# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 215A: Topology

**Approved:**2009-05-01 (revised 2024-08-26, )

**Units/Lecture:**

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

**Prerequisites:**

**Course Description:**

**Suggested Schedule:**

• (1 lecture) **Examples and constructions of topological spaces**: spheres, projective spaces, products, wedge, suspension, CW complexes and decompositions [Hatcher Ch 0]

• (2 lectures) **Homotopy**: definitions and examples of homotopy, homotopy equivalence, retracts, deformation retracts, contractible spaces, π0 and path connectedness, homotopy extension, homotopy invariance under quotient by a contractible subspace [Hatcher Ch 0]

• (2 lectures) **Fundamental group basics**: definition of π1, induced maps, homotopy invariance, group structure, changing base points [Hatcher 1.1]

• (1-2 lectures) **Basic examples of computing π1**: contractible spaces, S1, products [Hatcher 1.1]

• (3 lectures) **Seifert van Kampen**: theorem statement, applications in examples, proof of theorem [Hatcher 1.2]

• (1-2 lectures) **Basics of covering spaces**: definition and examples, homotopy lifting, injectivity of p∗, deck transformations, regular coverings [Hatcher 1.3]

• (3 lectures) **Classification of covering spaces**: statement and proof, universal covers [Hatcher 1.3]

• (2 lectures) **Basics of higher homotopy groups**: definitions, basic examples, group structure, changing basepoints, covering maps induce isomorphisms [Hatcher 4.1]

• (2-3 lectures) **Relative homotopy groups of pairs**: definition, long exact sequence [Hatcher 4.1]

• (3 lectures) **Fiber bundles**: Definitions, examples, homotopy extension lifting, Serre fibrations, homotopy long exact sequence and computational applications in examples [Fomenko-Fuchs Lecture 9]

• (if time allows) **Homotopy groups and cell complexes, Freudenthal suspension** [Fomenko-Fuchs Lecture 10-11]