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Exploring the full scope of differential topology, this
comprehensive account of geometric techniques for studying the
topology of smooth manifolds offers a wide perspective on the
field. Building up from first principles, concepts of manifolds are
introduced, supplemented by thorough appendices giving background
on topology and homotopy theory. Deep results are then developed
from these foundations through in-depth treatments of the notions
of general position and transversality, proper actions of Lie
groups, handles (up to the h-cobordism theorem), immersions and
embeddings, concluding with the surgery procedure and cobordism
theory. Fully illustrated and rigorous in its approach, little
prior knowledge is assumed, and yet growing complexity is instilled
throughout. This structure gives advanced students and researchers
an accessible route into the wide-ranging field of differential
topology.
The biennial meetings at Sao Carlos have helped create a worldwide
community of experts and young researchers working on singularity
theory, with a special focus on applications to a wide variety of
topics in both pure and applied mathematics. The tenth meeting,
celebrating the 60th birthdays of Terence Gaffney and Maria
Aparecida Soares Ruas, was a special occasion attracting the best
known names in the area. This volume contains contributions by the
attendees, including three articles written or co-authored by
Gaffney himself, and survey articles on the existence of Milnor
fibrations, global classifications and graphs, pairs of foliations
on surfaces, and Gaffney's work on equisingularity.
Even the simplest singularities of planar curves, e.g. where the
curve crosses itself, or where it forms a cusp, are best understood
in terms of complex numbers. The full treatment uses techniques
from algebra, algebraic geometry, complex analysis and topology and
makes an attractive chapter of mathematics, which can be used as an
introduction to any of these topics, or to singularity theory in
higher dimensions. This book is designed as an introduction for
graduate students and draws on the author's experience of teaching
MSc courses; moreover, by synthesising different perspectives, he
gives a novel view of the subject, and a number of new results.
Singularities arise naturally in a huge number of different areas
of mathematics and science. As a consequence, singularity theory
lies at the crossroads of paths that connect many of the most
important areas of applications of mathematics with some of its
most abstract regions. The main goal in most problems of
singularity theory is to understand the dependence of some objects
of analysis, geometry, physics, or other science (functions,
varieties, mappings, vector or tensor fields, differential
equations, models, etc.) on parameters. The articles collected here
can be grouped under three headings. (A) Singularities of real
maps; (B) Singular complex variables; and (C) Singularities of
homomorphic maps.
Singularities arise naturally in a huge number of different areas
of mathematics and science. As a consequence, singularity theory
lies at the crossroads of paths that connect many of the most
important areas of applications of mathematics with some of its
most abstract regions. The main goal in most problems of
singularity theory is to understand the dependence of some objects
of analysis, geometry, physics, or other science (functions,
varieties, mappings, vector or tensor fields, differential
equations, models, etc.) on parameters. The articles collected here
can be grouped under three headings. (A) Singularities of real
maps; (B) Singular complex variables; and (C) Singularities of
homomorphic maps.
In 1977 several eminent mathematicians were invited to Durham to
present papers at a short conference on homological and
combinatorial techniques in group theory. The lectures, published
here, aimed at presenting in a unified way new developments in the
area. Group theory is approached from a geometrical viewpoint and
much of the material has not previously been published. The various
ways in which topological ideas can be used in group theory are
also brought together. The volume concludes with an extensive set
of problems, ranging from explicit questions demanding detailed
calculation to fundamental questions motivating research in the
area. These lectures will be of interest mainly to researchers in
pure mathematics but will also prove useful in connection with
relevant postgraduate courses.
Even the simplest singularities of planar curves, e.g. where the
curve crosses itself, or where it forms a cusp, are best understood
in terms of complex numbers. The full treatment uses techniques
from algebra, algebraic geometry, complex analysis and topology and
makes an attractive chapter of mathematics, which can be used as an
introduction to any of these topics, or to singularity theory in
higher dimensions. This book is designed as an introduction for
graduate students and draws on the author's experience of teaching
MSc courses; moreover, by synthesising different perspectives, he
gives a novel view of the subject, and a number of new results.
The publication of this book in 1970 marked the culmination of a
particularly exciting period in the history of the topology of
manifolds. The world of high-dimensional manifolds had been opened
up to the classification methods of algebraic topology by Thom's
work in 1952 on transversality and cobordism, the signature theorem
of Hirzebruch in 1954, and by the discovery of exotic spheres by
Milnor in 1956. In the 1960s, there had been an explosive growth of
interest in the surgery method of understanding the homotopy types
of manifolds (initially in the differentiable category), including
results such as the $h$-cobordism theory of Smale (1960), the
classification of exotic spheres by Kervaire and Milnor (1962),
Browder's converse to the Hirzebruch signature theorem for the
existence of a manifold in a simply connected homotopy type
(1962).It also includes the $s$-cobordism theorem of Barden, Mazur,
and Stallings (1964), Novikov's proof of the topological invariance
of the rational Pontrjagin classes of differentiable manifolds
(1965), the fibering theorems of Browder and Levine (1966) and
Farrell (1967), Sullivan's exact sequence for the set of manifold
structures within a simply connected homotopy type (1966), Casson
and Sullivan's disproof of the Hauptvermutung for piecewise linear
manifolds (1967), Wall's classification of homotopy tori (1969),
and Kirby and Siebenmann's classification theory of topological
manifolds (1970). The original edition of the book fulfilled five
purposes by providing: a coherent framework for relating the
homotopy theory of manifolds to the algebraic theory of quadratic
forms, unifying many of the previous results; a surgery obstruction
theory for manifolds with arbitrary fundamental group, including
the exact sequence for the set of manifold structures within a
homotopy type, and many computations; the extension of surgery
theory from the differentiable and piecewise linear categories to
the topological category; a survey of most of the activity in
surgery up to 1970; and, a setting for the subsequent development
and applications of the surgery classification of manifolds. This
new edition of this classic book is supplemented by notes on
subsequent developments. References have been updated and numerous
commentaries have been added. The volume remains the single most
important book on surgery theory.
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