Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T21:23:20.870Z Has data issue: false hasContentIssue false
  • English
  • Français

On conjectures of Moore and Serre in the case of torsion-free suspensions

Published online by Cambridge University Press:  24 October 2008

Paul Selick
Paul Selick
Affiliation:
The University of Western Ontario, London, Ontario, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

Let p be a prime. A space X is said to have a homotopy exponent at p if multiplication by pr annihilates the p-torsion of πn(X) for some non-negative integer r independent of n. X is said to have totally finite rational homotopy if ⊕n πn (X) ⊗ ℚ is a finite vector space. Moore has conjectured that these properties are related for finite simply connected CW complexes.

Moore's Conjecture. A space having the homotopy type of a finite simply connected CW complex has a homotopy exponent at p if and only if it has totally finite rational homotopy.

For convenience I will divide the conjecture into its two implications so that I can refer to each separately.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 53 - 60
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Adams, J. F.Vector fields on spheres. Ann. of Math. 75 (1962), 603–632.CrossRefGoogle Scholar
(2)Adams, J. F.On the groups J(X) - IV. Topology 5 (1966), 21–71.CrossRefGoogle Scholar
(3)Adams, J. F. Lectures on generalized cohomology. Lecture Notes in Mathematics, no. 99 (Springer-Verlag, 1969), 1–138.Google Scholar
(4)Atiyah, M.Vector bundles and the Künneth formula. Topology 1 (1962), 245–248.CrossRefGoogle Scholar
(5)Cohen, F., Moore, J. and Neisendorfer, J.Torsion in homotopy groups. Ann. of Math. 109 (1979), 121–168.CrossRefGoogle Scholar
(6)Cohen, F., Moore, J. and Neisendorfer, J.The double suspension and exponents of the homotopy groups of spheres. Ann. of Math. 110 (1979), 549–565.CrossRefGoogle Scholar
(7)Cohen, F., Moore, J. and Neisendorfer, J. Decomposition of loop spaces and applications to exponents. Conference Proc., Aarhus 1978. Lecture Notes in Math. 763 (Springer-Verlag, 1979).Google Scholar
(8)Cohen, F., Moore, J. and Neisendorfer, J. Exponents in homotopy theory. (To appear.)Google Scholar
(9)Gray, B.On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8 (1969), 219–232.CrossRefGoogle Scholar
(10)James, I.Reduced product spaces. Ann. of Math. 62 (1955), 170–197.CrossRefGoogle Scholar
(11)James, I.On the suspension triad. Ann. of Math. 63 (1956), 191–247.CrossRefGoogle Scholar
(12)James, I.The suspension triad of a sphere. Ann. of Math. 63 (1956), 407–429.CrossRefGoogle Scholar
(13)Long, J. Thesis, Princeton University (1978).Google Scholar
(14)Mahowald, M.The image of J in the EHP sequence. Ann. of Math. 116 (1982), 65–112.CrossRefGoogle Scholar
(15)Mislin, G.The splitting of the Künneth sequence for generalized cohomology. Math. Zeit. 122 (1971), 237–245.CrossRefGoogle Scholar
(16)Neisendorfer, J.3-primary exponents. Math. Proc. Cambridge Philos. Soc. 90 (1981), 63–83.CrossRefGoogle Scholar
(17)Neisendorfer, J. Smaller exponents for Moore spaces. (To appear.)Google Scholar
(18)Neisendorfer, J. and Selick, P.Some examples of spaces with and without homotopy exponents. Proc. of Current Trends in Alg. Top., Canad. Conf. Proc. Series 2 (1982).Google Scholar
(19)Selick, P.Odd primary torsion in πk(S 3). Topology 17 (1978), 407–412.CrossRefGoogle Scholar
(20)Serre, J.-P.Cohomologie modulo 2 des complexes d'Eilenberg-MacLane. Comment. Math. Helv. 27 (1953), 198–232.CrossRefGoogle Scholar
(21)Toda, H.On the double suspension E 2. J. Inst. Polytech. Osaka City Univ., Ser. A 7 (1956), 103–145.Google Scholar
(22)Umeda, Y.A remark on a theorem of J. P. Serre. Proc. Japan Acad. 35 (1959), 563–566.Google Scholar