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
*

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)
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Sandro Buoncristiano

Dipartimento di Matematica

Universita di Roma Tor Vegata

00133 Roma, Italy

Email: buoncris@mat.uniroma2.it

GTM home page

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