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The Whitehead products and powers in $W$-topology

Author(s): K. A. Hardie; H. J. Marcum; N. Oda
Journal: Proc. Amer. Math. Soc. 131 (2003), 941-951.
MSC (2000): Primary 55Q15, 55Q20, 55Q25
Posted: October 18, 2002
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Abstract: We initiate a study of $W$-topology. This is a modification of ordinary topology in which morphisms exist after smashing with a fixed space $W$. Several classical topics are considered in this setting, inter alia Whitehead products, Hopf invariants and the $E$-$H$-$\Delta$ sequence. The main emphasis is on detection of non-trivial elements in the $W$-homotopy groups of spheres.

References:

[Bar]
M.G. Barratt, Track groups $({II})$, Proc. London Math. Soc. 5 (1955), 285-329. MR 17:395e

[Bau]
H.J. Baues, Relationen für primäre Homotopieoperationen und eine verallgemeinerte EHP -Sequenz, Ann. Sci. École Norm. Sup. 4 (1975), 509-533. MR 53:1586

[BH]
I. Berstein and P.J. Hilton, Category and generalized Hopf invariants, Illinois J. Math. 4 (1960), 473-451. MR 23:A3572

[H1]
K.A. Hardie, Higher Whitehead products, Quart. J. Math. Oxford (2) 12 (1961), 241-249. MR 25:4521

[H2]
-, On the Hopf-Toda invariant, Trans. Amer. Math. Soc. 112 (1964), 43-54. MR 29:4057

[H3]
-, Toda brackets and the Hopf construction $I$, Math. Colloq. Univ. of Cape Town 9 (1974), 61-74. MR 56:16625

[H4]
-, A note on products of the Whitehead type, Quaestiones Math. 11 (1988), 423-431. MR 90g:55012

[H5]
-, Approximating the homotopy sequence of a pair of spaces, Tsukuba J. Math. 15 (1991), 85-98. MR 92g:55016

[HJ1]
K.A. Hardie and A.V. Jansen, A Whitehead product for track groups, Algebraic Topology Proceedings, Arcata 1986, Lecture Notes in Math. vol. 1370, Springer-Verlag, 1989, pp. 163-170. MR 90j:55020

[HJ2]
-, A Whitehead product for track groups II, Publicacions Matemátiques 33 (1989), 205-212. MR 91j:55014

[M]
H.J. Marcum, Cone length of the exterior join, Glasgow Math. J. 40 (1998), 445-461. MR 99k:55012

[N]
J. Neisendorfer, Primary homotopy theory, Memoirs AMS vol. 25, no. 232, American Mathematical Society, 1980. MR 81b:55035

[O]
N. Oda, A generalization of the Whitehead product, Math. J. Okayama Univ. 39 (1997), 113-133. MR 2000a:55027

[OS1]
N. Oda and T. Shimizu, Natural maps between some $\Gamma$-suspension spaces and $\Gamma$-loop spaces, Fukuoka University Science Reports 29 (1999), 21-30. MR 2000e:55006

[OS2]
-, A $\Gamma$-Whitehead product for track groups and its dual, Quaestiones Math. 23 (2000), 113-128. MR 2002a:55015

[OS3]
-, Some formulas for the generalized Hardie-Jansen product and its dual, J. Korean Math. Soc. 36 (1999), 527-544. MR 2001a:55012

[P]
G.J. Porter, Higher order Whitehead products, Topology 3 (1965), 123-135. MR 30:4261

[T1]
H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies vol. 49, Princeton University Press, 1962. MR 26:777

[T2]
-, On double suspension $E^2$, J. Inst. Polytech. Osaka City Univ. 7 (1956), 102-145. MR 19:1188g

[T3]
-, Order of the identity class of a suspension space, Ann. of Math. (2) 78 (1963), 300-325. MR 27:6271

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Additional Information:

K. A. Hardie
Affiliation: Department of Mathematics, University of Cape Town, 7700 Rondebosch, South Africa
Email: hardieka@iafrica.com

H. J. Marcum
Affiliation: Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark, Ohio 43055
Email: marcum@math.ohio-state.edu

N. Oda
Affiliation: Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan
Email: odanobu@cis.fukuoka-u.ac.jp

DOI: 10.1090/S0002-9939-02-06833-8
PII: S 0002-9939(02)06833-8
Received by editor(s): April 8, 2000
Received by editor(s) in revised form: August 30, 2000
Posted: October 18, 2002
Additional Notes: The first author acknowledges support from the Foundation for Research Development and the University of Cape Town via the Topology and Category Theory Group.
Communicated by: Ralph Cohen
Copyright of article: Copyright 2002, American Mathematical Society

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