C. T. C. Wall’s contributions to the topology of manifolds. (English) Zbl 0940.57001

Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan

Cappell, Sylvain (ed.) et al., Surveys on surgery theory. Vol. 1: Papers dedicated to C. T. C. Wall on the occasion to his 60th birthday. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 145, 3-15 (2000).

From the paper: C. T. C. Wall spent the first half of his career, roughly from 1959 to 1977, working in topology and related areas of algebra. In this period, he produced more than 90 research papers and two books, covering cobordism groups, the Steenrod algebra, homological algebra, manifolds of dimensions 3, 4, \(\geq 5\), quadratic forms, finiteness obstructions, embeddings, bundles, Poincaré complexes, surgery obstruction theory, homology of groups, 2-dimensional complexes, the topological space form problem, computations of \(K\)- and \(L\)-groups, and more.
One quick measure of Wall’s influence is that there are two headings in the Mathematics Subject Classification that bear his name: 57Q12 (Wall finiteness obstruction for CW complexes); 57R67 (Surgery obstructions, Wall groups).
Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this survey is to give an overview of how his work has advanced our understanding of classification methods. Wall’s approaches to manifold theory may conveniently be divided into three phases, according to the scheme:
1. All manifolds at once, up to once, up to cobordism (1959-1961).
2. One manifold at a time, up to diffeomorphism (1962-1966).
3. All manifolds within a homotopy type (1967-1977).
For the entire collection see [Zbl 0933.00057].

MSC:

57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
01A70	Biographies, obituaries, personalia, bibliographies
57-03	History of manifolds and cell complexes
55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology