR**0402720** - Matthew Ando,
*Isogenies of formal group laws and power operations in the cohomology theories $E_n$*, Duke Math. J.**79**(1995), no. 2, 423–485. MR**1344767**, DOI https://doi.org/10.1215/S0012-7094-95-07911-3 - Nils Andreas Baas,
*Bordism theories with singularities*, Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970) Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 1–16. Various Publ. Ser., No. 13. MR**0346823** - Edgar H. Brown Jr.,
*Cohomology theories*, Ann. of Math. (2)**75**(1962), 467–484. MR**138104**, DOI https://doi.org/10.2307/1970209 - Edgar H. Brown Jr. and Franklin P. Peterson,
*A spectrum whose $Z_{p}$ cohomology is the algebra of reduced $p^{th}$ powers*, Topology**5**(1966), 149–154. MR**192494**, DOI https://doi.org/10.1016/0040-9383%2866%2990015-2 - 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** - A. D. Elmendorf. Stabilisation as a CW approximation. Available from http://hopf.math. purdue.edu, 1997.
- 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** - Jens Franke,
*On the construction of elliptic cohomology*, Math. Nachr.**158**(1992), 43–65. MR**1235295**, DOI https://doi.org/10.1002/mana.19921580104 - Mark Hovey, John H. Palmieri, and Neil P. Strickland,
*Axiomatic stable homotopy theory*, Mem. Amer. Math. Soc.**128**(1997), no. 610, x+114. MR**1388895**, DOI https://doi.org/10.1090/memo/0610 - M. Hovey and N. P. Strickland. Morava $K$-theories and localisation. To appear in the Memoirs of the American Mathematical Society, 1995.
- Peter S. Landweber,
*Homological properties of comodules over $M{\rm U}_ \ast (M{\rm U})$ and BP$_ \ast $(BP)*, Amer. J. Math.**98**(1976), no. 3, 591–610. MR**423332**, DOI https://doi.org/10.2307/2373808 - L. Gaunce Lewis Jr.,
*Is there a convenient category of spectra?*, J. Pure Appl. Algebra**73**(1991), no. 3, 233–246. MR**1124786**, DOI https://doi.org/10.1016/0022-4049%2891%2990030-6 - 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** - H. R. Margolis,
*Spectra and the Steenrod algebra*, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam, 1983. Modules over the Steenrod algebra and the stable homotopy category. MR**738973** - Haynes Miller,
*Finite localizations*, Bol. Soc. Mat. Mexicana (2)**37**(1992), no. 1-2, 383–389. Papers in honor of José Adem (Spanish). MR**1317588** - O. K. Mironov,
*Existence of multiplicative structures in the theory of cobordism with singularities*, Izv. Akad. Nauk SSR Ser. Mat.**39**(1975), no. 5, 1065–1092, 1219 (Russian). MR**0402784** - O. K. Mironov,
*Multiplications in cobordism theories with singularities and the Steenrod-tom Dieck operations*, Izv. Akad. Nauk SSSR Ser. Mat.**42**(1978), no. 4, 789–806 (Russian). MR**508827** - Jack Morava,
*A product for the odd-primary bordism of manifolds with singularities*, Topology**18**(1979), no. 3, 177–186. MR**546788**, DOI https://doi.org/10.1016/0040-9383%2879%2990001-6 - C. Nassau. On the structure of $P(n)_*(P(n))$ for $p=2$. preprint, 1996.
- C. Nassau.
*Eine nichtgeometrische Konstruktion der Spektren $P(n)$, Multiplikativen und antimultiplikativen Automorphismen von $K(n)$*. PhD thesis, Johann Wolfgang Goethe-Universität Frankfurt, October 1995. - Daniel Quillen,
*On the formal group laws of unoriented and complex cobordism theory*, Bull. Amer. Math. Soc.**75**(1969), 1293–1298. MR**253350**, DOI https://doi.org/10.1090/S0002-9904-1969-12401-8 - Daniel Quillen,
*Elementary proofs of some results of cobordism theory using Steenrod operations*, Advances in Math.**7**(1971), 29–56 (1971). MR**290382**, DOI https://doi.org/10.1016/0001-8708%2871%2990041-7 - Douglas C. Ravenel,
*Complex cobordism and stable homotopy groups of spheres*, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR**860042** - Douglas C. Ravenel,
*Nilpotence and periodicity in stable homotopy theory*, Annals of Mathematics Studies, vol. 128, Princeton University Press, Princeton, NJ, 1992. Appendix C by Jeff Smith. MR**1192553** - Nobuo Shimada and Nobuaki Yagita,
*Multiplications in the complex bordism theory with singularities*, Publ. Res. Inst. Math. Sci.**12**(1976/77), no. 1, 259–293. MR**0415642**, DOI https://doi.org/10.2977/prims/1195190968 - Tammo tom Dieck,
*Steenrod-Operationen in Kobordismen-Theorien*, Math. Z.**107**(1968), 380–401 (German). MR**244989**, DOI https://doi.org/10.1007/BF01110069 - W. Stephen Wilson,
*Brown-Peterson homology: an introduction and sampler*, CBMS Regional Conference Series in Mathematics, vol. 48, Conference Board of the Mathematical Sciences, Washington, D.C., 1982. MR**655040** - J. J. Wolbert. Classifying modules over ${K}$-theory spectra.
*J. Pure Appl. Algebra*, 124(1-3):289–323, 1998. - Urs Würgler,
*Cobordism theories of unitary manifolds with singularities and formal group laws*, Math. Z.**150**(1976), no. 3, 239–260. MR**418131**, DOI https://doi.org/10.1007/BF01221149 - Urs Würgler,
*On products in a family of cohomology theories associated to the invariant prime ideals of $\pi _ \ast ({\rm BP})$*, Comment. Math. Helv.**52**(1977), no. 4, 457–481. MR**478135**, DOI https://doi.org/10.1007/BF02567379 - Urs Würgler,
*Commutative ring-spectra of characteristic $2$*, Comment. Math. Helv.**61**(1986), no. 1, 33–45. MR**847518**, DOI https://doi.org/10.1007/BF02621900 - Z.-I. Yosimura. Universal coefficient sequences for cohomology theories of $\textrm {{C}{W}}$-spectra.
*Osaka J. Math.*, 12(2):305–323, 1975.

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Additional Information

**N. P. Strickland**

Affiliation:
Trinity College, Cambridge CB2 1TQ, England

Address at time of publication:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, United Kingdom

Email:
n.p.strickland@sheffield.ac.uk

Received by editor(s):
January 9, 1997

Published electronically:
March 1, 1999

Article copyright:
© Copyright 1999
American Mathematical Society