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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in algebraic topology
ⓘ
type of fibration ⓘ |
| appearsIn |
Serre’s thesis
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| namedAfter | Jean-Pierre Serre NERFINISHED ⓘ |
| relatedTo |
Serre spectral sequence
NERFINISHED
ⓘ
fiber bundle ⓘ principal bundle ⓘ |
| requires | continuity of the map between topological spaces ⓘ |
| roleIn | Quillen model structure on Top NERFINISHED ⓘ |
| typicalBase | CW complex ⓘ |
| typicalTotalSpace | topological space with homotopy-theoretic structure GENERATED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.