Tags: #spectral-sequences #homotopy #homotopy

Using the Path-Loop Fibration

\(\Omega X \to PX \xrightarrow{f} X\)

In particular the classifying space \(BG\) of a topological group \(G\) is the solution for a fibration \(BG \times_\tau G\) where the fiber space is the given group \(G\), the base space is the classifying space \(BG\) and the product \(BG \times G\) is twisted in such a way the total space \(BG \times_\tau G\) is contractible. The same idea where the base space \(X\) is given and the fibre space is unknown leads to the loop space \(\Omega X\) and the contractible total space \(X \times_\tau \Omega X\).

The handbooks of Algebraic Topology more or less explain the Eilenberg-Moore spectral sequence can be used to “compute” the homology groups of the new objects \(BG\) (BG) and \(\Omega X\) if the homology groups of \(G\) or \(X\) are known. In fact this spectral sequence is in general unable to give you the new homology groups, unless you are in a very special situation. The Serre spectral sequence works in the third situation, when you are looking for the homology groups of a total space \(B \times_\tau F\) if the homology groups of \(B\) and \(F\) are known; but in general you meet the same difficulties with the higher differentials and the extension problems at abutment.

In fact, by working the Serre spectral sequence backwards, we can compute the homology of the loop space \(\Omega S^n\).

Misc

Relation to Transgressions and spans: