Tags: #spectral-sequences #homotopy/fibrations

Filtrations and the Serre spectral sequence

References

• Rational Homotopy Theory and Differential Forms by Griffiths and Morgan
• Differential Forms in Algebraic Topology by Bott and Tu
• Differential Topology by Hirsch
• Comprehensive Introduction to Differential Geometry by Spivak
• Topology from the Differentiable Viewpoint by Milnor
• Topology and Geometry by Bredon
• User’s Guide to Spectral Sequences by Mcleary
  • View Here
  • Lots of technical details

Filtration from a SES

The standard Serre fibration: \(\Omega X \to PX \xrightarrow{f} X\) where \(\Omega X\) is the loop space, \(PX\) is the path space, and \(f\) is the “evaluation at the endpoint” map. Note that \(PX\) is contractible!

Consider a SES \(0 \to A \to B \to C \to C\), then look at it as a 2-step filtration of \(B\) so

• \(F^0B = B\),
• \(F^1B = A\),
• \(F^2B = 0\).

The sections are \(G_0 = C, G_1 = A\). Can use this to obtain LES from SS.

Homology in the ring-theoretic setting: If \(R\) is a Noetherian ring and \(I \subseteq R\), then if \(I\) can be generated by \(n\) elements then \(H_I^i(M) = 0\) for any \(R\)-module \(M\) and \(i > n\). Thus to prove \(I\) can not be generated by \(n\) elements, it suffices to find a module \(M\) where \(H_I^{n+1} \neq 0\).

Serre Spectral Sequence

Lots of good examples of computations here.

Some fibrations

• Hopf: \(S^1 \to S^3 \to S^2\)
• \(S^1 \to S^{2n+1} \to {\mathbf{CP}}^n\)
• Path space: \(\Omega S^n \to PS^n \to S^n\)

Serre Spectral Sequence Example: For the fibration \(S^1 \to S^3 \to S^2\), the \(E_2\) page:

Link to Diagram

Which is equal to Link to Diagram

And \(E_3 = E_\infty\), so \(d_2^{0,1}\) is an isomorphism.

Note: Probably a good starting point for basic calculations? Fill out the missing details for this table.