Serre spectral sequence

E256258

mathematical concept spectral sequence tool in algebraic topology

The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.

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All labels observed (3)

Label	Occurrences
Leray–Serre spectral sequence	2
Serre spectral sequence canonical	1
homology Serre spectral sequence	1

Statements (47)

Predicate	Object
instanceOf	mathematical concept ⓘ spectral sequence ⓘ tool in algebraic topology ⓘ
alternativeName	Serre spectral sequence ⓘ surface form: Leray–Serre spectral sequence
appearsIn	standard textbooks on algebraic topology ⓘ
appliesTo	Serre fibration ⓘ fibration of topological spaces ⓘ
convergenceType	converges to a filtration of the (co)homology of the total space ⓘ
convergesTo	(co)homology of the total space of the fibration ⓘ
E2Term	E2^{p,q} ≅ H^p(B; H^q(F)) in cohomology version ⓘ E2_{p,q} ≅ H_p(B; H_q(F)) in homology version ⓘ
field	algebraic topology ⓘ homological algebra ⓘ
generalizes	Leray spectral sequence in topological setting ⓘ
hasAssumption	base space is path-connected in standard formulations ⓘ fiber is path-connected in standard formulations ⓘ
hasDifferentials	dr maps of bidegree (r,1−r) in cohomology version ⓘ dr maps of bidegree (−r,r−1) in homology version ⓘ
hasStructure	bigraded groups with differentials ⓘ
hasVersion	cohomology Serre spectral sequence ⓘ Serre spectral sequence self-linksurface differs ⓘ surface form: homology Serre spectral sequence
introducedBy	Jean-Pierre Serre ⓘ
introducedIn	20th century ⓘ
isToolFor	inductive calculations on skeleta of CW-complexes ⓘ
namedAfter	Jean-Pierre Serre ⓘ
pageE2	E2 page is expressed in terms of (co)homology of base and fiber ⓘ
relatedConcept	Atiyah–Hirzebruch spectral sequence ⓘ Serre spectral sequence self-linksurface differs ⓘ surface form: Leray–Serre spectral sequence
relates	cohomology of a fibration ⓘ cohomology of the base space ⓘ cohomology of the fiber ⓘ homology of a fibration ⓘ homology of the base space ⓘ homology of the fiber ⓘ
requires	local coefficient systems in general form ⓘ spectral sequence formalism ⓘ
standardReference	Jean-Pierre Serre’s original papers on homotopy groups and fibrations ⓘ
usedFor	computing cohomology groups ⓘ computing homology groups ⓘ computing homotopy-invariant information of fibrations ⓘ
usedIn	algebraic K-theory ⓘ computation of cohomology of principal bundles ⓘ computation of homology of fiber bundles ⓘ computation of homology of loop spaces ⓘ rational homotopy theory ⓘ stable homotopy theory ⓘ
usedToProve	Serre’s finiteness theorem for homotopy groups of spheres (via related methods) ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Jean-Pierre Serre → notableWork → Serre spectral sequence ⓘ

Serre spectral sequence → hasVersion → Serre spectral sequence self-linksurface differs ⓘ

this entity surface form: homology Serre spectral sequence

Serre spectral sequence → relatedConcept → Serre spectral sequence self-linksurface differs ⓘ

this entity surface form: Leray–Serre spectral sequence

Serre spectral sequence → alternativeName → Serre spectral sequence ⓘ

this entity surface form: Leray–Serre spectral sequence