Math 527 - Homotopy Theory Cofiber sequences

The notion of cofiber sequence is dual to that of fiber sequence. Most constructions and statements about fiber sequences readily dualize to cofiber sequences, though there are differences to be mindful of. 1. Fiber sequences were defined in the category of pointed spaces Top * , so that taking the strict fiber (i.e. preimage of the basepoint) makes sense. No such restriction exists for cofiber sequences. There are two different notions: unpointed cofiber sequences in Top and pointed cofiber sequences in Top *. 2. Fiber sequences induce long exact sequences in homotopy, whereas cofiber sequences induce long exact sequences in (co)homology. Therefore the behavior of cofiber sequences with respect to weak homotopy equivalences is a more subtle point. Warning 0.1. In these notes, we introduce a non-standard notation with * for pointed notions. In the future, as in most of the literature, we will drop the * from the notation and rely on the context to distinguish between pointed and unpointed notions.