Algebraic topology from a homotopical viewpoint
The main novelty in this textbook on algebraic topology is the introduction of ordinary (co)homology theories via homotopical methods based on quasifibrations and infinite symmetric products. (The traditional approach, based on tools from homological algebra, is also presented, as a way of performing concrete computations, via cellular chain complexes.) The book is divided into twelve chapters, followed by two appendices. In Chapters 1-4, the authors introduce the basic concepts of homotopy theory, such as homotopy groups, fibrations and cofibrations. In the key Chapter 5, one finds the fundamental Dold-Thom theorem [\textit{A. Dold} and \textit{R. Thom}, Ann. Math. (2) 67, 239-281 (1958; Zbl 0091.37102)], whose proof is presented in Appendix A. The theorem guarantees that, under mild hypotheses, \(SP(X)\to SP(X/A)\) is a quasifibration, with fiber \(SP(A)\), where \(SP\) denotes the infinite symmetric product functor. Defining \(\widetilde H_n(X)\) to be \(\pi_nSP(X)\), the homotopy sequence of a quasifibration may then be used to check the Eilenberg-Steenrod axioms for \(\mathbb{Z}\)-homology. In Chapters 6 and 7, Moore and Eilenberg-MacLane spaces are constructed, and the homotopical definition of ordinary (co-)homology theories is completed. The multiplicative structure in cohomology is also introduced, via the Blakers-Massey homotopy excision theorem. Chapters 8-11 are devoted to topological \(K\)-theory. (A new proof of of the complex Bott periodicity theorem, based on quasifibrations, is presented in Appendix B.) Among the applications, the solution of the Hopf invariant problem using Adams operations is included. The final Chapter 12 contains a brief introduction to stable homotopy theory.
scientific article; zbMATH DE number 1776368