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

How this entity was disambiguated

This entity first appeared as the object of triple T2306391 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Serre spectral sequence
Context triple: [Jean-Pierre Serre, notableWork, Serre spectral sequence]

A. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
B. Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
C. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
D. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
E. Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Serre spectral sequence
Target entity description: 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.

A. Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
B. Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
C. Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
D. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
E. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
F. None of above. chosen

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) ⓘ

How these facts were elicited

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 ⓘ