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On fibre spaces and nilpotency. II

Published online by Cambridge University Press:  24 October 2008

I. M. James
Affiliation:
Mathematical Institute, University of Oxford

Extract

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: EE determines, for each point xX, a map fx: ExEx of the fibre over x. In a previous note (3) the situation was considered where fx is null-homotopic, for all x. In the present note we turn our attention to the situation where fx is homotopic to the identity on Ex, for all xX. If X admits a numerable categorical covering (as when X is an ANR) then such a fibre-preserving map f is a fibre homotopy equivalence, by the well-known theorem of Dold(1). Then the set Φ1(E) of fibre homotopy classes of such maps forms a normal subgroup of the group Φ*(E) of fibre homotopy classes of fibre homotopy equivalences. The purpose of this note is to prove

Theorem 1.1. Let X be a paracompact space of finite category. Let E be a fibre bundle over X of which the fibres are compact and path-connected ANR's. Then the Φ*(E)-growp Φ1(E) is Φ*(E)-nilpotent of class ≤ cat X.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(1)Dold, A.Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963), 223255.CrossRefGoogle Scholar
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