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The spectrum (P ⋀ bo) −∞

Published online by Cambridge University Press:  24 October 2008

Donald M. Davis  and
Mark Mahowald
Donald M. Davis
Affiliation:
Lehigh University, Bethlehem, PA 18015
Mark Mahowald
Affiliation:
Northwestern University, Evanston, IL 60201
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Extract

There are spectra P−k constructed from stunted real projective spaces as in [1] such that H*(P−k) is the span in ℤ/2[x, x−1] of those xi with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse system

which is similar to those of Lin ([15], p. 451). It is a corollary of Lin's work that there is an equivalence of spectra

where holim is the homotopy inverse limit ([3], ch. 5) and Ŝ–1 the 2-adic completion of a sphere spectrum. One may denote by this holim (P–κ), although one must constantly keep in mind that , but rather

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 85 - 93
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

[1] Adams, J. F.. Operations of the nth kind in K-theory and what we don't know about BP. In New Developments in Topology, L.M.S. Lecture Notes Series no. 11 (Cambridge University Press, 1974), 1–9.Google Scholar
[2] Atiyah, M. F., Bott, R. and Shapiro, A.. Clifford modules. Topology 3 (1964), 3–33.Google Scholar
[3] Bousfield, A. K.. The localization of spectra with respect homology. Topology 18 (1979), 257–281.CrossRefGoogle Scholar
[4] Crabb, M. C.. On the KOℤ2-Euler class. To appear.Google Scholar
[5] Davis, D. M.. Generalized homology and the generalized vector field problem. Quart. J. Math. Oxford Ser. (2) 25 (1974), 169–193.Google Scholar
[6] Davis, D. M.. On the cohomology of MO〈8〉. Proc. Adem Topology Conf., Contemporary Mathematics, vol. 12 (American Mathematical Society, 1982), 91–104.Google Scholar
[7] Davis, D. M., Gitler, S., Iberkleid, W. and Mahowald, M.. Orientability of bundles with respect to certain spectra. Bol. Soc. Mat. Mexicana (2) 24 (1979), 49–55.Google Scholar
[8] Davis, D. M., Gitler, S. and Mahowald, M.. The stable geometric dimension of vector bundles over real projective spaces. Trans. Amer. Math. Soc. 268 (1981), 39–61.Google Scholar
[9] Davis, D. M. and Mahowald, M.. Obstruction Theory and ko-theory, Leature Notes in Mathematics, vol. 658 (Springer-Verlag, 1978), 134–164.Google Scholar
[10] Davis, D. M. and Mahowald, M.. The Euler class for connective fco-theory and an application to immersions of quarternionic projective space. Indiana Univ. Math. J. 28 (1979), 1025–1034.Google Scholar
[11] Gitler, S., Mahowald, M. and Milgram, R. J.. The non-immersion problem for RP n and higher order cohomology operations. Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 432–437.CrossRefGoogle Scholar
[12] Johnson, D. C. and Wilson, W. S.. Projective dimension and Brown-Peterson homology. Topology 12 (1973), 327–353.CrossRefGoogle Scholar
[13] Jones, J. D. S. and Wegmann, S. A.. Limits of stable homotopy and cohomotopy groups. Math. Proc. Cambridge Philos. Soc. 94 (1983), 473–482.CrossRefGoogle Scholar
[14] Kahn, D. S. and Priddy, S. B.. Applications of the transfer to stable homotopy theory. Bull. Amer. Math. Soc. (N.S.) 78 (1972), 981–987.CrossRefGoogle Scholar
[15] Lin, W. H.. On conjectures of Mahowald, Segal and Sullivan. Math. Proc. Cambridge Philos. Soc. 87 (1980), 449–458.CrossRefGoogle Scholar
[16] Lin, W. H., Davis, D. M., Mahowald, M. E. and Adams, J. F.. Calculation of Lin's Ext groups. Math. Proc. Cambridge Philos. Soc. 87 (1980), 459–469.CrossRefGoogle Scholar
[17] Mahowald, M.. The metastable homotopy of Sn. Mem. Amer. Math. Soc. 72 (1967).Google Scholar
[18] Mahowald, M.. The image of J in the EHP sequence. Ann. of Math. (2) 116 (1982), 65–112.CrossRefGoogle Scholar
[19] Mahowald, M. and Milgram, R. J.. Operations which detect Sq 4 in connective K-theory and their applications. Quart. J. Math. Oxford Ser. (2) 27 (1976), 415–432.CrossRefGoogle Scholar
[20] Mahowald, M. and Ray, N.. A note on the Thom isomorphism. Proc. Amer. Math. Soc. 82 (1981), 307–308.CrossRefGoogle Scholar
[21] Wu, W. T.. Classes caractéristiques et i-carrés d'une variété. C.R. Acad. Sci. Paris 230 (1950), 508–511.Google Scholar