Introduction to Homotopy Theory

1. Auflage 2011
ISBN: 978-1-4419-7328-3
Verlag: Springer

Buch, Englisch, 344 Seiten, Format (B × H): 154 mm x 237 mm, Gewicht: 522 g

Reihe: Universitext

This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory.

The underlying theme of the entire book is the Eckmann-Hilton duality theory. It is assumed that the reader has had some exposure to the rudiments of homology theory and fundamental group theory. These topics are discussed in the appendices. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.

Zielgruppe

Graduate

Autoren/Hrsg.

Arkowitz, Martin

Fachgebiete

• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

1 Basic Homotopy.- 1.1 Introduction.- 1.2 Spaces, Maps, Products and Wedges.- 1.3 Homotopy I.- 1.4 Homotopy II.- 1.5 CW Complexes.- 1.6 Why Study Homotopy Theory?.- Exercises.- 2 H-Spaces and Co-H-Spaces.- 2.1 Introduction.- 2.2. H-Spaces and Co-H-Spaces.- 2.3 Loop Spaces and Suspensions.- 2.4 Homotopy Groups I.- 2.5 Moore Spaces and Eilenberg-Mac Lane Spaces.- 2.6 Eckmann-Hilton Duality I.- Exercises.- 3 Cofibrations and Fibrations.- 3.1 Introduction.- 3.2 Cofibrations.- 3.3 Fibrations.- 3.4 Examples of Fiber Bundles.- 3.5 Replacing a Map by a Cofiber or Fiber Map.- Exercises.- 4 Exact Sequences.- 4.1 Introduction.- 4.2 The Coexact and Exact Sequence of a Map.- 4.3 Actions and Coactions.- 4.4 Operations.- 4.5 Homotopy Groups II.- Exercises.- 5 Applications of Exactness.- 5.1 Introduction.- 5.2 Universal Coefficient Theorems.- 5.3 Homotopical Cohomology Groups.- 5.4 Applications to Fiber and Cofiber Sequences.- 5.5 The Operation of the Fundamental Group.- 5.6 Calculation of Homotopy Groups.-Exercises.- 6 Homotopy Pushouts and Pullbacks.- 6.1 Introduction.- 6.2 Homotopy Pushouts and Pullbacks I.- 6.3 Homotopy Pushouts and Pullbacks II.- 6.4 Theorems of Serre, Hurewicz and Blakers-Massey.- 6.5 Eckmann-Hilton Duality II.- Exercises.- 7 Homotopy and Homology Decompositions.- 7.1 Introduction.- 7.2 Homotopy Decompositions of Spaces.- 7.3 Homology Decompositions of Spaces.- 7.4 Homotopy and Homology Decompositions of Maps.- Exercises.- 8 Homotopy Sets.- 8.1 Introduction.- 8.2 The Set [X, Y].- 8.3 Category.- 8.4 Loop and Group Structure in [X, Y].-Exercises.- 9 Obstruction Theory.- 9.1 Introduction.- 9.2 Obstructions Using Homotopy Decompositions.- 9.3 Lifts and Extensions.- 9.4 Obstruction Miscellany.- Exercises.- A Point-Set Topology.- B The Fundamental Group.- C Homology and Cohomology.- D Homotopy Groups of the n-Sphere.- E Homotopy Pushouts and Pullbacks.- F Categories and Functors.- Hints to Some of the Exercises.- References.- Index.-

Über Autor(innen)

Arkowitz, Martin
Martin Arkowitz is currently a professor of mathematics at Dartmouth College. He received his Ph.D. in mathematics at Cornell University. His area of expertise is algebraic topology.

Martin Arkowitz is currently a professor of mathematics at Dartmouth College. He received his Ph.D. in mathematics at Cornell University. His area of expertise is algebraic topology.