Geometry & Topology Monographs

Volume 6: Fragments of geometric topology from the sixties

Sandro Buoncristiano


This book presents some of the main themes in the development of the combinatorial topology of high-dimensional manifolds, which took place roughly during the decade 1960-70 when new ideas and new techniques allowed the discipline to emerge from a long period of lethargy.
The first great results came at the beginning of the decade. I am referring here to the weak Poincaré conjecture and to the uniqueness of the PL and differentiable structures of Euclidean spaces, which follow from the work of J Stallings and EC Zeeman. Part I is devoted to these results, with the exception of the first two sections, which offer a historical picture of the salient questions which kept the topologists busy in those days. It should be noted that Smale proved a strong version of the Poincaré conjecture also near the beginning of the decade. Smale's proof (his h-cobordism theorem) will not be covered in this book.
The principal theme of the book is the problem of the existence and the uniqueness of triangulations of a topological manifold, which was solved by R Kirby and L Siebenmann towards the end of the decade.
This topic is treated using the `immersion theory machine' due to Haefliger and Poenaru. Using this machine the geometric problem is converted into a bundle lifting problem. The obstructions to lifting are identified and their calculation is carried out by a geometric method which is known as Handle-Straightening.
The treatment of the Kirby-Siebenmann theory occupies the second, the third and the fourth part, and requires the introduction of various other topics such as the theory of microbundles and their classifying spaces and the theory of immersions and submersions, both in the topological and PL contexts.
The fifth part deals with the problem of smoothing PL manifolds, and with related subjects including the group of diffeomorphisms of a differentiable manifold.
The sixth and last part is devoted to the bordism of pseudomanifolds a topic which is connected with the representation of homology classes according to Thom and Steenrod. For the main part it describes some of Sullivan's ideas on topological resolution of singularities.
The monograph is necessarily incomplete and fragmentary, for example the important topics of h-cobordism and surgery are only stated and for these the reader will have to consult the bibliography. However the book does aim to present a few of the wide variety of issues which made the decade 1960-70 one of the richest and most exciting periods in the history of manifold topology.

Sandro Buoncristiano
December 2003

AMS subject classification. 57NXX, 57QXX, 57RXX

Note: This book is being published in stages starting with parts I to III (December 2003). Other parts will be added as they are finalised.

Complete book to date (Frontmatter, Parts I-III, Bibliography)

Frontmatter (pages i-iv) preliminary version Part I : PL Topology (pages 1-21) Part II : Microbundles (pages 41-72) Part III : The differential (pages 77-107) Bibliography (preliminary version) Notes on file formats

Sandro Buoncristiano
Dipartimento di Matematica
Universita di Roma Tor Vegata
00133 Roma, Italy

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