Showing papers on "Equivariant map published in 1969"

On Equivariant Maps Between Spheres with Involutions

Abstract: P: w~nn 1 -* Z (see [7, p. 271] and [8, Th. A]), thereby verifying a conjecture of Bredon. In view of [8, Th. A], it only remains to improve a divisibility relation by a factor of 2 in case n _ 0(4), n > 0. However this is a delicate matter and is accomplished by means of Adams operations in equivariant KO-theory. The main theorem of [7] is a periodicity for the groups wn(r, q; t). A second aim of this note is to show that these groups may be viewed as cobordism groups, and that the periodicity then follows directly. This point of view also leads to an isomorphism of wn(r, q; t) with a suitable stable homotopy (or cohomotopy) group of a stunted real projective space. I would like to thank Professors M. Atiyah and G. Bredon for helpful conversations.

Bordism and Involutions

Abstract: Let X be a topological space with A c X a subspace, and let z-: (X, A) (X, A) be an involution; i.e., a continuous map z-: X X with square the identity and such that MA c A. Combining the notions of bordism (Atiyah [1]) and of differentiable periodic maps (Conner and Floyd [4]), one may define bordism groups of the involution (X, A, z). Specifically, a (free) equivariant bordism class of (X, A, z) is an equivalence class of triples (M, A, f) with M a compact differentiable manifold with boundary, A: M e M a differentiable (fixed-point free) involution on M, and f: (M, AM) (X, A) a continuous equivariant map [zf = fte] sending AM into A. Two triples (M, a, f ) and (M', a', f') are equivalent, or bordant, if there is a 4-tuple (W, V, v, g) such that W and V are compact differentiable manifolds with boundary, a V = AM U AM'and a W = M U M' U V/DM U AM' a V, v: (W, V) (W, V) is a differentiable (fixed-point free) involution restricting to , on M and ,t' on M', and g: (W, V) (X, A) is a continuous equivariant map [zg = gv] restricting to f on M and f ' on M'. The disjoint union of triples induces an operation on the set of (free) equivariant bordism classes of (X, A, z) making this set into an abelian group. This is a graded group, where the grading is given by the dimension of the manifold M, and one lets 9Z,(X, A, z) be the group of equivariant bordism classes of (X, A, z), and one lets 9Z*(X, A, z) be the group of free equivariant bordism classes of (X, A, 4). If A is empty, one writes W*(X, z) and 9M(X, z) for these groups. Letting X be a point, with A empty and z the identity map, this reduces to the situation studied by Conner and Floyd [4], with 9Z*(pt, 1) = I*(Z2) and 9Z*(pt, 1) = 9*(Z2) in their notation. The main purpose of this paper is to compute the groups 9Z*(X, z) and 9Z*(X, z) and to explore their interrelationships. As always, however, the study of an involution (X, z) is really the study of a pair (X, FT, z), with FT the fixed-point set of z, and one is forced to consider pairs in order to study the absolute case. To study more general pairs tends to force a study of 4tuples, and this additional complication will naturally be avoided. The major portion of this paper is a geometric analysis of bordism of in-

On Invariants with the Novikov Additive Property

Abstract: The present paper is the third in a series of notes ([6, 73) concerning this type of additivity. In [63 I proved, that any real valued invariant with this additive property coincides (up to a factor in each dimension, of course) with the signature on all closed manifolds. In Section 2 of the present note, we will generalize this result in three ways: (i) We admit the invariant to take values in any abelian group G (in [6] I made much use of the fact that IR has no elements of order two), (ii) we also consider the unoriented case, and (iii) we formulate the additive property in terms of closed manifolds only, we will not assume that the invariant is defined for bounded manifolds at all. in Section 3 we will prove a similar theorem for real valued invariants of the equivariant oriented diffeomorphism type of orientation preserving involutions. The proof is based on Section 2 and on [73.

Invariantie in de schattingstheorie

Abstract: Summary A number of methods of estimation have been proposed on relatively intuitive grounds. A “best” method does not exist. Sometimes the principle of invariance or, better termed, equivariance is very useful. By use of the group structure on hand it is possible to characterize equivariant estimators. Lezing gehouden op de Statistische Dag 1969.