Ioan Mackenzie James

On equivariant homotopy type

LET G BE a topological group. By a G-ANR we mean a separable and metrizable G-space X with the following property: whenever B is a normal G-space and A is an invariant closed subspace of B. any G-map A +X can be extended over an invariant neighbourhood of A in B. When G is a compact Lie group it is known (see (1.4) and (1.6) of [Sl) that any smooth compact manifold on which G acts smoothly is a G-A NR. The fixed point set of a G-space X with respect to a subgroup H of G is denoted by XH. For any space A. the maps A *X” correspond precisely to the G-maps G/H x A + X. It follows, in particular, that if X is a G-ANR then XH is an ANR for all H C G. When X and Y are G-spaces a G-map f: X + Y determines a map f”: X” + Y H of the fixed point sets for all subgroups H C G. If f is a G-homotopy equivalence then f” is a homotopy equivalence for all H. In Ch. II of [l] Bredon gives a converse of this in the case when G is finite, subject to certain restrictions on X and Y (cf. also Matumoto[5]). The purpose of the present note is to establish

Topological and uniform spaces

This book is based on lectures I have given to undergraduate and graduate audiences at Oxford and elsewhere over the years. My aim has been to provide an outline of both the topological theory and the uniform theory, with an emphasis on the relation between the two. Although I hope that the prospec- tive specialist may find it useful as an introduction it is the non-specialist I have had more in mind in selecting the contents. Thus I have tended to avoid the ingenious examples and counterexamples which often occupy much ofthe space in books on general topology, and I have tried to keep the number of definitions down to the essential minimum. There are no particular pre- requisites but I have worked on the assumption that a potential reader will already have had some experience of working with sets and functions and will also be familiar with the basic concepts of algebra and analysis. There are a number of fine books on general topology, some of which I have listed in the Select Bibliography at the end of this volume. Of course I have benefited greatly from this previous work in writing my own account.Undoubtedly the strongest influence is that of Bourbakis Topologie Generale [2], the definitive treatment of the subject which first appeared over a genera- tion ago.

Homotopy-abelian Lie groups

where h* denotes the induced homomorphism. Note that #* is an isomorphism if A is a covering map and p, a ^ 2 . Hence if two topological groups have a common universal covering group then their higher homotopy groups are related by an isomorphism which is compatible with the Samelson product. Let <nrq(G), where g ^ l , denote the subset of w2q(G) consisting of elements ((3, fi), where fiCz7Tq(G). We assert the following

Autism in mathematicians

The cause of autism is mysterious, but genetic factors are important. It takes a variety of forms; the expression autism spectrum, which is often used, gives a false impression that it is just the severity of the disorder that varies. Different people are affected in different ways, but the core problems are impairments of communication, social.

An Introduction to Fibrewise Homotopy Theory

Let us work over a (topological) base space B. A fibrewise space over B consists of a space X together with a map p: X → B, called the projection. Usually X alone is sufficient notation. We regard any subspace of X as a fibrewise space over B by restricting the projection. When p is a fibration we describe X as fibrant.

Aspects of topology : in memory of Hugh Dowker 1912-1982

1. Obituary: Clifford Hugh Dowker Dona Strauss 2. Knot tabulations and related topics Morwen B. Thistlethwaite 3. How general is a generalized space? Peter T. Johnstone 3. A survey of metrization theory J. Nagata 4. Some thoughts on lattice valued functions and relations M. W. Warner 5. General topology over a base I. M. James 6. K-Dowker spaces M. E. Rudin 7. Graduation and dimension in locales J. Isbell 8. A geometrical approach to degree theory and the Leray-Schauder index J. Dugundji 9. On dimension theory B. A. Pasynkov 10. An equivariant theory of retracts Sergey Antonian 11. P-embedding, LCn spaces and the homotopy extension property Kiiti Morita 12. Special group automorphisms and special self-homotopy equivalences Peter Hilton 13. Rational homotopy and torus actions Stephen Halperin 14. Remarks on stars and independent sets P. Erdos and J. Pach 15. Compact and compact Hausdorff A. H. Stone 16. T1 - and T2 axioms for frames C. H. Dowker and D. Strauss.

La mente matematica

Negli ultimi anni sono stati pubblicati numerosi libri divulgativi sulla matematica. I migliori tra questi hanno aiutato il grande pubblico a capire come sono fatti i matematici e cosa fanno di professione. Ogni scienza ha la sua cultura e quella della matematica e abbastanza sui generis. Come ha detto Henri Poincare: La matematica e quell’attivita in cui la mente umana sembra assorbire di meno dal mondo esterno, in cui agisce o sembra agire da e per se stessa, cosicche studiando il pensiero geometrico possiamo sperare di raggiungere cio che e essenziale alla mente dell’uomo”.

Fixed-point Methods

In this somewhat technical section we look at the theory of fibrewise ENRs and ANRs. The results are mostly due to Dold [47]. Our restriction to base spaces which are ENRs allows us to simplify the exposition at several points. We begin with a discussion of some of the properties of ENRs and ANRs which we have already used in earlier sections.

The Pointed Theory

In this chapter we work over a pointed base space B. A fibrewise pointed space over B consists of a space X together with maps