[DRAFT] A Peripatetic Course in Algebraic Topology

These notes are based on lectures in algebraic topology taught by Peter May and Henry Chan at the 2016 University of Chicago Math REU. They are loosely chronological, having been reorganized for my benefit and significantly annotated by my personal exposition, plus solutions to in-class/HW exercises, plus content from readings (from May’s Finite Book), books (e.g. May’s Concise Course, Munkres’ Elements of Algebraic Topology, and Hatcher’s Algebraic Topology), Wikipedia, etc. I Foundations + Weeks 1 to 3 3 1 Topological notions 3 1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Separation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Continuity and operations on spaces . . . . . . . . . . . . . . . . . . . . . . 4 2 Algebraic notions 5 2.1 Rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Categorical notions 11 3.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 [DRAFT] Universal properties . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20