Some Techniques in Homotopy Theory

This chapter deals with five topics which help in understanding homotopy type and how to alter a CW complex within its homotopy type; for example, to reduce the number of cells in a dimension of interest. The most important theorems are 4.1.7 and 4.1.8 which are the key ingredients in the Rebuilding Lemma 6.1.4. That in turn tells us much about topological finiteness properties of groups (in Chapter 7). The last topic, the Hurewicz Theorem, is of fundamental importance in algebraic topology.