Tweet Mathematics Books For Simplicial Structures in Topology

By: Davide L. Ferrario, Renzo A. Piccinini
Publisher: Springer, Pages: 243, Date: 2010
ISBN-10 / ASIN: 1441972358

Foreword to the English Edition
Except for a few added comments, this is a faithful translation of the book Strutture simpliciali in topologia, published by Pitagora Editrice, Bologna 2009, as part of the collection Quaderni of the Italian Mathematical Union. It should be noted that this book is neither a comprehensive text in algebraic topology nor is it a monograph on simplicial objects in the modern sense. Its focus is instead on the role of finite simplicial structures, and the algebraic topology deriving from them.

Contents
This book consists of six chapters. In the first one, we present the basic concepts in Topology, Group Actions and Category Theory needed for developing the remainder of the book. The part concerning Category Theory is especially important since this book has been set in categorial terminology. This chapter could also serve as a basic text for a mini-course on General Topology.

In the second chapter, we study the category of simplicial complexes Csim, and two important covariant functors: 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

The homology of polyhedra is defined in Chap. III which is, therefore, more geometric in nature than the previous one. The Simplicial Approximation Theorem provides the means by which one can pass from a simplicial approach to a geometric one. In practice, this theorem states that every map between two polyhedra may be “approached” by the geometric realization of a simplicial function between two triangulations of the polyhedra (reminding that any continuous curve on a plane can be approached by a polygonal line). The Long Exact Sequence Theorem appears also in this chapter/

This chapter also contains an important application of the homology of polyhedra to the Theory of Fixed Points, namely, the Lefschetz Fixed Point Theorem, as well as several corollaries such as the Brouwer Fixed Point Theorem and the Fundamental Theorem of Algebra.

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