×

Simplicial structures in topology. (English) Zbl 1211.55001

CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York, NY: Springer (ISBN 978-1-4419-7235-4/hbk; 978-1-4614-2698-1/pbk; 978-1-4419-7236-1/ebook). xvi, 243 p. (2011).
This book focuses on the role of finite simplicial structures and the algebraic topology deriving from them. The book consists of six chapters. The first one contains fundamental concepts in topology, group actions and category theory, needed for developing the remainder of the book. This chapter could be a basic text for a mini-course on general topology. In the second chapter the authors present the category of simplicial complexes Csim, the geometric realization functor from Csim to the category of topological spaces and the homology functor from Csim to the category of graded abelian groups. This chapter also contains an introduction to homological algebra, results on simplicial homology and reduced homology.
The homology of polyhedra is defined in Chapter III, with more geometric results: the simplicial approximation theorem, the long exact sequence theorem. This chapter contains interesting applications of the homology of polyhedra: the Lefschetz fixed point theorem, the Brouwer fixed point theorem, the fundamental theorem of algebra. There is a paragraph dedicated to the homology of real projective spaces: block homology, homology of \(\mathbb RP_n\) for \(n\) greater or equal to 4. The last part of this chapter is about the homology of the product of two polyhedra and the acyclic models theorem.
In the fourth chapter the cohomology of polyhedra is studied. The cohomology groups of a polyhedron (which is an invariant stronger than the homology since the cohomology with coefficients in a commutative ring with identity element is also a ring) are related to its homology groups by the universal coefficient theorem. Chapter V has three sections: manifolds, closed surfaces and Poincaré duality (the Poincaré duality theorem for connected, triangulable and orientable \(n\)-manifolds is proved). The last chapter, homotopy groups, focuses on fundamental groups of polyhedra, polyhedra with a given fundamental group, action of the fundamental group on the higher homotopy groups, homotopy groups of spheres, and obstruction theory.
This book on algebraic topology is interesting and can be a basis for a course on homological algebra or on homotopy theory or for a course on cellular structures in topology. It is a clear and comprehensive introduction to simplicial structures in topology with illustrative examples throughout and extensive exercises at the end of each chapter. This book is also a very good text for advanced undergraduate and beginning graduate students, researchers and professionals interested in topology and applications in mathematics.

MSC:

55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI