Algebraic Topology - Allen Hatcher

This book was written to be a readable introduction to Algebraic Topology with
rather broad coverage of the subject. Our viewpoint is quite classical in spirit, and
stays largely within the confines of pure Algebraic Topology. In a sense, the book
could have been written thirty years ago since virtually all its content is at least that
old. However, the passage of the intervening years has helped clarify what the most
important results and techniques are. For example, CW complexes have proved over
time to be the most natural class of spaces for Algebraic Topology, so they are emphasized
here much more than in the books of an earlier generation. This emphasis
also illustrates the book’s general slant towards geometric, rather than algebraic, aspects
of the subject. The geometry of Algebraic Topology is so pretty, it would seem
a pity to slight it and to miss all the intuition that it provides. At deeper levels, algebra
becomes increasingly important, so for the sake of balance it seems only fair to
emphasize geometry at the beginning.
Let us say something about the organization of the book. At the elementary level,
Algebraic Topology divides naturally into two channels, with the broad topic of Homotopy
on the one side and Homology on the other. We have divided this material
into four chapters, roughly according to increasing sophistication, with Homotopy
split between Chapters 1 and 4, and Homology and its mirror variant Cohomology
in Chapters 2 and 3. These four chapters do not have to be read in this order, however.
One could begin with Homology and perhaps continue on with Cohomology
before turning to Homotopy. In the other direction, one could postpone Homology
and Cohomology until after parts of Chapter 4. However, we have not pushed this
latter approach to its natural limit, in which Homology and Cohomology arise just as
branches of Homotopy Theory. Appealing as this approach is from a strictly logical
point of view, it places more demands on the reader, and since readability is one of
our first priorities, we have delayed introducing this unifying viewpoint until later in
the book.
There is also a preliminary Chapter 0 introducing some of the basic geometric
concepts and constructions that play a central role in both the homological and homotopical
sides of the subject.
Each of the four main chapters concludes with a selection of Additional Topics
that the reader can sample at will, independent of the basic core of the book contained
in the earlier parts of the chapters. Many of these extra topics are in fact rather
important in the overall scheme of Algebraic Topology, though they might not fit into
the time constraints of a first course. Altogether, these Additional Topics amount
to nearly half the book, and we have included them both to make the book more
comprehensive and to give the reader who takes the time to delve into them a more
substantial sample of the true richness and beauty of the subject.
Not included in this book is the important but somewhat more sophisticated
topic of spectral sequences. It was very tempting to include something about this
marvelous tool here, but spectral sequences are such a big topic that it seemed best
to start with them afresh in a new volume. This is tentatively titled ‘Spectral Sequences
in Algebraic Topology’ and referred to herein as [SSAT]. There is also a third book in
progress, on vector bundles, characteristic classes, and K–theory, which will be largely
independent of [SSAT] and also of much of the present book. This is referred to as
[VBKT], its provisional title being ‘Vector Bundles and K–Theory.’
In terms of prerequisites, the present book assumes the reader has some familiarity
with the content of the standard undergraduate courses in algebra and point-set
topology. One topic that is not always a part of a first point-set topology course but
which is quite important for Algebraic Topology is quotient spaces, or identification
spaces as they are sometimes called. Good sources for this are the textbooks by Armstrong
and J¨anich listed in the Bibliography.
A book such as this one, whose aim is to present classical material from a fairly
classical viewpoint, is not the place to indulge in wild innovation. Nevertheless there is
one new feature of the exposition that may be worth commenting upon, even though
in the book as a whole it plays a relatively minor role. This is a modest extension
of the classical notion of simplicial complexes, which we call Ñ complexes. These
have made brief appearances in the literature previously, without a standard name
emerging. The idea is to weaken the condition that each simplex be embedded, to
require only that the interiors of simplices are embedded. (In addition, an ordering
of the vertices of each simplex is also part of the structure of a Ñ complex.) For
example, if one takes the standard picture of the torus as a square with opposite
edges identified and divides the square into two triangles by cutting along a diagonal,
then the result is a Ñ complex structure on the torus having 2 triangles, 3 edges, and
1 vertex. By contrast, it is known that a simplicial complex structure on the torus
must have at least 14 triangles, 21 edges, and 7 vertices. So Ñ complexes provide
a significant improvement in efficiency, which is nice from a pedagogical viewpoint
since it cuts down on tedious calculations in examples. A more fundamental reason
for considering Ñ complexes is that they just seem to be very natural objects from
the viewpoint of Algebraic Topology. They are the natural domain of definition for
simplicial homology, and a number of standard constructions produce Ñ complexes
rather than simplicial complexes, for instance the singular complex of a space, or the
classifying space of a discrete group or category.

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