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This is a review, written by Don Davis for Math Reviews,
of the book, {\it Algebraic Topology from a Homotopical Viewpoint},
written by Marcelo Aguilar, Samuel Gitler, and Carlos Prieto,
published by Springer-Verlag Universitext series, in 2000.
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This carefully written text presents a new approach to a first
course in algebraic topology. The main novelty is defining the
homology groups of a (pointed path-connected) CW complex as the
homotopy groups of its infinite symmetric product SP $X$.
Cohomology groups are defined as homotopy classes of maps into
Eilenberg-MacLane spaces, which are defined as infinite symmetric
products of Moore spaces.
This differs dramatically from most standard texts by making no
mention of singular or simplicial homology. Cellular homology
is discussed, in order to allow calculations and to approach the
usual Kunneth and Universal Coefficient theorems.
It is somewhat closer to the 1975 texts of Gray and Switzer, but
very different from them, as well. Both of them introduced homology
and cohomology using spectra, and defined Eilenberg-MacLane spaces
by attaching cells to kill unwanted homotopy classes.
The Dold-Thom theorem that underlies the symmetric product approach
to homology states that SP $X\to$ SP$(X/A)$ is a quasifibration.
A thorough proof of this is given in an appendix, apparently the
first proof in English of this fundamental result. By the way,
this book is a translated and expanded version of the authors' original
Spanish text,
but shows no signs of being a translation. The English is excellent,
and the misprints are few and minor.
The first third of the book is fairly standard, dealing with the
fundamental group, homotopy groups, cofibrations and fibrations,
and covering spaces, in this order. There is a thorough discussion
of the duality between cofibrations and fibrations. Covering spaces
are treated as a special kind of fibration, and their relationship
with $\pi_1(-)$ is not emphasized as much as in most books. Much
of the theory of covering spaces is relegated to exercises.
Throughout, exercises are embedded in the text; there are no sets
of exercises at the end of chapters. The purpose of exercises in
this book seems to be primarily for exposition of material.
One nice topic here, not found in most texts, is a classifying
space for $n$-fold covering spaces, namely $F_n({\bold R}^\infty)
/\Sigma_n$. This is illustrative of the sophisticated level of
this text. In comparison, the associative property of $\pi_1(-)$ is
established by just a formula for the homotopy between $(\omega_1
\omega_2)\omega_3$ and $\omega_1(\omega_2\omega_3)$, without the
usual picture as an aid.
The novel approach to homology used in this book can be
disconcerting to one who is used to the standard treatment.
One must wait a long time (or know to skip over a lot
of material) before getting to the homology groups of familiar
spaces such as surfaces and real projective spaces.
Nowhere does this book mention that the first homology group
is the abelianization of the fundamental group.
No explanation is made for why the ubiquitous sign
in the anticommutativity of the cup product is present.
Another big omission compared to standard texts
is the treatment of Kunneth and Universal Coefficient
theorems. Here the reader is directed to other texts. Statements
of results are given, but Tor and Ext are not even defined.
This book never even tells what a chain complex is or how its
homology groups are defined. The reader is referred to a text
such as MacLane's for such essentials.
After the chapters on homology and cohomology are four thorough
chapters on vector bundles, K-theory, Adams operations, and
characteristic classes. A very nice and thorough proof of the
Adams-Atiyah proof of nonexistence of division algebras is given.
In these chapters, the text again expects a lot of the reader.
In the first section (9 pages), where a fairly standard definition
of vector bundle is given, there are no examples. In the next
section, it is shown how a map from $X$ into the space Pr$(V)$
of all projections (idempotent endomorphisms) of a vector space
$V$ gives rise to a vector bundle. Some of the standard examples
are then given from this sophisticated perspective.
The final chapter deals with spectra and the Brown Representability
Theorem. It treats the distinction between prespectra and spectra
as in May's \lq\lq Concise'' 1999 text.
In addition to the appendix on the Dold-Thom theorem, there is one
with a thorough topological proof of the complex Bott Periodicity Theorem,
following a 1999 paper of the first and third authors. This
utilizes the quasifibration theme of the book.
In conclusion, this text has developed a truly new approach to
introductory algebraic topology. The introduction explains that
this approach is in line with the recent work of Voevodsky.
But a reader with no background in homological algebra will
need to refer to a supplementary text.
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