Serre fibration
E889911
A Serre fibration is a continuous map between topological spaces that satisfies a homotopy lifting property for CW complexes, making it a central tool in algebraic topology for studying the homotopy and homology of fiber bundles.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Ehresmann fibrations | 1 |
| Serre fibration canonical | 1 |
| Serre fibrations | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10855490 — 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 fibration Context triple: [Serre spectral sequence, appliesTo, Serre fibration]
-
A.
Serre spectral sequence
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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B.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
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C.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
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D.
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.
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E.
Hopf fibration
The Hopf fibration is a fundamental construction in topology that describes the 3-sphere as a fiber bundle of circles over the 2-sphere, revealing deep connections between geometry, algebra, and higher-dimensional spaces.
- 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 fibration Target entity description: A Serre fibration is a continuous map between topological spaces that satisfies a homotopy lifting property for CW complexes, making it a central tool in algebraic topology for studying the homotopy and homology of fiber bundles.
-
A.
Serre spectral sequence
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.
-
B.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
-
C.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
-
D.
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.
-
E.
Hopf fibration
The Hopf fibration is a fundamental construction in topology that describes the 3-sphere as a fiber bundle of circles over the 2-sphere, revealing deep connections between geometry, algebra, and higher-dimensional spaces.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in algebraic topology
ⓘ
type of fibration ⓘ |
| appearsIn |
Serre’s thesis
ⓘ
Serre’s work on homotopy groups of spheres ⓘ |
| assumptionIn | many spectral sequence computations ⓘ |
| belongsTo | homotopical algebra framework ⓘ |
| characterizedBy |
right lifting property with respect to CW pair inclusions
ⓘ
right lifting property with respect to inclusions S^{n-1} → D^n ⓘ |
| closedUnder |
composition
ⓘ
pullbacks ⓘ |
| codomain | topological space ⓘ |
| comparedTo | Hurewicz fibration ⓘ |
| context | model category structures on topological spaces ⓘ |
| definedAs | continuous map p : E → B between topological spaces with a homotopy lifting property ⓘ |
| domain | topological space ⓘ |
| ensures | long exact sequence of homotopy groups for fiber, total space, and base ⓘ |
| field | algebraic topology ⓘ |
| generalizes | Hurewicz fibration in a weaker sense ⓘ |
| hasCondition | homotopy lifting property only required for CW complexes or disks, not all spaces ⓘ |
| hasConsequence |
base-change properties for homotopy groups
ⓘ
homotopy exactness of certain sequences ⓘ |
| hasExample |
evaluation map from path space to base space
ⓘ
path-space fibration PX → X ⓘ projection map of a Serre fiber bundle ⓘ |
| hasFiber | homotopy fiber of the map ⓘ |
| hasProperty |
homotopy lifting property for CW complexes
ⓘ
homotopy lifting property for disks D^n ⓘ homotopy lifting property for pairs (D^n,S^{n-1}) ⓘ |
| implies | weak homotopy equivalence on fibers under suitable conditions ⓘ |
| isToolFor |
computing homology of fiber bundles
ⓘ
computing homotopy groups of spheres ⓘ |
| isWeakerThan | Hurewicz fibration ⓘ |
| namedAfter | Jean-Pierre Serre ⓘ |
| relatedTo |
Serre spectral sequence
ⓘ
fiber bundle ⓘ principal bundle ⓘ |
| requires | continuity of the map between topological spaces ⓘ |
| roleIn |
Quillen model structure on Top
ⓘ
linked to:
Quillen model category
|
| typicalBase | CW complex ⓘ |
| typicalTotalSpace | topological space with homotopy-theoretic structure ⓘ |
| usedFor |
constructing long exact sequences of homotopy groups
ⓘ
defining Serre spectral sequence ⓘ studying homology via spectral sequences ⓘ studying homotopy groups ⓘ |
| usedIn |
construction of Postnikov towers
ⓘ
homotopy theory of CW complexes ⓘ obstruction theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Serre fibration Description of subject: A Serre fibration is a continuous map between topological spaces that satisfies a homotopy lifting property for CW complexes, making it a central tool in algebraic topology for studying the homotopy and homology of fiber bundles.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Serre fibration
linked to: Serre fibration