Spectral Sequences

Spectral Sequences

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become an important research tool, particularly in homotopy theory.

Read more about Spectral Sequences:  Discovery and Motivation, Formal Definition, Exact Couples, Visualization, Convergence, Degeneration, and Abutment, Further Examples

Other articles related to "spectral sequences, spectral sequence, sequence, sequences":

Spectral Sequence - Further Examples
... Some notable spectral sequences are Adams spectral sequence in stable homotopy theory Adams–Novikov spectral sequence, a generalization to extraordinary cohomology theories ... Atiyah–Hirzebruch spectral sequence of an extraordinary cohomology theory Bar spectral sequence for the homology of the classifying space of a group ... Barratt spectral sequence converging to the homotopy of the initial space of a cofibration ...
Spectral Sequence - Convergence, Degeneration, and Abutment
... that we began with, the sheets of the spectral sequence were constant once r was at least 1 ... In that setup it makes sense to take the limit of the sequence of sheets Since nothing happens after the zeroth sheet, the limiting sheet E∞ is the same as E1 ... They are one of the most powerful aspects of spectral sequences ...
Spectral Sequence - Exact Couples
... The most powerful technique for the construction of spectral sequences is William Massey's method of exact couples ... Exact couples are particularly common in algebraic topology, where there are many spectral sequences for which no other construction is known ... In fact, all known spectral sequences can be constructed using exact couples ...
Spectral Sequences - Further Examples
... Some notable spectral sequences are Adams spectral sequence in stable homotopy theory Adams–Novikov spectral sequence, a generalization to extraordinary cohomology theories ... Atiyah–Hirzebruch spectral sequence of an extraordinary cohomology theory Bar spectral sequence for the homology of the classifying space of a group ... Barratt spectral sequence converging to the homotopy of the initial space of a cofibration ...
Spectral Sequence - Formal Definition
... A spectral sequence is a choice of a nonnegative integer r0 and a collection of three sequences For all integers r ≥ r0, an object Er, called a sheet (as in a ... Putting the zero differential on all the rest of our sheets gives a spectral sequence whose terms are E0 = C• Er = H(C•) for all r ≥ 1 ... The terms of this spectral sequence stabilize at the first sheet because its only nontrivial differential was on the zeroth sheet ...

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