Significant Spectral Sequences

Significant spectral sequences are essential tools in algebraic topology, helping to compute homology and cohomology of complex spaces. They connect various concepts, from fibrations to stable homotopy, revealing deep relationships between algebraic structures and topological properties.

1. Serre spectral sequence

   • Relates the homology of a fibration to the homology of its base and fiber.
   • Provides a systematic way to compute the homology of a space that is built from simpler pieces.
   • Utilizes the concept of a double complex to organize the computation of spectral sequences.

2. Adams spectral sequence

   • Primarily used in stable homotopy theory to compute the stable homotopy groups of spheres.
   • Builds on the idea of the Serre spectral sequence but focuses on the category of spectra.
   • Involves the use of Ext groups in the context of stable homotopy categories.

3. Atiyah-Hirzebruch spectral sequence

   • Connects the cohomology of a space with its characteristic classes, particularly in the context of complex vector bundles.
   • Useful for computing the topological invariants of manifolds.
   • Provides a framework for understanding the relationship between homotopy and cohomology theories.

4. Leray spectral sequence

   • Applies to sheaves and is used to compute the cohomology of a space from the cohomology of its open covers.
   • Particularly important in algebraic geometry and the study of sheaf cohomology.
   • Relies on the concept of a filtered complex to organize the computation.

5. Eilenberg-Moore spectral sequence

   • Focuses on the homology of a fibration in the context of category theory and homotopy theory.
   • Provides a way to compute the homology of a space that is a fibration over another space.
   • Useful in the study of operads and higher category theory.

6. Bockstein spectral sequence

   • Arises in the context of cohomology with coefficients in a group, particularly in relation to the Bockstein homomorphism.
   • Helps in computing the cohomology of spaces with twisted coefficients.
   • Connects the ordinary cohomology with the mod p cohomology.

7. Grothendieck spectral sequence

   • A powerful tool in algebraic geometry, particularly in the context of derived categories.
   • Relates the derived functors of sheaves to the cohomology of schemes.
   • Useful for understanding the relationships between different cohomological dimensions.

8. Hochschild-Serre spectral sequence

   • Connects the homology of a group with the homology of its normal subgroups and quotient groups.
   • Useful in the study of group cohomology and representation theory.
   • Provides a way to compute the cohomology of groups using spectral sequences.

9. Lyndon-Hochschild-Serre spectral sequence

   • A refinement of the Hochschild-Serre spectral sequence, focusing on the case of groups with a more complex structure.
   • Useful for computing the cohomology of groups with respect to their subgroups.
   • Provides insights into the relationships between different layers of group extensions.

10. May spectral sequence

    • A spectral sequence that arises in the context of stable homotopy theory and the study of operads.
    • Useful for computing the homology of operads and their associated categories.
    • Connects the algebraic structures of operads with topological spaces.