Hurewicz homomorphism
E911367
The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
concept in algebraic topology
ⓘ
homomorphism ⓘ |
| alsoKnownAs | Hurewicz map NERFINISHED ⓘ |
| appearsIn | Hurewicz theorem NERFINISHED ⓘ |
| assumes | basepoint choice for homotopy groups ⓘ |
| codomain | homology group H_n(X) of a topological space X ⓘ |
| construction | sends a homotopy class of maps S^n → X to the induced homology class of the fundamental cycle of S^n ⓘ |
| context |
CW-complexes
ⓘ
Postnikov towers NERFINISHED ⓘ stable homotopy theory ⓘ |
| definedBy | induced map on homology from representing sphere map ⓘ |
| domain | homotopy group π_n(X) of a topological space X ⓘ |
| field | algebraic topology ⓘ |
| generalizationOf | map from fundamental group to first homology group ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| isDefinedFor |
n ≥ 1
ⓘ
path-connected spaces ⓘ pointed topological spaces ⓘ |
| mathematicalDiscipline |
homological algebra
ⓘ
homotopy theory ⓘ |
| namedAfter | Witold Hurewicz NERFINISHED ⓘ |
| property |
for simply connected spaces, first nonzero Hurewicz homomorphism is an isomorphism under Hurewicz theorem hypotheses
ⓘ
is a group homomorphism ⓘ is compatible with long exact sequences of pairs ⓘ is natural with respect to continuous maps ⓘ |
| relatedTo |
Freudenthal suspension theorem
NERFINISHED
ⓘ
Whitehead theorem NERFINISHED ⓘ |
| relates |
homology groups
ⓘ
homotopy groups ⓘ |
| roleIn |
bridge between geometric and algebraic invariants of spaces
ⓘ
identifies first nontrivial homotopy group with corresponding homology group under connectivity assumptions ⓘ provides algebraic approximation to homotopy groups ⓘ |
| satisfies |
compatibility with suspension
ⓘ
naturality with respect to maps of spaces ⓘ |
| sourceStructure | homotopy group π_n(X) ⓘ |
| specialCase | for n = 1 coincides with abelianization map from π_1(X) to H_1(X) ⓘ |
| symbol | h_n ⓘ |
| targetStructure | abelian group H_n(X) ⓘ |
| usedIn |
classification of spaces up to homotopy type
ⓘ
computation of homotopy groups via homology ⓘ obstruction theory ⓘ spectral sequence arguments in homotopy theory ⓘ |
| usedToProve | relationships between connectivity and vanishing of homology groups ⓘ |
| uses |
homotopy classes of maps from spheres
ⓘ
singular homology ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.