Algebraic Topology I

Graduate-level entry into algebraic topology, where the central move is turning geometric questions about spaces and continuous maps into algebraic ones about groups and their morphisms via functors like the fundamental group, higher homotopy groups, and singular and cellular homology. You will spend the term computing these invariants on concrete examples (CW-complexes, simplicial complexes, manifolds), working through homological-algebra machinery like chain homotopies and Mayer-Vietoris, and proving things on regular problem sets backed by two midterms and a final. It assumes solid point-set topology and group theory, and is the standard prerequisite for MATH 612, differential topology, and any later work in geometry, K-theory, or algebraic geometry where you need to tell spaces apart by algebraic means.