Spectral sequence

In homological algebra, especially applied to algebraic topology or group cohomology, a spectral sequence is a sequence of differential modules (En,dn) such that

Overall explanation

One way to visualise what is occurring in a spectral sequence is by means of a notebook?spreadsheet metaphor. The initial E1 being the first sheet of data, the E2 sheet is derived from it by a definite process; and so on. The 'end result' of the calculation would be a final sheet.

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The spreadsheet talk here is fairly appropriate, because in practice the Ei tend to carry some grading data, often a double grading. Each sheet is then ruled into cells, indexed by row and column, with an abelian group in each cell. Each sheet also has mappings called differentials, acting from each cell on the sheet to some other cell in a way referred to pictorially as knight's moves. The 'definite process' mentioned above is then a way to calculate each cell in the next sheet using the previous sheet's data and differentials. The process often stabilizes at the final sheet, and then repeats itself eternally because all the differentials from that sheet onwards are identically zero.

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Notationally the En would therefore carry two index numbers

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:Enp,q

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with differentials dnp,q acting from each Enp,q to some Enp+a,q+b, with a and b depending only on n. That is, the spectral sequence as process is analogous to a book with pages ruled out into grids, one for each En. (As David Mumford writes, it becomes easier to work it out on one's own, rather than try to follow someone else's notations.)

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The spectral sequence is often used to derive some data about the final sheet knowing the data from initial sheets, or vice versa. Take, for instance, the Leray-Serre spectral sequence of a fibration in algebraic topology. For many fibrations, on the second sheet the first column is cohomology of the fiber, and the first row is cohomology of the base space, while the final sheet is determined in a certain way by cohomology of the total space of the fibration. One might, for example, use this spectral sequence to calculate cohomology of the group SU(3) from the fibration SU(3) → S5. This fibration has total space SU(3), base space S5, and fiber SU(2) which is the same as the 3-sphere S3. So it is possible to calculate the cohomology of SU(3) knowing the cohomology of spheres.

Related Topics:
Leray-Serre spectral sequence - Fibration - Algebraic topology - Cohomology

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