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The group of homotopy self-equivalence classes of CW complexes

Published online by Cambridge University Press:  24 October 2008

John W. Rutter
Affiliation:
University of Liverpool

Extract

Exact sequences, which were subsequently used for calculating the group ℰ(X) of homotopy classes of self-homotopy equivalences of a space X, were given by Barcus and Barratt (§5 of (1)) in the case where X is obtained from a simply-connected (q + 1 > 1)-dimensional complex by adding one (q + l)-cell (q ≥ 3): these were later extended by Kudo and Tsuchida (theorems 2·2 and 2·8 of (6)) and by the author (theorem 3·1* of (15)), who also obtained a related sequence (theorem 2·3* of (15)). In the case of a two-cell complex, one or more of these sequences has been shown to split by Oka, Sawashita and Sugawara (theorems 3·9, 3·13 and 3·15 of (11)). The sequences have been used to calculate ℰ (X) for a number of complexes having two, three or more cells by various authors, including Oka (8), Oka, Sawashita and Sugawara (11), Rutter (17) and Sawashita (18). However the aforementioned sequences are only applicable to the addition of top-dimensional cells if the complex has no cells in its penultimate dimension. In this article I obtain sequences which are applicable without this latter restriction, show that one of them is generally split, and in special cases where there is only one top-dimensional cell obtain a further splitting: sequences are given which are valid without the assumption that A is simply connected. Also I give a new formula for calculating ℰ(X)in the case where X is not 2-connected.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Barcus, W. D. and Barratt, M. G.On the homotopy classification of the extensions of a fixed map. Trans. Amer. Math. Soc. 88 (1958), 5774.CrossRefGoogle Scholar
(2)Dror, E. and Rutter, J. W. On skeleton preserving self homotopy equivalences of CW complexes. (Preprint.)Google Scholar
(3)Eckmann, B. and Hilton, P. J.Operators and cooperators in homotopy theory. Math. Ann. 141 (1960), 121.CrossRefGoogle Scholar
(4)Ganea, T.A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39 (1965), 295322.CrossRefGoogle Scholar
(5)Hilton, P. J.An introduction to homotopy theory (Cambridge University Press, 1953).CrossRefGoogle Scholar
(6)Kudo, Y. and Tsuchida, K.On the generalized Barcus-Barratt sequence. Sci. Rep. Hirosaki Univ. 13 (1967), 19.Google Scholar
(7)Mimura, M. and Sawashita, N.On the group of self homotopy equivalences of H-spaces of rank 2. J. Math. Kyoto Univ. 21 (1981), 331–49.Google Scholar
(8)Oka, S.Groups of self equivalences of certain complexes. Hiroshima Math. J. 2 (1972), 285298.CrossRefGoogle Scholar
(9)Oka, S.Finite complexes whose self homotopy equivalences form cyclic groups. Mem. Fac. Sci. Kyushu Univ. Ser. A 34 (1980), 171–81.Google Scholar
(10)Oka, S.On the group of self-homotopy equivalences of H-spaces of low rank I, II. Mem. Fac. Sci. Kyushu Univ. Ser. A 35 (1981), 247282, 307323.Google Scholar
(11)Oka, S., Sawashita, N. and Sugawara, M.On the group of self-equivalences of a mapping cone. Hiroshima Math. J. 4 (1974), 928.CrossRefGoogle Scholar
(12)Rutter, J. W.A homotopy classification of maps into an induced fibre space. Topology 6 (1967), 379403.CrossRefGoogle Scholar
(13)Rutter, J. W.Maps and equivalences into equalizing fibrations and from coequalizing cofibrations. Math. Z. 122 (1971), 125141.CrossRefGoogle Scholar
(14)Rutter, J. W.Self-equivalences and principal morphisms. Proc. London Math. Soc. 20 (1970), 644658.CrossRefGoogle Scholar
(15)Rutter, J. W.Groups of self homotopy equivalences of induced spaces. Comment. Math. Helv. 45 (1970), 236255.CrossRefGoogle Scholar
(16)Rutter, J. W.The suspension of the loops on a space with comultiplication. Math. Ann. 209 (1974), 6982.CrossRefGoogle Scholar
(17)Rutter, J. W.The group of self-homotopy equivalences of principal three sphere bundles over the seven sphere. Math. Proc. Cambridge Phil. Soc. 84 (1978), 303311.CrossRefGoogle Scholar
(18)Sawashita, N.On the group of self equivalences of the product of spheres. Hiroshima Math. J. 5 (1975), 6986.CrossRefGoogle Scholar
(19)Sieradski, A. J.Stabilization of self equivalences of the pseudo-projective spaces. Michigan Math. J. 19 (1972), 109119.CrossRefGoogle Scholar
(20)Toda, H.Composition methods in homotopy groups of spheres. Ann. of Math. Studies 49 (1962).Google Scholar