Fibrations
A generalization of a product of spaces is given by a fibration,
Here the total space E is a sort of "twisted product" of the base space B and the fiber F. In general the fundamental groups of B, E and F are terms in a long exact sequence involving higher homotopy groups. When all the spaces are connected, this has the following consequences for the fundamental groups:
- π1(B) and π1(E) are isomorphic if F is simply connected
- πn+1(B) and πn(F) are isomorphic if E is contractible
The latter is often applied to the situation E = path space of B, F = loop space of B or B = classifying space BG of a topological group G, E = universal G-bundle EG.
Read more about this topic: Fundamental Group