Algebraic Topology: Lecture 1

We introduced (higher) homotopy groups and showed that they are functors from the category of pointed topological spaces and base-point preserving homotopy classes of base-point preserving maps to the category of pointed sets and base-point preserving maps (resp. groups and group-homomorphism, resp. abelian groups and group-homomorphisms) as n = 0 (resp. n = 1, resp. n > 1). We further defined a weak equivalence to be a map f : X --> Y such that the induced map of homotopy groups is a bijection, for all non-negative integers n and for all choices of base-point x in X. For reading see Hatcher, Algebraic Topology, the beginning of chapters 0 and 4.