John Milnor

On the Structure of Hopf Algebras

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras and some surrounding topics. New proofs of the classical theorems are given, as well as some new results. The paper is divided into eight sections with the following titles: 1. Algebras and modules. 2. Coalgebras and comodules. 3. Algebras, coalgebras, and duality. 4. Elementary properties of Hopf algebras. 5. Universal algebras of Lie algebras. 6. Lie algebras and restricted Lie algebras. 7. Some classical theorems. 8. Morphisms of connected coalgebras into connected algebras. The first four sections are introductory in nature. Section 5 shows that, over a field of characteristic zero, the category of graded Lie algebras is isomorphic with the category of primitively generated Hopf algebras. In ? 6, a similar result is obtained in the case of characteristic p # 0, but with graded Lie algebras replaced by graded restricted Lie algebras. Section 7 studies conditions when a Hopf algebra with commutative multiplication splits either as a tensor product of algebras with a single generator or a tensor product of

Groups of Homotopy Spheres: I

DEFINITION. Two closed n-manifolds M, and M2 are h-cobordant1 if the disjoint sum M, + (- M2) is the boundary of some manifold W, where both M1 and (-M2) are deformation retracts of W. It is clear that this is an equivalence relation. The connected sum of two connected n-manifolds is obtained by removing a small n-cell from each, and then pasting together the resulting boundaries. Details will be given in ? 2.

Curvatures of left invariant metrics on lie groups

This article outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a Riemannian metric invariant under left translation.

On the concept of attractor

This note proposes a definition for the concept of “attractor,” based on the probable asymptotic behavior of orbits. The definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor.

AN AXIOMATIC APPROACH TO MEASURABLE UTILITY

Abstract : Previous treatments of this approach brought topological considerations of the prospects space into the axioms. In this paper considerations of the topology of the prospect space itself are removed, the previous axioms are weakened an infinite number of sure prospects are allowed. On the basis of these axioms the existence of a measurable utility is established.

Construction of Universal Bundles, II

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THE STEENROD ALGEBRA AND ITS DUAL1

1. Summary Let 57 * denote the Steenrod algebra corrresponding to an odd prime p. (See ? 2 for definitions.) Our basic results (? 3) is that 5i* is a Hopf

On spaces having the homotopy type of a CW-complex

have become important in homotopy theory, and our basic objective is to show that such constructions do not lead outside the class SW (Theorem 3). The first section is concerned with the smaller class Wo, consisting of all spaces which have the homotopy type of a countable CW-complex. ??1 and 2 are independent of each other. The basic reference for this paper is J. H. C. Whitehead [15]. 1. The class Wvo.

Dynamical properties of plane polynomial automorphisms

This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

Geometry and Dynamics of Quadratic Rational Maps

This article is an expository description of quadratic rational maps from the Riemann sphere to itself