Fundamental Group - Fibrations

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
• π_n+1(B) and π_n(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.