Algebraic Topology

Clark Bray is an Associate Professor of the Practice in the Department of Mathematics at Duke University. He graduated from Rice University in 1993, cum laude, with majors in Mathematics and Physics, and finished his Ph.D. in Mathematics at Stanford University in 1999, studying algebraic topology with Professor Gunnar Carlsson. While subsequently a Lecturer at Stanford, he created and taught a program in algebraic topology for talented high school math students at the Stanford University Mathematics Camp (SUMaC) in 2001-2003. Adrian Butscher received his B.Sc. in mathematics and physics at the University of Toronto and his Ph.D. in mathematics at Stanford University.  After a fifteen-year period in academia, conducting research in pure and applied differential geometry and teaching at all levels of the undergraduate and graduate curricula, Adrian now works as a research scientist with Autodesk, Inc., a global software company that creates digital design tools for the manufacturing, architecture, building, construction, media, and entertainment industries.  While at Stanford, Adrian served as a counsellor and instructor for SuMAC, the Stanford University Mathematics Camp, a month-long, immersive educational experience for mathematically gifted high school students.Simon Rubinstein-Salzedo received his PhD in mathematics from Stanford University in 2012. Afterwards, he taught at Dartmouth College and Stanford University. In 2015, he founded Euler Circle, a mathematics institute in the San Francisco Bay Area, dedicated to teaching college-level mathematics classes to advanced high-school students, as well as mentoring them on mathematics research. His research interests include number theory, algebraic geometry, combinatorics, probability, and game theory.

“Algebraic topology provides a self-contained introduction to the field … . the book thus provides a particularly well-organized, interesting, and smooth exposition of its subject. … This particular book unique is that it provides a clear, elementary, but mathematically solid introduction to algebraic topology that keeps the subject interesting throughout. … provides a clear, readable, and detailed treatment of the ideas and proofs in the subject … .” (Thomas Mack, Mathematical Reviews, July, 2022)“The book could easily be used in an undergraduate course or read by a bright high school student. It should certainly be in any high school library.” (Jonathan Hodgson, zbMATH 1481.55001, 2022)

• Introduction.- 1. Surface Preliminaries.- 2. Surfaces.- 3. The Euler Characteristic and Identification Spaces.- 4. Classification Theorem of Compact Surfaces.- 5. Introduction to Group Theory.- 6. Structure of Groups.- 7. Cosets, Normal Subgroups, and Quotient Groups.- 8. The Fundamental Group.- 9. Computing the Fundamental Group.- 10. Tools for Fundamental Groups.- 11. Applications of Fundamental Groups.- 12. The Seifert-Van Kampen Theorem.- 13. Introduction to Homology.- 14. The Mayer-Vietoris Sequence.- A. Topological Notions.- Bibliography.- Index.