EAN: 9783030983123

COMPACT TEXTBOOKS IN MATHEMATICS INTRODUCTION...

This textbook provides a succinct introduction to algebraic topology. It follows a modern
categorical approach from the beginning and gives ample motivation throughout so that students
will find this an ideal first encounter to the field. Topics are treated in a self-contained
manner  making this a convenient resource for instructors searching for a comprehensive
overview of the area. It begins with an outline of category theory  establishing the concepts
of functors  natural transformations  adjunction  limits  and colimits. As a first application
van Kampen's theorem is proven in the groupoid version. Following this  an excursion to
cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts
the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology
is then defined  motivating the Eilenberg-Steenrod axioms  and the simplicial approximation
theorem is proven. After verifying the axioms for singular homology  various versions of the
Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of
spheres are classified by degree.The final chapter discusses cellular homology of CW complexes
culminating in the uniqueness theorem for ordinary homology. Introduction to Algebraic Topology
is suitable for a single-semester graduate course on algebraic topology. It can also be used
for self-study  with numerous examples  exercises  and motivating remarks included.

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