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Basic algebraic topology and its applications. (English) Zbl 1354.55001

New Delhi: Springer (ISBN 978-81-322-2841-7/hbk; 978-81-322-2843-1/ebook). xxix, 615 p. (2016).
This is a comprehensive textbook on algebraic topology. It is designed to be accessible to students of all levels of mathematics, so suitable for anyone wanting and needing to learn about algebraic topology. It can also offer a valuable resource for advanced students with a specialized knowledge in other areas who want to pursue their interest in this area. This book covers all the basic material necessary for understanding the fundamentals of algebraic topology: homotopy, homology and cohomology theories. Each chapter and appendix begins with a brief historical background information of its central theme, which is useful in particular for beginner students. Moreover, further readings are provided at the end of each of them, which also enables students to study the subject discussed therein in more depth. The author tries here to give a detailed explanation of what problems have been studied in algebraic topology and how they have been solved. In particular, with respect to the latter point, the author attempts to convey the methods used there through doing exercises. The aim of algebraic topology is to describe topological phenomena of spaces with the use of invariants which are provided by homotopy, homology and cohomology groups. We can say more conclusively that its final goal consists in finding such invariants. In addition to the above three theories, this book includes \(K\)-theory. It is a research discipline independent of them. Its definition is simple, but it provides a useful tool for analysis of the topological phenomena of spaces referred to above. Incidentally, furthermore we have homological algebra. It is also a separate branch drawn similarly from studying the invariants mentioned above. It is closely related to the appearance of category theory which is referred to in an appendix.
This book consists of eighteen chapters and two appendices with a list of symbols and both author and subject indexes. Each of these chapters and appendices includes examples, exercises, and in addition to that, applications, except a few of them. Besides each of them provides lists of references and additional readings. Below in order to capture features of the book we list the contents of this book and give an enumeration of exercises by choosing a simple but effective one from the exercises provided in each chapter and appendix.
Contents. Prerequisite concepts and notations; Homotopy theory: elementary basic concepts; The fundamental groups; Covering spaces; Fiber bundles, vector bundles and \(K\)-theory; Geometry of simplicial complexes and fundamental groups of polyhedra; Higher homotopy groups; \(CW\)-complexes and homotopy; Products in homotopy theory; Homology and cohomology theories; Eilenberg-MacLane spaces; Eilenberg-Steenrod axioms for homology and cohomology theories; Consequences of the Eilenberg-Steenrod axioms; Applications; Spectral homology and cohomology theories; Obstruction theory; More relations between homology and homotopy; A brief history of algebraic topology; Topological groups and Lie groups; Categories, functors and natural transformations. (Here the last two headings are the titles of the appendices.)
Extract from Exercises. Show that the product of two compact spaces is compact; Show that a retract of a contractible space is contractible; Show that \(\pi_1({\mathbb R}^2-{\mathbb Q}^2)\) is uncountable; If \(B\) is an \(H\)-space, prove that every covering space of \(B\) is regular; Show that the tangent bundle \(T(S^n)\) to \(S^n\) is trivial only if \(n=1, 3\) or 7; Show that a group \(G\) is finitely presented iff there exists a polyhedron \(X\) such that \(G \cong \pi_1(X, x_0)\); Show that the 4-manifold \(S^2\times S^2\) is simply connected, but it is not homeomorphic to \(S^4\); If \(X\) is a \(CW\)-complex and \(A\) is a subcomplex of \(X\), show that the quotient space \(X/A\) is also a \(CW\)-complex; If \(X\) is an \(H\)-space, show that all Whitehead products are trivial on \(X\); Calculate the cohomology groups of the Klein bottle; Show that for \(n > 1\) the spaces \(\Omega K(G, n)\) and \(K(G, n-1)\) are homotopy equivalent; If \(X\) is a space consisting of a single point, show that \(H^0(X)=G_X\) and \(\tilde{H}^0(X)=0\); Show that for \(n\neq m\), the spheres \(S^n\) and \(S^m\) cannot be homeomorphic; Show that every nullhomotopic map \(f : S^n\to S^n\) has at least one fixed point; Show that the sphere spectrum \(\underline{S}\) is a ring spectrum and every spectrum \(\underline{E}\) is a module over \(\underline{S}\); Show that the extension index of a continuous map \(f : A\to Y\) is a topological invariant; Show that two continuous maps from \(S^3\) to \(S^2\) are homotopic iff they have the same Hopf invariant; Show that the 3-dimensional projective space \({\mathbb R}P^3\) and \(SO(3, {\mathbb R})\) are homeomorphic; If \(\alpha : A\to B\) is a retraction and also a monomorphism in a category \({\mathcal C}\), prove that \(\alpha\) is an isomorphism.

MSC:

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