We would like to understand homotopy classes of maps between spaces. Recall an important construction, the mapping cylinder of := . Equivalently, it is pushout of (sending to , inclusion) and ; Note that is a deformation retract of , and . Have a closed inclusion sending , and there exists an open neighborhood of such that deformation retracts onto . Finally, we can factor an arbitrary map into closed inclusion into followed by homotopy equivalence.
Now . If we ask that the other copy of (i.e. ) sent to a point , the data amounts to a map and a homotopy from (null-homotopy), this amounts to a continuous map from the mapping cone into .
Given a map , we can extend it to iff is nullhomotopic; has fiber; A CW complex is a sequence of spaces (inductive limit) where is a discrete set of points; is obtained from by forming pushout with with attaching maps .
Understanding in terms of mapping cones: is a map and for each copy of and homotopy (the image of the orgin in .)
Conclusion: is inverse limit of . For , .
Note that is an element of . We get a function , extending to a homomorphism from the free abelian group (cellular cochain). The map is extensible iff this cochain is zero, which is not a homological condition. However it is easy to prove two things:
This cochain is a cocycle.
If is homotopic to , then their corresponding cochains (cocycles by (1)) differ by a coboundary.
Thus if the cohomology class vanishes, we can choose homotopic to such that can be extended. We might worry about that there is a tree of possibilities if we increase . The miraculous thing about obstruction theory is that this is not the case.