Note that is coexact, since We can iterate this construction, forming the mapping cone of mapping cone, etc. Note that , so the LES is just forming iterated suspension.
Cofibrations are good embeddings and satisfies the homotopy extension property. The prototypical examples are inclusion to mapping cylinder for . Equivalently, has HEP iff is a retract of . Since (since is compact the Hom-tensor adjunction works fine), it is the same as the following diagram.
Fibrations are like fiber bundles and satisfies the homotopy lifting property. The prototypical examples are the path space fibration and pullback along any continous map .
Essentially the theory of model category gives something like factorization system (surjectives follow by injectives) but slightly weaker (not requiring the factorization to be functorial), i.e. the notion of weak factorization system. Via it we can formulate the notion of a model category succinctly.
We can check that has HEP, hence so is by closure under pushout. We can use it to show that is a homotopy equivalence by proving a more general lemma that if is contractible and has HEP, then is a homotopy equivalence.
Where does group structure of higher homotopy group come from? We can check is a cogroup object in the homotopy category (co--group), and that if is any pointed space and is any co--group, so is (the smash product). This is similar to the fact that is a group for any set and group .
Recall the Eckmann-Hilton argument: If is a co--group in two ways, say and , and is a cohomomorphism for the -structure (and vice versa) then and is cocommutative. A corollary is that is a cocommutative group. The idea behind the Eckmann-Hilton argument is the very simple observation that group objects in the category of groups are precisely abelian groups. For more on Eckmann-Hilton argument, see this.
We see that the homotopy category is very closed to being a triangulated category, except that is not generally invertible (e.g. see this for an example). Inverting gives the category of spectra which is triangulated.
Since has a -coaction, i.e. there is a map by crushing the middle circle of the cone. As a consequence, is a -set, and the map is a map as -sets.
Definition of homotopy group: Starting from Define and , and note that .
One can show by perturbation argument. By the path lifting property, is bijective for . In particular, the higher homotopy groups of all orientable surface vanish.
Interpretation of : If represents zero in , then where . (Proof is by interpreting the homotopy as a homotopy from inverted can to .)