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Homotopical topology. Translated from the Russian. 2nd edition. (English) Zbl 1346.55001

Graduate Texts in Mathematics 273. Cham: Springer (ISBN 978-3-319-23487-8/hbk; 978-3-319-23488-5/ebook). xi, 627 p. (2016).
Reading the synopsis and preface of this book, we can see that this second edition is a reissue of the classical text of the Moscow State University after a gap of 45 years with some alterations to chapters. This book covers all the basic material necessary for complete understanding of the fundamentals of algebraic topology: covering spaces and the fundamental group, homotopy groups, homology and cohomology; in addition to these topics, obstruction theory, spectral sequences, Steenrod squares, \(K\)-functor and cobordisms. Perhaps the most major update to the first edition is an addition of a new chapter on these last two topics in which the relationship between these two theories and homotopy theory originally given becomes naturally enough discussed. This increase in the number of topics has made the book more convenient for serious students not only to extend their knowledge but also to gain insight into the interplay between these three subjects.
This book also includes the explicit calculation of stable homotopy groups of spheres, which is the most important subject in algebraic topology. In fact we come across it in the next to last chapter and arrive finally at the table of the 2-components of the first 13 stable homotopy groups of spheres. The authors show practically how they can be computed explicitly using the Adams spectral sequence to which the prefix \(\lq\lq\)classical” is nowadays attached. The above computation itself may possibly go beyond the common basics, but this book focuses on understanding the underlying idea of this spectral sequence through its computation. It has been shown that the Serre spectral sequence for homology provides a systematic technique for computation of homotopy groups via the Hurewicz theorem. The authors present an example of the computations of the homotopy groups of a space \(X\) in two different ways using these two spectral sequences such that in a certain situation, the Adams method allows us to proceed to the next step, but the Serre method does not allow us to do so. In this respect this implies that the former is more powerful than the latter. The difference between these two methods occurs in the process of successively killing the cohomology groups of \(X\). Even so the above computation has come to a standstill after 13 steps because we are faced with a difficulty in determining its differentials; indeed the authors remark that nontrivial differentials appear in the Adams spectral sequence. The newly added chapters on \(K\)-theory and cobordism provide key tools in algebraic topology which lead to an improvement of the techniques of computing the homotopy groups of \(X\) above after further development in the future.
This book is designed to help students to select the level of learning subjects they want to reach; however it is certain that it will have the power of bringing students to the forefront of their branch of research. It consists of an introduction and 6 chapters with captions accompanying illustrations and name and subject indexes, which are also divided into 44 lectures. Besides it contains a large number of exercises (about 500 according to the preface) arranged appropriately. Below we make a list of some of them in order to extract features of the book by selecting from the exercises presented in the introduction and each of the chapters two or three which, however, are as simple as possible in the description.
Examples of Exercises: Prove that projective lines \(\mathbb{C}P^1\) and \(\mathbb{H}P^1\) are homeomorphic, respectively, to \(S^2\) and \(S^4\). Show that the group \(\mathrm{SO}(4)\) is homeomorphic to \(S^3\times \mathrm{SO}(3)\), that is, to \(S^3\times\mathbb{R}P^3\). Prove that for any nonempty space \(X\), \(\mathrm{cat } \Sigma X \leq 2\). Prove that if the fiber of the Serre fibration \(E\), \(B\), \(p\) is contractible in \(E\), then \(\pi_n(B)\cong \pi_n(E)\oplus \pi_{n-1}(F)\). Prove that \(\Omega K(\pi, n)=K(\pi, n-1)\). Find the Euler characteristics of classical surfaces. Find Stiefel-Whitney numbers of classical surfaces. Prove that stably equivalent bundles have equal Ponryagin classes. Compute \(\pi_{n+3}(S^n)\). Prove the equality \(T(\xi\times\eta)=T(\xi)\sharp T(\eta)\). Prove that if \(c\) is a cocycle of the class \(\gamma \in H^r(X; \mathbb{Z}_2)\), then \(D_q(c\times c)\) is a cocycle of the class \(Sq^{r-q}\gamma\). If the algebra \(A\) is commutative, then \(\mathrm{Tor}^A_n(M, N) \cong \mathrm{Tor}^A_n(N, M)\). If the module \(M\) is projective, then \(\mathrm{Ext}^n_A(M, N)=0\) for any \(N\) and any \(n >0\). Prove that in the Adams spectral sequence \(d_2y_2=z\). Prove that the order of the torsion in the groups \(K^0(X)\), \(K^1(X)\) divides the order of the torsion in the groups \(H^{\mathrm{even}}(X; \mathbb{Z})\), \(H^{\mathrm{odd}}(X; \mathbb{Z})\), respectively. Show that \(d_3 : H^n(X, \mathbb{Z}) \to H^{n+3}(X; \mathbb{Z})\) is a stable cohomology operation. Prove that \(e : \pi_{2N+2n-1}(S^{2N}) \to \mathbb{Q}/\mathbb{Z}\) is a homomorphism. Describe the composition map \(\Omega^\ast_U \rightarrow{\mu} K \rightarrow\mathrm{ch } H^\ast(-; \mathbb{Q})\).
Finally, we refer again to the stable homotopy groups \(\pi^S_r(S^0)\) \((r\leq 13)\) of spheres. The result of their computation is listed at the end of Section 36.3 (Chapter 5, Lecture 36); however, the computation of the 3-components is left to interested students as an exercise, which provides an example of computation where using the Adams spectral sequence is crucial.

MSC:

55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55Nxx Homology and cohomology theories in algebraic topology
55Pxx Homotopy theory
55Sxx Operations and obstructions in algebraic topology
55Txx Spectral sequences in algebraic topology
19Lxx Topological \(K\)-theory
55Rxx Fiber spaces and bundles in algebraic topology
57Rxx Differential topology
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