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Mary Warner, as she was mainly known in the mathematical world, died in April 1998. At a time when few women mathematicians reached the top in their profession, she succeeded in doing so through her ability and determination. Her research contributions were commemorated at a recent international conference on fuzzy topology, the field in which she was one of the pioneers and recognized as one of the leading figures for the past thirty years. She was also an outstanding teacher. But to understand her achievements properly it is necessary to know something of her life.

SynopsisThe purpose of this paper is to introduce a fibrewise generalisation of category, in the sense of Lusternik–Schnirelmann. This reduces to the classical concept when the space is a point. Fibrewise category may be compared with equivariant category, which has been the subject of some recent research [1,7,8]. Many variations on the basic idea of category have been discussed in the literature, for example the concept of category of a map, but since the generalisations to the fibrewise case are fairly routine they are not considered here.

1. Introduction. Let X be a space and let E be a fibre space over E. A fibre-preserving map f: E → E determines, for each point x ∈ X, a map fx: Ex → Ex 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 x ∈ X. 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 proveTheorem 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.

Consider a topological n–sphere Sn, where n ≥ 1. By a commutative product I mean a continuous function f: Sn × Sn → Sn such that

1. Introduction. We say that a smooth compact manifold is embedded in q–space when there is given a regular one-one map of the manifold into q–dimensional Euclidean space. Whitney (5) has shown that embeddings always exist when q is not less than twice the dimension of the manifold.

We prove a theorem which facilitates homotopy classification of maps into a topological group G. Some information about homotopy groups of G is obtained, including the following two results. Consider the Samelson product, as defined in (7), which constitutes a bilinear pairing of πp(G) with πq(G) to πp+q(G). The product of a α ∈ πp(G) with β ∈ πq(G) is written in the form 〈α, β〉. There exist groups having Samelson products of infinite order. Homotopy-commutative groups have zero Samelson products. We shall proveTheorem (1·1). If G is a connected Lie group then there exists a positive integer n such that n〈α, β〉 = 0 for every pair α, β of elements in the homotopy groups of G.

In the theory of vector-spaces an ordered, ortho-normal set of r vectors is called an r-frame. Let Sn denote the unit sphere in euclidean (n+ 1)-space, where n ≥ 1. By an r-field on Sn is meant a continuous function which assigns to each point of Sn an r-frame in the tangent space at that point. If q < r we obtain a q-field from an r-field by suppressing the first r – q vectors of each r-frame. Certainly Sn admits a 0-field, and does not admit an (n+ 1)-field, since the tangent space is n-dimensional. An n-field on Sn is called a parallelism. Notice that an (n − 1)-field on Sn can always be extended to an n-field, since spheres are orientable. The problem is to determine the greatest value of r such that Sn admits an r-field.

Suppose that we have a fibre bundle where the total space has the same homotopy type as the product of the fibre and the base. When can we conclude that the bundle is trivial, in the sense of fibre bundle theory? This question arises in the classification theory of Hopf homogeneous spaces, especially in relation to Stiefel manifolds. Results are proved, using cohomology operations, which answer the question in some cases.