The amount of algebraic topology a graduate student specializing in
topology must learn can be intimidating. Moreover, by their second year of
graduate studies, students must make the transition from understanding simple
proofs line-by-line to understanding the overall structure of proofs of
difficult theorems.
To help students make this transition, the material in this book is
presented in an increasingly sophisticated manner. It is intended to bridge the
gap
between algebraic and geometric topology, both by providing the algebraic tools
that a geometric topologist needs and by concentrating on those areas of
algebraic topology that are geometrically motivated.
Prerequisites for using this book include basic set-theoretic topology, the
definition of CW-complexes, some knowledge of the fundamental group/covering
space theory, and the construction of singular homology. Most of this material
is briefly reviewed at the beginning of the book.
The topics discussed by the authors include typical material for first- and
second-year graduate courses. The core of the exposition consists of chapters
on homotopy groups and on spectral sequences. There is also material that would
interest students of geometric topology (homology with local coefficients and
obstruction theory) and algebraic topology (spectra and generalized homology),
as well as preparation for more advanced topics such as algebraic
$K$-theory and the s-cobordism theorem.
A unique feature of the book is the inclusion, at the end of each chapter,
of several projects that require students to present proofs of substantial
theorems and to write notes accompanying their explanations. Working on these
projects allows students to grapple with the “big picture”, teaches
them how to give mathematical lectures, and prepares them for participating in
research seminars.
The book is designed as a textbook for graduate students studying algebraic
and geometric topology and homotopy theory. It will also be useful for students
from other fields such as differential geometry, algebraic geometry, and
homological algebra. The exposition in the text is clear; special cases are
presented over complex general statements.
Readership
Graduate students and research mathematicians interested in
geometric topology and homotopy theory.