INTRODUCTION TO ALGEBRAIC TOPOLOGY

Prerequisites & course information

Requisites: ( MATH 541 and 551 ), graduate/professional standing, or declared in Mathematics VISP (undergraduate or graduate) Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Spring 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in algebraic topology (e.g., covering spaces, universal covering spaces, fundamental group, CW-complexes, closed oriented surfaces, homology, euler characteristic, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., a set is simply connected, two sets are homotopic, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of algebraic topology (e.g., the classification of closed oriented surfaces, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises. Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in algebraic topology (e.g., covering spaces, universal covering spaces, fundamental group, CW-complexes, closed oriented surfaces, homology, euler characteristic, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g., a set is simply connected, two sets are homotopic, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of algebraic topology (e.g., the classification of closed oriented surfaces, etc.), and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises. Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate