Spectral sequences are powerful tools in homological algebra, connecting different theories. They allow us to compute complex structures by breaking them down into simpler pieces and using successive approximations.
From Leray and Serre to Eilenberg-Moore and Adams, these sequences tackle various algebraic and topological problems. They're essential for understanding , fibrations, and even of spheres.
Spectral Sequences in Homological Algebra
Key Spectral Sequences for Homology and Cohomology
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relates the cohomology of a space with the cohomology of a on that space
Enables computing using a F→E→B by successive approximations
Starts with E2p,q=Hp(B;Hq(F)) and converges to Hp+q(E)
is a that relates the of the base space and the homology of the fiber to the homology of the total space in a fibration
For a fibration F→E→B, it has Ep,q2=Hp(B;Hq(F)) and converges to Hp+q(E)
Powerful tool for computing of a space when it's realized as a fibration (Hopf fibration of spheres)
relates a of a space to its
For a spectrum h and CW complex X, it starts with E2p,q=Hp(X;πq(h)) and converges to hp+q(X)
Allows computing groups like or from ordinary cohomology (stable homotopy groups of spheres)
Spectral Sequences for Ring and Module Structures
relates the cohomology of a fiber space to the cohomology of the base and total space for a fibration of spaces with additional structure
Applies when cohomology of fiber, total space and base have compatible ring or module structures
Starts with E2p,q=Torp,qH∗(B)(H∗(E),k) and converges to Hp+q(F) as modules over H∗(B)
computes stable homotopy groups of spheres from Ext groups in the category of modules over the
E2s,t=ExtAs,t(Fp,Fp)⇒πt−sS(S0)(p)
Connects stable homotopy theory to homological algebra of the Steenrod algebra ( encoding cohomology operations)
Group Cohomology Spectral Sequences
Spectral Sequences for Group Extensions
relates the cohomology of a group extension to the cohomology of the subgroup and quotient group
For a group extension 1→H→G→K→1 it has E2p,q=Hp(K;Hq(H;M)) converging to Hp+q(G;M)
Allows computing group cohomology using the cohomology of smaller pieces H and K (when G is a semidirect product)
is a specific case of the Hochschild-Serre spectral sequence
Applies to group cohomology with coefficients in a trivial module
Has E2p,q=Hp(K;Hq(H;Z)) and converges to Hp+q(G;Z)
Useful for analyzing the cohomological dimension of solvable groups (successive extensions by abelian groups)