Algebraic Topology: Lecture 10

We proved that the category T of topological spaces and continuous maps has a cofibrantly generated model structure, where the weak equivalences are the maps f : X --> Y such that the induced map

     f_* : pi_q(X,x) --> pi_q(Y,f(x))

is a bijection, for all non-negative integers q and all points x in X, and where

     I = { S^{n-1} --> D^n | n non-negative integer}

     J = { D^n --> D^n x [0,1] | n non-negative integer}

are sets of generating cofibrations and generating trivial cofibrations, respectively.

This model structure is often called the Serre model structure since the definition of the class of fibrations was given by Serre in his thesis.